Open Access

On the rigid cohomology of certain Shimura varieties

Research in the Mathematical Sciences20163:37

https://doi.org/10.1186/s40687-016-0078-5

Received: 30 June 2015

Accepted: 20 June 2016

Published: 26 October 2016

Abstract

We construct the compatible system of l-adic representations associated to a regular algebraic cuspidal automorphic representation of \(GL_n\) over a CM (or totally real) field and check local–global compatibility for the l-adic representation away from l and a finite number of rational primes above which the CM field or the automorphic representation ramifies. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.

1 Introduction

Our main theorem is as follows (see Corollary 7.14).

Theorem A

Let p denote a rational prime and let \(\imath : \overline{{{\mathbb {Q}}}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {C}}\). Suppose that E is a CM (or totally real) field and that \(\pi \) is a cuspidal automorphic representation of \(GL_n({\mathbb {A}}_E)\) such that \(\pi _{\infty }\) has the same infinitesimal character as an irreducible algebraic representation \(\rho _\pi \) of \({{\text {RS}}}^E_{\mathbb {Q}}GL_n\). Then there is a unique continuous semi-simple representation
$$\begin{aligned} r_{p,\imath }(\pi ): G_E \longrightarrow GL_{n}(\overline{{{\mathbb {Q}}}}_p) \end{aligned}$$
such that, if \(q \ne p\) is a rational prime above which \(\pi \) is unramified and if v|q is a prime of E, then \(r_{p,\imath }(\pi )\) is unramified at v and
$$r_{p,\imath }(\pi )|_{W_{E_v}}^{{\text {ss}}}=\imath ^{-1} {\text {rec}}_{E_v}\left( \pi _{v}|\det |_v^{(1-n)/2}\right) .$$

Here \({\text {rec}}_{E_v}\) denotes the local Langlands correspondence for \(E_v\). It may be possible to extend the local–global compatibility to other primes v. Ila Varma is considering this question.

The key point is that we make no self-duality assumption on \(\pi \). In the presence of such a self-duality assumption (‘polarizability’, see [9]) the existence of \(r_{p,\imath }(\pi )\) has been known for some years (see [18, 51]). In almost all polarizable cases \(r_{p,\imath }(\pi )\) is realized in the cohomology of a Shimura variety, and in all polarizable cases \(r_{p,\imath }(\pi )^{\otimes 2}\) is realized in the cohomology of a Shimura variety (see [16]). In contrast, according to unpublished computations of one of us (M.H.) and of Laurent Clozel, in the non-polarizable case the representation \(r_{p,\imath }(\pi )\) will never occur in the Betti or etale cohomology of a Shimura variety. Rather we construct it as a p-adic limit of representations which do occur in such cohomology groups.

We sketch our argument. We may easily reduce to the case of an imaginary CM field F which contains an imaginary quadratic field in which p splits. For all sufficiently large integers N, we construct a 2n-dimensional representation \(R_p(\imath ^{-1}(\pi ||\det ||^N)^\infty )\) such that for good primes v we have
$$\begin{array}{l} R_p\left( \imath ^{-1}\left( \pi ||\det ||^N\right) ^\infty \right) |_{W_{F_v}}^{{\text {ss}}}\\ \quad \cong \imath ^{-1} {\text {rec}}_{F_v}\left( \pi _v |\det |_v^{N+(1-n)/2}\right) \oplus \imath ^{-1} {\text {rec}}_{F_v}\left( \pi _v |\det |_v^{N+(1-n)/2}\right) ^{\vee ,c} \epsilon _p^{1-2n}, \end{array}$$
as a p-adic limit of (presumably irreducible) p-adic representations associated to polarizable, regular algebraic cuspidal automorphic representations of \(GL_{2n}({\mathbb {A}}_F)\). It is then elementary algebra to reconstruct \(r_{p,\imath }(\pi )\).
We work on the quasi-split unitary similitude group \(G_n\) associated to \(F^{2n}\). Note that \(G_n\) has a maximal parabolic subgroup \(P_{n,(n)}^+\) with Levi component
$$\begin{aligned} L_{n,(n)}\cong GL_1 \times RS^F_{\mathbb {Q}}GL_n. \end{aligned}$$
(We will give all these groups integral structures.) We set
$$\varPi (N)={{\text {Ind}}}_{P_{n,(n)}^+({\mathbb {A}}^{\infty ,p})}^{G_n({\mathbb {A}}^{\infty ,p})} \left( 1 \times \imath ^{-1}\left( \pi ||\det ||^N\right) ^{\infty ,p}\right) .$$
Then our strategy is to realize \(\varPi (N)\), for sufficiently large N, in a space of overconvergent p-adic cusp forms for \(G_n\) of finite slope. Once we have done this, we can use an argument of Katz (see [35]) to find congruences modulo arbitrarily high powers of p to classical (holomorphic) cusp forms on \(G_n\) (of other weights). (Alternatively it is presumably possible to construct an eigenvariety in this setting, but we have not carried this out.) One can attach Galois representations to these classical cusp forms by using the trace formula to lift them to polarizable, regular algebraic, discrete automorphic representations of \(GL_{2n}({\mathbb {A}}_F)\) (see, e.g., [52]) and then applying the results of Chenevier and Harris [18], Shin [51].

We learnt the idea that one might try to realize \(\varPi (N)\) in a space of overconvergent p-adic cusp forms for \(G_n\) (of finite slope) from Chris Skinner. The key problem was how to achieve such a realization. To sketch our approach we must first establish some more notation.

To a neat open compact subgroup U of \(G_n\) we can associate a Shimura variety \(X_{n,U}/{{\text {Spec}}\,}{\mathbb {Q}}\). It is a moduli space for abelian \(n[F: {\mathbb {Q}}]\)-folds with an isogeny action of F and certain additional structures. It is not proper. It has a canonical normal compactification \(X_{n,U}^{{\text {min}}}\) and, to certain auxiliary data \(\varDelta \), one can attach a smooth compactification \(X_{n,U,\varDelta }\) which naturally lies over \(X_{n,U}^{{\text {min}}}\) and whose boundary is a simple normal crossings divisor. To a representation \(\rho \) of \(L_{n,(n)}\) (over \({\mathbb {Q}}\)) one can attach a locally free sheaf \({\mathcal {E}}_{U,\rho }/X_{n,U}\) together with a canonical (locally free) extension \({\mathcal {E}}_{U,\varDelta ,\rho }\) to \(X_{n,U,\varDelta }\), whose global sections are holomorphic automorphic forms on \(G_n\) ‘of weight \(\rho \) and level U’. (The space of global sections does not depend on \(\varDelta \).) The product of \({\mathcal {E}}_{U,\varDelta ,\rho }\) with the ideal sheaf of the boundary of \(X_{n,U,\varDelta }\), which we denote \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\), is again locally free, and its global sections are holomorphic cusp forms on \(G_n\) ‘of weight \(\rho \) and level U’ (and again the space of global sections does not depend on \(\varDelta \)).

To the schemes \(X_{n,U}\), \(X_{n,U}^{{\text {min}}}\) and \(X_{n,U,\varDelta }\) one can naturally attach dagger spaces \(X_{n,U}^\dag \), \(X_{n,U}^{{{\text {min}}},\dag }\) and \(X_{n,U,\varDelta }^\dag \) in the sense of Grosse-Klönne [27]. These are like rigid analytic spaces except that one consistently works with overconvergent sections. If U is the product of a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) and a suitable open compact subgroup of \(G_n({\mathbb {Q}}_p)\), then one can define admissible open subdagger spaces (‘the ordinary loci’)
$$X_{n,U}^{{{\text {ord}}},\dag } \subset X_{n,U}^\dag $$
and
$$X_{n,U}^{{{\text {ord}}},{{\text {min}}},\dag } \subset X_{n,U}^{{{\text {min}}},\dag } $$
and
$$X_{n,U,\varDelta }^{{{\text {ord}}},\dag } \subset X_{n,U,\varDelta }^\dag . $$
By an overconvergent cusp form of weight \(\rho \) and level U one means a section of \({\mathcal {E}}_{U,\rho }^{{\text {sub}}}\) over \(X_{n,U,\varDelta }^{{{\text {ord}}},\dag }\). (Again this definition does not depend on the choice of \(\varDelta \).)
We write \(G_n^{(m)}\) for the semi-direct product of \(G_n\) with the additive group with \({\mathbb {Q}}\)-points \({{\text {Hom}}}_F(F^m,F^{2n})\), and \(P_{n,(n)}^{(m),+}\) for the pre-image of \(P_{n,(n)}^+\) in \(G_n^{(m)}\). We also write \(L_{n,(n)}^{(m)}\) for the semi-direct product of \(L_{n,(n)}\) with the additive group with \({\mathbb {Q}}\)-points \({{\text {Hom}}}_F(F^m,F^n)\), which is naturally a quotient of \(P_{n,(n)}^{(m),+}\). (Again we will give these groups integral structures.) To a neat open compact subgroup \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) with projection \(U'\) in \(G_n({\mathbb {A}}^\infty )\) one can attach a (relatively smooth, projective) Kuga–Sato variety \(A_{n,U}^{(m)}/X_{n,U'}\). For a cofinal set of U it is an abelian scheme isogenous to the m-fold self-product of the universal abelian scheme over \(X_{n,U'}\). To certain auxiliary data \(\varSigma \) one can attach a smooth compactification \(A_{n,U,\varSigma }^{(m)}\) of \(A_{n,U}^{(m)}\) whose boundary is a simple normal crossings divisor; which lies over \(X_{n,U}^{{\text {min}}}\); and which, for suitable \(\varSigma \) depending on \(\varDelta \), lies over \(X_{n,U',\varDelta }\). Thus
$$\begin{array}{ccc} A_{n,U}^{(m)} &{} \hookrightarrow &{} A_{n,U,\varSigma }^{(m)} \\ \downarrow &{}&{} \downarrow \\ X_{n,U'} &{} \hookrightarrow &{} X_{n,U',\varDelta } \\ || &{} &{} \downarrow \\ X_{n,U'} &{} \hookrightarrow &{} X_{n,U'}^{{\text {min}}}. \end{array} $$
We define \(A_{n,U}^{(m),{{\text {ord}}},\dag }\) and \(A_{n,U,\varSigma }^{(m),{{\text {ord}}},\dag }\) to be the pre-image of \(X_{n,U'}^{{{\text {ord}}},{{\text {min}}},\dag }\) in the dagger spaces associated to \(A_{n,U}^{(m)}\) and \(A_{n,U,\varSigma }^{(m)}\).
We will define
$$H^i_{c-\partial }\left( \overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p\right) $$
to be the hypercohomology of the complex on \(A^{(m),{{\text {ord}}},\dag }_{n,U,\varSigma }\) which is the tensor product of the de Rham complex with log poles towards the boundary, \(A^{(m),{{\text {ord}}},\dag }_{n,U,\varSigma }-A^{(m),{{\text {ord}}},\dag }_{n,U}\), and the ideal sheaf defining the boundary. We believe it is a sort of rigid cohomology of the ordinary locus \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}\) in the special fibre of an integral model of \(A_{n,U}^{(m)}\); more specifically, cohomology with compact support towards the toroidal boundary, but not towards the non-ordinary locus, hence our notation. However we have not bothered to verify that this group only depends on ordinary locus in the special fibre. The theory of Shimura varieties provides us with sufficiently canonical lifts that this will not matter to us. Our proof that for N sufficiently large \(\varPi (N)\) occurs in the space of overconvergent p-adic cusp forms for \(G_n\) proceeds by evaluating \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) in two ways.
Firstly we use the usual Hodge spectral sequence. The higher direct images from \(A_{n,U,\varSigma }^{(m)}\) to \(X_{n,U',\varDelta }\) of the tensor product of the ideal sheaf of the boundary and the sheaf of differentials of any degree with log poles along the boundary, are canonically filtered with graded pieces sheaves of the form \({\mathcal {E}}_{U',\varDelta ,\rho }^{{\text {sub}}}\). Thus \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) can be computed in terms of the groups
$$H^j\left( X_{n,U,\varDelta }^{{{\text {ord}}},\dag }, {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\right) . $$
A crucial observation for us is that for \(j>0\) this group vanishes (see Theorem 5.4 and Proposition 6.12). This observation seems to have been made independently, at about the same time, by Andreatta, Iovita and Pilloni (see [1, 2]). It seems quite surprising to us. It is false if one replaces \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\) with \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {can}}}\). Its proof depends on a number of apparently unrelated facts, including:
  • \(X_{n,U}^{{{\text {ord}}},{{\text {min}}},\dag }\) is affinoid.

  • The stabilizer in \(GL_n({\mathcal {O}}_F)\) of a positive definite hermitian \(n \times n\) matrix over F is finite.

  • Certain line bundles on self-products A of the universal abelian scheme over \(X_{n',U'}\) (for \(n'<n\)) are relatively ample for \(A/X_{n',U'}\).

This observation implies that \(H_{c-\partial }^i(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) can be computed by a complex whose terms are spaces of overconvergent cusp forms. Hence it suffices for us to show that, for N sufficiently large, \(\varPi (N)\) occurs in
$$H^i_{c-\partial }\left( \overline{{A}}^{(m),{{\text {ord}}}}_{n}, \overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U, \varSigma } H^i_{c-\partial }\left( \overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p\right) $$
for some m and i (depending on N).
To achieve this we use a second spectral sequence which computes the cohomology group \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) in terms of the rigid cohomology of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) and its various boundary strata. See Sect. 6.5. This is an analogue of the spectral sequence
$$E_1^{i,j}=H^j(Y^{(i)},{\mathbb {C}}) \Rightarrow H^{i+j}_c(Y-\partial Y,{\mathbb {C}}), $$
where Y is a proper smooth variety over \({\mathbb {C}}\), where \(\partial Y\) is a simple normal crossings divisor on Y, and where \(Y^{(i)}\) is the disjoint union of the i-fold intersections of irreducible components of \(\partial Y\). (So \(Y^{(0)}=Y\).) Some of the terms in this spectral sequence seem a priori to be hard to control, e.g. \(H^i_{{\text {rig}}}(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })\). However employing theorems about rigid cohomology due to Berthelot and Chiarellotto, we see that the eigenvalues of Frobenius on \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }, \overline{{{\mathbb {Q}}}}_p)\) are all Weil \(p^j\)-numbers for \(j\ge 0\). Moreover the weight 0 part, \(W_0 H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\), equals the cohomology of a complex only involving the rigid cohomology in degree 0 of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}\) and its various boundary strata. (See Proposition 6.24.) This should have a purely combinatorial description. More precisely we define a simplicial complex \({\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })\) whose vertices correspond to boundary components of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) and whose j-faces correspond to j-boundary components with non-trivial intersection. For \(i>0\) we obtain an isomorphism
$$H^i\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\right) \right| ,\overline{{{\mathbb {Q}}}}_p\right) \cong W_0 H_{c-\partial }^{i+1}\left( \overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p\right) . $$
Thus it suffices to show that for N sufficiently large \(\varPi (N)\) occurs in
$$H^i\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n}\right) \right| ,\overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U, \varSigma } H^i\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\right) \right| ,\overline{{{\mathbb {Q}}}}_p\right) $$
for some m and some \(i>0\) (possibly depending on N).
The boundary of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) comes in pieces indexed by the conjugacy classes of maximal parabolic subgroups of \(G_n\). We shall be interested in the union of the irreducible components which are associated to \(P_{n,(n)}^+\). These correspond to an open subset \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|_{=n}\) of \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|\). As \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|\) is compact, the interior cohomology
$$H^i_{{\text {Int}}}\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n}\right) \right| _{=n},\overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U, \varSigma } H^i_{{\text {Int}}}\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\right) \right| _{=n},\overline{{{\mathbb {Q}}}}_p\right) $$
is naturally a subquotient of \(H^i(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})|,\overline{{{\mathbb {Q}}}}_p)\). (By interior cohomology we mean the image of the cohomology with compact support in the cohomology. The interior cohomology of an open subset of an ambient compact Hausdorff space is naturally a subquotient of the cohomology of that ambient space.) Thus it even suffices to show that for N sufficiently large \(\varPi (N)\) occurs in
$$H^i_{{\text {Int}}}\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n}\right) \right| _{=n},\overline{{{\mathbb {Q}}}}_p\right) $$
for some m and some \(i>0\) (possibly depending on N).
However the data \(\varSigma \) are a \(G_n^{(m)}({\mathbb {Q}})\)-invariant (glued) collection of polyhedral cone decompositions and \({\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})\) is obtained from \(\varSigma \) by replacing 1-cones by vertices, 2-cones by edges, etc. The cones corresponding to \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})|_{=n}\) are a disjoint union of polyhedral cones in the space of positive definite hermitian forms on \(F^n\). From this one obtains an equality
$$\left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n}\right) \right| _{=n}= \coprod _{h \in P_{n,(n)}^{+}({\mathbb {A}}^{\infty ,p}) \backslash G_n({\mathbb {A}}^{\infty ,p})/U^p} {\mathfrak {T}}^{(m)}_{(n), hUh^{-1} \cap P_{n,(n)}^{(m),+}({\mathbb {A}}^\infty )}, $$
where
with U(n) denoting the usual \(n \times n\) compact unitary group. We deduce that
$$H^i_{{\text {Int}}}\left( \left| {\mathcal {S}}\left( \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n}\right) \right| _{=n},\overline{{{\mathbb {Q}}}}_p\right) ={{\text {Ind}}}_{P_{n,(n)}^{+}({\mathbb {A}}^{\infty ,p})}^{G_n({\mathbb {A}}^{\infty ,p})} H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p\right) ^{{\mathbb {Z}}_p^\times }, $$
where
$$H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U'} H^i_{{\text {Int}}}\left( {\mathfrak {T}}^{(m)}_{(n), U'}, \overline{{{\mathbb {Q}}}}_p\right) $$
as \(U'\) runs over neat open compact subgroups of \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\). (The \({\mathbb {Z}}_p^\times \)-invariants result from a restriction on the open compact subgroups of \(G_n({\mathbb {A}}^\infty )\) that we are considering.) Thus it suffices to show that for all sufficiently large N, the representation \(1 \times (\pi ||\det ||^N)^{\infty ,p}\) occurs in \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}^{(m)}, {\mathbb {C}})\) for some \(i>0\) and some m (possibly depending on N).
We will write simply \({\mathfrak {T}}_{(n),U'}\) for \({\mathfrak {T}}^{(0)}_{(n),U'}\), a locally symmetric space associated to \(L_{n,(n)}\cong GL_1 \times {{\text {RS}}}_{\mathbb {Q}}^F GL_n\). If \(\rho \) is a representation of \(L_{n,(n)}\) over \({\mathbb {C}}\), then it gives rise to a locally constant sheaf \({\mathcal {L}}_{\rho ,U'}\) over \({\mathfrak {T}}_{(n),U'}\). We set
$$H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)},{\mathcal {L}}_\rho \right) =\lim _{\rightarrow U'} H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n),U'},{\mathcal {L}}_{\rho ,U'}\right) , $$
a smooth \(L_{n,(n)}({\mathbb {A}}^\infty )\)-module. The space \({\mathfrak {T}}^{(m)}_{(n), U'}\) is an \((S^1)^{nm[F: {\mathbb {Q}}]}\)-bundle over the locally symmetric space \({\mathfrak {T}}^{(0)}_{(n),U'}\) and if \(\pi ^{(m)}\) denotes the fibre map then
$$R^j\pi ^{(m)}_* {\mathbb {C}}\cong {\mathcal {L}}_{\wedge ^j{{\text {Hom}}}_F(F^m,F^n)^\vee \otimes _{\mathbb {Q}}{\mathbb {C}},U'}, $$
where \(L_{n,(n)}\) acts on \({{\text {Hom}}}_F(F^m,F^n)\) via projection to \({{\text {RS}}}^F_{\mathbb {Q}}GL_n\). Moreover the Leray spectral sequence
$$E_2^{i,j}=H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)},{\mathcal {L}}_{\wedge ^j{{\text {Hom}}}_F(F^m,F^n)^\vee \otimes _{\mathbb {Q}}{\mathbb {C}}}\right) \Rightarrow H^{i+j}_{{\text {Int}}}\left( {\mathfrak {T}}^{(m)}_{(n)} ,{\mathbb {C}}\right) $$
degenerates at the second page. (This can be seen by considering the action of the centre of \(L_{n,(n)}({\mathbb {A}}^\infty )\).) Thus it suffices to show that for all sufficiently large N, we can find non-negative integers j and m and an irreducible constituent \(\rho \) of \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n)^\vee \otimes _{\mathbb {Q}}{\mathbb {C}}\) such that the representation \(1 \times (\pi ||\det ||^N)^{\infty ,p}\) occurs in \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}, {\mathcal {L}}_\rho )\) for some \(i \in {\mathbb {Z}}_{>0}\). Clozel [20] checked that (for \(n>1\)) this will be the case as long as \(1 \times (\pi ||\det ||^N)_\infty \) has the same infinitesimal character as some irreducible constituent of \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n) \otimes _{\mathbb {Q}}{\mathbb {C}}\), i.e. if \(\rho _\pi \otimes ({\text{ N }}_{F/{\mathbb {Q}}} \circ \det )^N\) occurs in \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n) \otimes _{\mathbb {Q}}{\mathbb {C}}\). From Weyl’s construction of the irreducible representations of \(GL_n\), for large enough N this will indeed be the case for some m and j.

We remark it is essential to work with N sufficiently large. It is not an artefact of the fact that we are working with Kuga–Sato varieties rather than local systems on the Shimura variety. We can twist a local system on the Shimura variety by a power of the multiplier character of \(G_n\). However the restriction of the multiplier factor of \(G_n\) to \(L_{n,(n)} \cong GL_1 \times {{\text {RS}}}^F_{\mathbb {Q}}GL_n\) factors through the \(GL_1\)-factor and does not involve the \({{\text {RS}}}^F_{\mathbb {Q}}GL_n\) factor.

We learnt from the series of papers [3032] the key observation that \(|{\mathcal {S}}(\partial A_{n,U,\varSigma }^{(m)})|\) has a nice geometric interpretation involving the locally symmetric space for \(L_{n,(n)}\) and that this could be used to calculate cohomology.

Although the central argument we have sketched above is not long, this paper has unfortunately become very long. If we had only wanted to construct \(r_{p,\imath }(\pi )\) for all but finitely many primes p, then the argument would have been significantly shorter as we could have worked only with Shimura varieties \(X_{n,U}\) which have good integral models at p. The fact that we want to construct \(r_{p,\imath }(\pi )\) for all p adds considerable technical complications and also requires appeal to the recent work [44]. (Otherwise we would only need to appeal to [41, 42].)

Another reason this paper has grown in length is the desire to use a language to describe toroidal compactifications of mixed Shimura varieties that is different from the language used in [41, 42, 44]. We do this because at least one of us (R.T.) finds this language clearer. In any case it would be necessary to establish a substantial amount of notation regarding toroidal compactifications of Shimura varieties, which would require significant space. We hope that the length of the paper, and the technicalities with which we have to deal, won’t obscure the main line of the argument. On a first reading the reader might like to start with “Appendix A”, which summarizes the extensive notation we use, and then turn to Sects. (5 and) 6 and 7. These sections will provide reference back to the key results from earlier sections. We have added “Appendix B” to help comparison between the notation of this paper and the notation of [41, 42, 44], which we hope will make life easier for those readers that want to follow-up on our many references to these papers.

After we announced these results, but while we were writing up this paper, Scholze found another proof of Theorem A, relying on his theory of perfectoid spaces. His arguments seem to be in many ways more robust. For instance, he can handle torsion in the cohomology of the locally symmetric varieties associated to \(GL_n\) over a CM field. Scholze’s methods have some similarities with ours. Both methods first realize the Hecke eigenvalues of interest in the cohomology with compact support of the open Shimura variety by an analysis of the boundary and then show that they also occur in some space of p-adic cusp forms. We work with the ordinary locus of the Shimura variety, which for the minimal compactification is affinoid. Scholze works with the whole Shimura variety, but at infinite level. He (very surprisingly) shows that at infinite level, as a perfectoid space, (some compactification of) the Shimura variety has a Hecke invariant affinoid cover.

1.1 Notation

If \(G \twoheadrightarrow H\) is a surjective group homomorphism and if U is a subgroup of G we will sometimes use U to also denote the image of U in H.

If \(f: X \rightarrow Y\) and \(f': Y \rightarrow Z\) then we will denote by \(f' \circ f: X \rightarrow Z\) the composite map f followed by \(f'\). In this paper we will use both left and right actions. Suppose that G is a group acting on a set X and that \(g,h \in G\). If G acts on X on the left we will write gh for \(g \circ h\). If G acts on X on the right we will write hg for \(g \circ h\).

If f is an automorphism of \({{\text {Hom}}}(X,Y)\) we will sometimes use \((\circ f)\) to denote the map
$$\begin{array}{rcl} {{\text {Hom}}}(X,Y) &{} \longrightarrow &{} {{\text {Hom}}}(X,Y) \\ h &{} \longmapsto &{} h \circ f. \end{array} $$
We will sometimes use  /  to denote a quotient, and sometimes we will use it to denote the fact that the object to the left lives ‘over’ the object to the right. Both these usages are standard, and we hope no confusion will arise.

If G is a group (or group scheme) then Z(G) will denote its centre.

We will write \(S_n\) for the symmetric group on n letters. We will write U(n) for the group of \(n \times n\) complex matrices h with \({}^th({}^ch)=1_n\).

If G is an abelian group we will write \(G[\infty ]\) for the torsion subgroup of G, \(G[\infty ^p]\) for the subgroup of elements of order prime to p, and \(G^{{\text {TF}}}=G/G[\infty ]\). We will write \(TG= \lim _{\leftarrow N} G[N]\) and \(T^pG=\lim _{\leftarrow p {\not | }N}G[N]\). We will also write \(VG=TG \otimes _{\mathbb {Z}}{\mathbb {Q}}\) and \(V^pG=T^pG \otimes _{\mathbb {Z}}{\mathbb {Q}}\).

If A is a ring, if B is a locally free, finite A-algebra, and if \(X/{{\text {Spec}}\,}B\) is a quasi-projective scheme; then we will let \({{\text {RS}}}^B_A X\) denote the restriction of scalars (or Weil restriction) of X from B to A. (See, for instance, section 7.6 of [12].)

By a p-adic formal scheme we mean a formal scheme such that p generates an ideal of definition.

If X is an A-module and B is an A-algebra, we will sometimes write \(X_B\) for \(X \otimes _A B\). If X is reflexive over A, then we will also use X to denote the additive group scheme over A defined by
$$X(B)=X \otimes _A B=X_B $$
for all A-algebras B.
If X is a locally free \({\mathcal {O}}_F\)-module we will write \(GL(X/{\mathcal {O}}_F)\) for the group scheme over \({\mathbb {Z}}\) defined by
$$GL(X/{\mathcal {O}}_F)(A)={{\text {Aut}}}\left( \left( X \otimes _{\mathbb {Z}}A\right) /\left( {\mathcal {O}}_F \otimes _{\mathbb {Z}}A\right) \right) . $$
If Y is a scheme and if \(G_1,G_2/Y\) are group schemes then we will let
$${\mathcal {H}}{{o}}{{m}}(G_1,G_2) $$
denote the Zariski sheaf on Y whose sections over an open W are
$${{\text {Hom}}}(G_1|_W,G_2|_W). $$
If in addition R is a ring then we will let \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)_R\) denote the tensor product of sheaves \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)\otimes _{\mathbb {Z}}R\) and we will let \({{\text {Hom}}}(G_1,G_2)_R\) denote the R-module of global sections of \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)_R\). If Y is noetherian this is the same as \({{\text {Hom}}}(G_1,G_2) \otimes _{\mathbb {Z}}R\), but for a general base Y it may differ.

If \({\mathcal {S}}\) is a simplicial complex we will write \(|{\mathcal {S}}|\) for the corresponding topological space.

If F is a field then \(G_F\) will denote its absolute Galois group. If F is a number field and \(F_0 \subset F\) is a subfield and S is a finite set of primes of \(F_0\), then we will denote by \(G_F^S\) the maximal continuous quotient of \(G_F\) in which all primes of F not lying above an element of S are unramified.

Suppose that F is a number field and that v is a place of F. If v is finite we will write \(\varpi _v\) for a uniformizer in \(F_v\) and k(v) for the residue field of v. We will write \(| \,\,\,|_v\) for the absolute value on F associated to v and normalized as follows:
  • if v is finite then \(|\varpi _v|_v=(\# k(v))^{-1}\);

  • if v is real then \(|x|_v=\pm x\);

  • if v is complex then \(|x|_v={}^cxx\).

We write
$$||\,\,\,||_F=\prod _v |\,\,\,|_v: {\mathbb {A}}_F^\times \longrightarrow {\mathbb {R}}_{>0}^\times . $$
We will write \({\mathcal {D}}_F^{-1}\) for the inverse different of \({\mathcal {O}}_F\).

If \(w \in {\mathbb {Z}}\) and p is a prime number then by a Weil \(p^w\)-number we mean an element \(\alpha \in \overline{{{\mathbb {Q}}}}\) which is an integer away from p and such that for each infinite place v of \(\overline{{{\mathbb {Q}}}}\) we have \(|\alpha |_v=p^w\).

Suppose that v is finite and that
$$r: G_{F_v} \longrightarrow GL_n(\overline{{{\mathbb {Q}}}}_l) $$
is a continuous representation, which in the case v|l we assume to be de Rham. Then we will write \({{\text {WD}}}(r)\) for the corresponding Weil–Deligne representation of the Weil group \(W_{F_v}\) of \(F_v\) (see, for instance, section 1 of [56]). If \(\pi \) is an irreducible smooth representation of \(GL_n(F_v)\) over \({\mathbb {C}}\) we will write \({\text {rec}}_{F_v}(\pi )\) for the Weil–Deligne representation of \(W_{F_v}\) corresponding to \(\pi \) by the local Langlands conjecture (see, for instance, the introduction to [29]). If \(\pi _i\) is an irreducible smooth representation of \(GL_{n_i}(F_v)\) over \({\mathbb {C}}\) for \(i=1,2\) then there is an irreducible smooth representation \(\pi _1 \boxplus \pi _2\) of \(GL_{n_1+n_2}(F_v)\) over \({\mathbb {C}}\) satisfying
$${\text {rec}}_{F_v}(\pi _1 \boxplus \pi _2)={\text {rec}}_{F_v}(\pi _1) \oplus {\text {rec}}_{F_v}(\pi _2). $$
Suppose that G is a reductive group over \(F_v\) and that P is a parabolic subgroup of G with unipotent radical N and Levi component L. Suppose also that \(\pi \) is a smooth representation of \(L(F_v)\) on a vector space \(W_\pi \) over a field \(\varOmega \) of characteristic 0. We will define
$${{\text {Ind}}}_{P(F_v)}^{G(F_v)} \pi $$
to be the representation of \(G(F_v)\) by right translation on the set of locally constant functions
$$\varphi : G(F_v) \longrightarrow W_\pi $$
such that
$$\varphi (hg)=\pi (h) \varphi (g) $$
for all \(h \in P(F_v)\) and \(g \in G(F_v)\). In the case \(\varOmega ={\mathbb {C}}\) we also define
$${{\text {n-Ind}}}_{P(F_v)}^{G(F_v)} \pi ={{\text {Ind}}}_{P(F_v)}^{G(F_v)} \left( \pi \otimes \delta _P^{1/2}\right) $$
where
$$\delta _P(h)^{1/2}=\left| \det \left( {{\text {ad}}}(h)|_{{{\text {Lie}}\,}N}\right) \right| ^{1/2}_v. $$
If G is a linear algebraic group over F then the concept of a neat open compact subgroup of \(G({\mathbb {A}}_F^\infty )\) is defined, for instance, in section 0.6 of [49].

2 Some algebraic groups and automorphic forms

For the rest of this paper fix the following notation. Let \(F^+\) be a totally real field and \(F_0\) an imaginary quadratic field, and set \(F=F_0F^+\). Write c for the non-trivial element of \({{\text {Gal}}}(F/F^+)\). Also choose a rational prime p which splits in \(F_0\) and choose an element \(\delta _F \in {\mathcal {O}}_{F,(p)}\) with \({{\text {tr}}}_{F/F^+} \delta _F=1\) (which is possible as p is unramified in \(F/F^+\)).

Fix an isomorphism \(\imath : \overline{{{\mathbb {Q}}}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {C}}\). Fix a choice of \(\sqrt{p} \in \overline{{{\mathbb {Q}}}}_p\) by \(\imath \sqrt{p} >0\). If v is a prime of F and \(\pi \) an irreducible admissible representation of \(GL_m(F_v)\) over \(\overline{{{\mathbb {Q}}}}_p\) define
$${\text {rec}}_{F_v}(\pi )=\imath ^{-1} {\text {rec}}_{F_v}(\imath \pi ) $$
a Weil–Deligne representation of \(W_{F_v}\) over \(\overline{{{\mathbb {Q}}}}_p\).

Let n be a non-negative integer. We will often attach n as a subscript to other notation, when we need to record the particular choice of n we are working with, but, at other times when the choice of n is clear, we may drop it from the notation.

2.1 Three algebraic groups

Write \(\varPsi _n\) for the \(n \times n\)-matrix with 1’s on the anti-diagonal and 0’s elsewhere, and set
$$\begin{aligned} J_{n}= \left( \begin{array}{cc} {0} &{} {\quad \varPsi _n} \\ {-\varPsi _n} &{} {\quad 0} \end{array} \right) \in GL_{2n}({\mathbb {Z}}). \end{aligned}$$
Let
$$\begin{aligned} \varLambda _n=\left( {\mathcal {D}}^{-1}_F\right) ^n \oplus {\mathcal {O}}_{F}^{n}, \end{aligned}$$
and define a perfect pairing
$$\begin{aligned} \langle \,\,\, ,\,\,\, \rangle _n: \varLambda _n \times \varLambda _n \longrightarrow {\mathbb {Z}}\end{aligned}$$
by
$$\begin{aligned} \langle x,y \rangle _n={{\text {tr}}}_{F/{\mathbb {Q}}}\left( {}^tx J_{n} {}^cy\right) . \end{aligned}$$
We will write \(V_n\) for \(\varLambda _n \otimes {\mathbb {Q}}\). Let \(G_n\) denote the group scheme over \({\mathbb {Z}}\) defined by
$$G_n(R) =\left\{ (g,\mu ) \in {{\text {Aut}}}((\varLambda _n \otimes _{{\mathbb {Z}}} R)/({\mathcal {O}}_F \otimes _{\mathbb {Z}}R)) \times R^\times : {}^tg J_n {}^cg=\mu J_n\right\} ,$$
for any ring R, and let \(\nu : G_n \rightarrow GL_1\) denote the multiplier character which sends \((g,\mu )\) to \(\mu \). Then \(G_n\) is a quasi-split connected reductive group scheme over \({\mathbb {Z}}[1/D_{F/{\mathbb {Q}}}]\) (where \(D_{F/{\mathbb {Q}}}\) denotes the discriminant of \(F/{\mathbb {Q}}\)) and splits over \({\mathcal {O}}_{F^{{\text {nc}}}}[1/D_{F/{\mathbb {Q}}}]\) (where \(F^{{{\text {nc}}}}\) denotes the normal closure of \(F/{\mathbb {Q}}\)). In particular \(G_0\) will denote \(GL_1\) and \(\nu : G_0 \rightarrow GL_1\) is the identity map.
If \(n>0\) set
$$\begin{aligned} C_n={\mathbb {G}}_m \times \ker \left( N_{F/F^+}: {{\text {RS}}}_{{\mathbb {Z}}}^{{\mathcal {O}}_F} {\mathbb {G}}_m \longrightarrow {{\text {RS}}}^{{\mathcal {O}}_{F^+}}_{{\mathbb {Z}}} {\mathbb {G}}_m\right) . \end{aligned}$$
Then there is a natural map
$$\begin{array}{rcl} G_n &{} \longrightarrow &{} C_n \\ (g,\mu ) &{} \longmapsto &{} (\mu , \mu ^{-n} \det g). \end{array}$$
If \(n=0\) we set \(C_0={\mathbb {G}}_m\) and let \(G_0 \longrightarrow C_0\) denote the map \(\nu \). In either case this map identifies \(C_n\) with \(G_n/[G_n,G_n]\).
We will write \(\varLambda _{n,(i)}\) for the submodule of \(\varLambda _n\) consisting of elements whose last \(2n-i\) entries are 0, and \(V_{n,(i)}\) for \(\varLambda _{n,(i)} \otimes {\mathbb {Q}}\). If W is a submodule of \(\varLambda _n\) we will write \(W^\perp \) for its orthogonal complement with respect to \(\langle \,\,\,,\,\,\,\rangle _n\). Thus \(\varLambda _{n,(i)}^\perp \) is the submodule of \(\varLambda _n\) consisting of vectors whose last i entries are 0. Also write
$$\begin{aligned} \varLambda _n^{(m)}={{\text {Hom}}}\left( {\mathcal {O}}_F^m,{\mathbb {Z}}\right) \oplus \varLambda _n, \end{aligned}$$
and set \(V_n^{(m)}=\varLambda _n^{(m)} \otimes _{\mathbb {Z}}{\mathbb {Q}}\). Throughout this paper there will be various objects indexed by a superscript \({}^{(m)}\). In the case \(m=0\) we will sometimes simply drop it from the notation. For example, \(\varLambda _n=\varLambda _n^{(0)}\).
Define an additive group scheme \({{\text {Hom}}}_n^{(m)}\) over \({\mathbb {Z}}\) by
$$\begin{aligned} {{\text {Hom}}}^{(m)}_n(R)={{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^m,\varLambda _n\right) \otimes _{\mathbb {Z}}R. \end{aligned}$$
Then \({{\text {Hom}}}_n^{(m)}\) has an action of \(G_n \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) given by
$$\begin{aligned} (g,h)f=g \circ f \circ h^{-1}. \end{aligned}$$
Also define a perfect pairing
$$\begin{aligned} \langle \,\,\,,\,\,\, \rangle _n^{(m)}: {{\text {Hom}}}_n^{(m)}(R) \times {{\text {Hom}}}_n^{(m)}(R) \longrightarrow R \end{aligned}$$
by
$$\begin{aligned} \left\langle f,f'\right\rangle _n^{(m)}=\sum _{i=1}^m \left\langle fe_i, f'e_i \right\rangle _n, \end{aligned}$$
where \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_F^m\). We have
$$\left\langle (g,h)f,f'\right\rangle _n^{(m)}= \nu (g) \left\langle f, \left( g^{-1},{}^{c,t}h\right) f'\right\rangle _n^{(m)}.$$
Moreover \(G_n(R)\) is identified with the set of pairs
$$(g,\mu ) \in GL\left( {{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^m, \varLambda _n\right) /{\mathcal {O}}_F\right) (R) \times R^\times $$
such that g commutes with the action of \(GL_m({\mathcal {O}}_F \otimes _{{\mathbb {Z}}} R)\) and such that
$$\begin{aligned} \langle gf,gf'\rangle _n^{(m)}=\mu \langle f,f'\rangle _n^{(m)} \end{aligned}$$
for all \(f,f' \in {{\text {Hom}}}_{{\mathcal {O}}_F}({\mathcal {O}}_F^m, \varLambda _n)(R)\). We set
$$\begin{aligned} G_n^{(m)}=G_n \ltimes {{\text {Hom}}}_n^{(m)}. \end{aligned}$$
Then \(G_n^{(m)}\) has an action of \({{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) by
$$\begin{aligned} h(g,f)=(g,(1,h)f). \end{aligned}$$
Moreover \(G_n^{(m)}\) acts on \(\varLambda _n^{(m)}\), by letting \(f \in {{\text {Hom}}}_n^{(m)}\) act by
$$\begin{aligned} f: (h,x) \longmapsto (h+\langle x, f \rangle _n, x) \end{aligned}$$
and \(g \in G_n\) act by
$$\begin{aligned} g: (h,x) \longmapsto (h,gx). \end{aligned}$$
Moreover \({{\text {RS}}}_{\mathbb {Z}}^{{\mathcal {O}}_F} GL_m\) acts on \(\varLambda _n^{(m)}\) by
$$\begin{aligned} \gamma : (h ,x) \longmapsto (h \circ \gamma ^{-1}, x). \end{aligned}$$
We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \(m_1 \ge m_2\) we embed \({\mathcal {O}}_F^{m_2} \hookrightarrow {\mathcal {O}}_F^{m_1}\) via
$$\begin{aligned} i_{m_2,m_1}: (x_1,\ldots ,x_{m_2}) \longmapsto (x_1,\ldots ,x_{m_2},0,\ldots ,0). \end{aligned}$$
This gives rise to maps
$$\begin{aligned} i_{m_2,m_1}^*: {{\text {Hom}}}_n^{(m_1)} \longrightarrow {{\text {Hom}}}_n^{(m_2)} \end{aligned}$$
and
$$\begin{aligned} i_{m_2,m_1}^*: G_n^{(m_1)} \longrightarrow G_n^{(m_2)}. \end{aligned}$$
It also gives rise to
$$\begin{aligned} i_{m_2,m_1}^*: \varLambda _n^{(m_1)} \twoheadrightarrow \varLambda _n^{(m_2)}. \end{aligned}$$
Suppose that R is a ring and that X is an \({\mathcal {O}}_F \otimes _{\mathbb {Z}}R\)-module. We will write \({{\text {Herm}}}_X\) for the R-module of R-bilinear pairings
$$\begin{aligned} (\,\,\, ,\,\,\,): X \times X \longrightarrow R \end{aligned}$$
which satisfy
  1. (1)

    \((ax,y)=(x,{}^cay)\) for all \(a \in {\mathcal {O}}_F\) and \(x,y \in X\);

     
  2. (2)

    \((x,y)=(y,x)\) for all \(x,y \in X\).

     
If \(z \in {{\text {Herm}}}_X\) we will sometimes denote the corresponding pairing \((\,\,\,,\,\,\,)_z\). If S is an R-algebra we have a natural map
$$\begin{aligned} {{\text {Herm}}}_X \otimes _R S \longrightarrow {{\text {Herm}}}_{X \otimes _R S}. \end{aligned}$$
If \(X={\mathcal {O}}_F^m \otimes _{\mathbb {Z}}R\) then we will write
$$\begin{aligned} {{\text {Herm}}}^{(m)}(R)={{\text {Herm}}}_{{\mathcal {O}}_F^m} \otimes _{\mathbb {Z}}R \longrightarrow {{\text {Herm}}}_{{\mathcal {O}}_F^m \otimes _{\mathbb {Z}}R}. \end{aligned}$$
If \(X \rightarrow Y\) then there is a natural map \({{\text {Herm}}}_Y \rightarrow {{\text {Herm}}}_X\). In particular if \(m_1 \ge m_2\), then there is a natural map
$$\begin{aligned} {{\text {Herm}}}^{(m_1)} \longrightarrow {{\text {Herm}}}^{(m_2)} \end{aligned}$$
induced by the map \({\mathcal {O}}_F^{(m_2)} \hookrightarrow {\mathcal {O}}_F^{(m_1)}\) described in the last paragraph. The group \(GL(X/{\mathcal {O}}_F)\) acts on the left on \({{\text {Herm}}}_X\) by
$$\begin{aligned} (x,y)_{h z}=(h^{-1}x,h^{-1}y)_z. \end{aligned}$$
There is a natural isomorphism
$${{\text {Herm}}}_{X \oplus Y} \cong {{\text {Herm}}}_X \oplus {{\text {Hom}}}_R\left( X \otimes _{{\mathcal {O}}_F \otimes R, c \otimes 1} Y, R\right) \oplus {{\text {Herm}}}_Y,$$
under which an element (zfw) of the right hand side corresponds to
$$\begin{aligned} ((x,y),(x',y'))_{(z,f,w)}= (x,x')_z + f(x \otimes y')+f(x' \otimes y)+(y,y')_w. \end{aligned}$$
If X is an \({\mathcal {O}}_F \otimes _{\mathbb {Z}}R\)-module, there is a natural pairing
$$\begin{aligned} \begin{array}{rcl} (X \otimes _{{\mathcal {O}}_F\otimes R,c \otimes 1} X) \times {{\text {Herm}}}_{X} &{} \longrightarrow &{} R \\ (x \otimes y, z)&{} \longmapsto &{} (x,y)_z. \end{array} \end{aligned}$$
We further define
$$\begin{aligned} \begin{array}{rcl} {{\text {sw}}}: (X \otimes _{{\mathcal {O}}_F \otimes R,c \otimes 1} X) &{} \longrightarrow &{} (X \otimes _{{\mathcal {O}}_F \otimes R,c \otimes 1} X) \\ x \otimes y &{} \longmapsto &{} y \otimes x, \end{array} \end{aligned}$$
and
$$\begin{aligned} S(X)=(X \otimes _{{\mathcal {O}}_F\otimes R,c \otimes 1} X)/({{\text {sw}}}-1). \end{aligned}$$
There is a natural map in the other direction
$$\begin{aligned} \begin{array}{rcl} S(X) &{} \longrightarrow &{} X \otimes _{{\mathcal {O}}_F \otimes R,c \otimes 1} X \\ w &{} \longmapsto &{} w+{{\text {sw}}}(w), \end{array} \end{aligned}$$
such that the composite \(S(X) \rightarrow X \otimes _{{\mathcal {O}}_F \otimes R,c \otimes 1} X \rightarrow S(X)\) is multiplication by 2. Note that if \(F/F^+\) is ramified above 2 then \(S({\mathcal {O}}_F^m)\) can have 2-torsion, but that \(S({\mathcal {O}}_{F,(p)}^m)\) is torsion free. (Either \(p>2\) or by assumption \(F/F^+\) is not ramified above 2.) There is a perfect duality
$$\begin{aligned} S({\mathcal {O}}_F^m)^{{\text {TF}}}\times {{\text {Herm}}}^{(m)}({\mathbb {Z}}) \longrightarrow {\mathbb {Z}}. \end{aligned}$$
We will write
$$\begin{aligned} e=\sum _{i=1}^m e_i \otimes e_i \in {\mathcal {O}}_F^m \otimes _{{\mathcal {O}}_F,c} {\mathcal {O}}_F^m, \end{aligned}$$
where \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_F^m\).
Set \(N_n^{(m)}({\mathbb {Z}})\) to be the set of pairs
$$(f,z) \in {{\text {Hom}}}_{{\mathcal {O}}_F}({\mathcal {O}}_F^m,\varLambda _n) \oplus \left( \frac{1}{2}{{\text {Herm}}}^{(m)}({\mathbb {Z}})\right) $$
such that
$$\begin{aligned} (x,y)_z - \frac{1}{2} \langle f x ,fy \rangle _n\in {\mathbb {Z}}\end{aligned}$$
for all \(x,y \in {\mathcal {O}}_F^m\). We define a group scheme \(N_{n}^{(m)}/{{\text {Spec}}\,}{\mathbb {Z}}\) by setting \(N_{n}^{(m)}(R)\) to be the set of pairs
$$\begin{aligned} (f,z) \in N_n^{(m)}({\mathbb {Z}}) \otimes _{\mathbb {Z}}R \end{aligned}$$
with group law given by
$$(f,z) (f',z')=\left( f+f', z+z'+\frac{1}{2}\left( \langle f\,\,\,,f'\,\,\,\rangle _n - \langle f'\,\,\,,f\,\,\,\rangle _n \right) \right) ,$$
where by \(\langle f\,\,\,,f'\,\,\,\rangle _n - \langle f'\,\,\,,f\,\,\,\rangle _n\) we mean the hermitian form
$$\begin{aligned} (x,y) \longmapsto \langle f(x),f'(y)\rangle _n - \langle f'(x),f(y)\rangle _n. \end{aligned}$$
Note that \((f,z)^{-1}=(-f,-z)\). Thus there is an exact sequence
$$\begin{aligned} (0) \longrightarrow {{\text {Herm}}}^{(m)} \longrightarrow N_{n}^{(m)} \longrightarrow {{\text {Hom}}}^{(m)}_n \longrightarrow (0). \end{aligned}$$
In fact, \(Z(N_{n}^{(m)})={{\text {Herm}}}^{(m)}\). The commutator in \(N_n^{(m)}\) induces an alternating map
$$\begin{aligned} {{\text {Hom}}}_n^{(m)}(R) \times {{\text {Hom}}}_n^{(m)}(R) \longrightarrow {{\text {Herm}}}^{(m)}(R) \end{aligned}$$
under which \((f,f')\) maps to the pairing
$$(x,y) \longmapsto \left\langle f(x),f'(y)\right\rangle _n- \left\langle f'(x),f(y)\right\rangle _n.$$
If \(m_1 \ge m_2\) there is a natural map
$$\begin{aligned} N_n^{(m_1)} \longrightarrow N_n^{(m_2)} \end{aligned}$$
compatible with the previously described maps
$$\begin{aligned} {{\text {Hom}}}_n^{(m_1)} \rightarrow {{\text {Hom}}}_n^{(m_2)} \end{aligned}$$
and
$$\begin{aligned} {{\text {Herm}}}^{(m_1)} \rightarrow {{\text {Herm}}}^{(m_2)}. \end{aligned}$$
Note that \(G_n \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) acts on \(N_{n}^{(m)}\) from the left by
$$\begin{aligned} (g,h)(f,z)=\left( g \circ f \circ h^{-1}, \nu (g) hz\right) . \end{aligned}$$
If 2 is invertible in R we see that
$$\begin{aligned} {{\text {Herm}}}^{(m)}(R)=\left\{ g\in N_n^{(m)}(R): (-1_m)(g)=g \right\} \end{aligned}$$
and
$$\begin{aligned} {{\text {Hom}}}_n^{(m)}(R)=\left\{ g\in N_n^{(m)}(R): (-1_m)(g)=g^{-1} \right\} . \end{aligned}$$
Set
$$\begin{aligned} \widetilde{{G}}^{(m)}_n=G_n \ltimes N_n^{(m)}, \end{aligned}$$
which has an \({{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\)-action via
$$\begin{aligned} h(g,u)=(g,h(u)). \end{aligned}$$
If \(m_1 \ge m_2\) then we get a natural map \(\widetilde{{G}}^{(m_1)}_n \rightarrow \widetilde{{G}}_n^{(m_2)}\). Note that
$$\begin{aligned} G^{(m)}_n\cong \widetilde{{G}}^{(m)}_n/{{\text {Herm}}}^{(m)}. \end{aligned}$$
Let \(B_n\) denote the subgroup of \(G_n\) consisting of elements which preserve the chain \(\varLambda _{n,(n)} \supset \varLambda _{n,(n-1)} \supset \cdots \supset \varLambda _{n,(1)} \supset \varLambda _{n,(0)}\) and let \(N_n\) denote the normal subgroup of \(B_n\) consisting of elements with \(\nu =1\), which also act trivially on \(\varLambda _{n,(i)}/\varLambda _{n,(i-1)}\) for all \(i=1,\ldots ,n\). Let \(T_n\) denote the group consisting of the diagonal elements of \(G_n\) and let \(A_n\) denote the image of \({\mathbb {G}}_m\) in \(G_n\) via the embedding that sends t onto \(t1_{2n}\). Over \({\mathbb {Q}}\) we see that \(T_n\) is a maximal torus in a Borel subgroup \(B_n\) of \(G_n\) and that \(N_n\) is the unipotent radical of \(B_n\). Moreover \(A_n\) is a maximal split torus in the centre of \(G_n\).
If \(\varOmega \) is an algebraically closed field of characteristic 0 then set
$$\begin{aligned} X^*(T_{n,/\varOmega })={{\text {Hom}}}\left( T_n \times {{\text {Spec}}\,}\varOmega , {\mathbb {G}}_m \times {{\text {Spec}}\,}\varOmega \right) . \end{aligned}$$
Also let \(\varPhi _n \subset X^*(T_{n,/\varOmega })\) denote the set of roots of \(T_n\) on \({{\text {Lie}}\,}G_n\); let \(\varPhi ^+_n \subset \varPhi _n\) denote the set of positive roots with respect to \(B_n\) and let \(\varDelta _n \subset \varPhi _n^+\) denote the set of simple positive roots. We will write \(\varrho _n\) for half the sum of the elements of \(\varPhi _n^+\). If \(R \subset {\mathbb {R}}\) is a subring then \(X^*(T_{n,/\varOmega })^+_R\) will denote the subset of \(X^*(T_{n,/\varOmega })_R\) consisting of elements which pair non-negatively with the coroot \(\check{\alpha } \in X_*(T_{n,/\varOmega })\) corresponding to each \(\alpha \in \varDelta _n\). We will write simply \(X^*(T_{n,/\varOmega })^+\) for \(X^*(T_{n,/\varOmega })_{\mathbb {Z}}^+\). If \(\lambda \in X^*(T_{n,/\varOmega })^+\) we will let \(\rho _{n,\lambda }\) (or simply \(\rho _\lambda \)) denote the irreducible representation of \(G_n\) with highest weight \(\lambda \). When \(\rho _\lambda \) is used as a subscript we will sometimes replace it by just \(\lambda \).
There is a natural identification
$$G_n \times {{\text {Spec}}\,}\varOmega \cong \left\{ (\mu ,g_\tau ) \in {\mathbb {G}}_m \times GL_{2n}^{{{\text {Hom}}}(F,\varOmega )}: g_{\tau c} =\mu J_n {}^tg_\tau ^{-1} J_n \,\, \forall \tau \right\} .$$
This gives rise to the further identification
$$\begin{aligned} T_n \times {{\text {Spec}}\,}\varOmega \cong \left\{ (t_0,(t_{\tau ,i})) \in {\mathbb {G}}_m \times \left( {\mathbb {G}}_m^{2n}\right) ^{{{\text {Hom}}}(F,\varOmega )}: t_{\tau ,i}t_{\tau c, 2n+1-i}=t_0 \,\, \forall \tau , i\right\} . \end{aligned}$$
We will use this to identify \(X^*(T_{n,/\varOmega })\) with a quotient of
$$\begin{aligned} X^*\left( {\mathbb {G}}_m \times \left( {\mathbb {G}}_m^{2n}\right) ^{{{\text {Hom}}}(F,\varOmega )}\right) \cong {\mathbb {Z}}\oplus ({\mathbb {Z}}^{2n})^{{{\text {Hom}}}(F,\varOmega )}. \end{aligned}$$
Under this identification \(X^*(T_{n,/\varOmega })^+\) is identified to the image of the set of
$$\begin{aligned} (a_0,(a_{\tau ,i})) \in {\mathbb {Z}}\oplus ({\mathbb {Z}}^{2n})^{{{\text {Hom}}}(F,\varOmega )} \end{aligned}$$
with
$$\begin{aligned} a_{\tau ,1} \ge a_{\tau ,2} \ge \cdots \ge a_{\tau ,2n} \end{aligned}$$
for all \(\tau \).

If R is a subring of \({\mathbb {R}}\) and H an algebraic subgroup of \(\widetilde{{G}}_n^{(m)}\) we will write \(H(R)^+\) for the subgroup of H(R) consisting of elements with positive multiplier. Thus \(G_n({\mathbb {R}})^+\) (resp. \(G_n^{(m)}({\mathbb {R}})^+\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})^+\)) is the connected component of the identity in \(G_n({\mathbb {R}})\) (resp. \(G_n^{(m)}({\mathbb {R}})\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})\)).

Let
$$\begin{aligned} U_{n,\infty }=(U(n)^2)^{{{\text {Hom}}}(F^+,{\mathbb {R}})} \rtimes \{ 1,j \} \end{aligned}$$
with \(j^2=1\) and \(j(A_\tau ,B_\tau )j=(B_\tau ,A_\tau )\). Embed \(U_{n,\infty }\) in \(G_n({\mathbb {R}})\) by sending \((A_\tau ,B_\tau ) \in (U(n)^2)^{{{\text {Hom}}}(F^+,{\mathbb {R}})}\) to
$$\begin{array}{l} \left( 1,\left( \left( \begin{array}{cc} {(A_\tau +B_\tau )/2} &{} {\quad (A_\tau -B_\tau )\varPsi _n/2i} \\ {\varPsi _n(B_\tau -A_\tau )/2i} &{} {\quad \varPsi _n(A_\tau +B_\tau )\varPsi _n/2} \end{array} \right) \right) _\tau \right) \\ \\ \quad \in G_n({\mathbb {R}}) \subset {\mathbb {R}}^\times \times \prod _{\tau \in {{\text {Hom}}}(F^+,{\mathbb {R}})} GL_{2n}(F \otimes _{F^+,\tau } {\mathbb {R}}), \end{array}$$
and sending j to
$$\begin{aligned} \left( -1,\left( \left( \begin{array}{cc} {-1_n} &{} {\quad 0} \\ {0} &{} {\quad 1_n} \end{array} \right) \right) _\tau \right) . \end{aligned}$$
(This map depends on identifications \(F \otimes _{F^+,\tau } {\mathbb {R}}\cong {\mathbb {C}}\), but the image of the map does not, and this image is all that will concern us.) Then \(U_{n,\infty }\) is a maximal compact subgroup of \(G_n({\mathbb {R}})\) (and even of \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})\)). If \(L \supset T_n \times {{\text {Spec}}\,}{\mathbb {R}}\) is a Levi component of a parabolic subgroup \(P \supset B_n \times {{\text {Spec}}\,}{\mathbb {R}}\) then \(U_{n,\infty } \cap L({\mathbb {R}})\) is a maximal compact subgroup of \(L({\mathbb {R}})\). The connected component of the identity of \(U_{n,\infty }\) is \(U_{n,\infty }^0=U_{n,\infty } \cap G_n({\mathbb {R}})^+\).
We will write \({\mathfrak {p}}_n\) for the set of elements of \({{\text {Lie}}\,}G_n({\mathbb {R}})\) of the form
$$\begin{aligned} \left( 0, \left( \left( \begin{array}{cc} {A_\tau } &{} {\quad B_\tau \varPsi _n} \\ {\varPsi _nB_\tau } &{} {\quad \varPsi _nA_\tau \varPsi _n} \end{array} \right) \right) _{\tau \in {{\text {Hom}}}(F^+,{\mathbb {R}})} \right) , \end{aligned}$$
where \({}^{c,t}A_\tau =A_\tau \) and \({}^{c,t}B_\tau =B_\tau \) for all \(\tau \). Then
$$\begin{aligned} {{\text {Lie}}\,}G_n({\mathbb {R}})={\mathfrak {p}}_n \oplus {{\text {Lie}}\,}(U_{n,\infty } A_n({\mathbb {R}})). \end{aligned}$$
We give the real vector space \({\mathfrak {p}}_n\) a complex structure by letting i act by
$$\begin{aligned} i_0: (A_\tau ,B_\tau )_{\tau \in {{\text {Hom}}}(F^+,{\mathbb {R}})} \longmapsto (B_\tau ,-A_\tau )_{\tau \in {{\text {Hom}}}(F^+,{\mathbb {R}})} . \end{aligned}$$
We decompose
$$\begin{aligned} {\mathfrak {p}}_n \otimes _{\mathbb {R}}{\mathbb {C}}={\mathfrak {p}}_n^+ \oplus {\mathfrak {p}}_n^- \end{aligned}$$
by setting
$$\begin{aligned} {\mathfrak {p}}_n^{\pm }=({\mathfrak {p}}_n \otimes _{\mathbb {R}}{\mathbb {C}})^{i_0\otimes 1=\pm 1 \otimes i}. \end{aligned}$$
We also set
$$\begin{aligned} {\mathfrak {q}}_n= {\mathfrak {p}}_n^- \oplus {{\text {Lie}}\,}(U_{n,\infty } A_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}. \end{aligned}$$
It is a parabolic subalgebra of \(({{\text {Lie}}\,}G_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\) with unipotent radical \({\mathfrak {p}}_n^-\) and Levi component \({{\text {Lie}}\,}(U_{n,\infty } A_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\). We will write \({\mathfrak {Q}}_n\) for the parabolic subgroup of \(G_n \times _{\mathbb {Q}}{\mathbb {C}}\) with Lie algebra \({\mathfrak {q}}_n\). Note that
$$\begin{aligned} {\mathfrak {Q}}_n({\mathbb {C}}) \cap G_n({\mathbb {R}})=U_{n,\infty }^0 A_n({\mathbb {R}})^0. \end{aligned}$$
Let \({\mathfrak {H}}_n^+\) (resp. \({\mathfrak {H}}_n^{\pm }\)) denote the set of I in \(G_n({\mathbb {R}})\) with multiplier 1 such that \(I^2=-1_{2n}\) and such that the symmetric bilinear form \(\langle I \,\,\, ,\,\,\, \rangle _n\) on \(\varLambda _n \otimes _{{\mathbb {Z}}} {\mathbb {R}}\) is positive definite (respectively positive or negative definite). Then \(G_n({\mathbb {R}})\) (resp. \(G_n({\mathbb {R}})^+\)) acts transitively on \({\mathfrak {H}}_n^\pm \) (resp. \({\mathfrak {H}}_n^+\)) by conjugation. Moreover \(J_n \in {\mathfrak {H}}_n^+\) has stabilizer \(U_{n,\infty }^0A_n({\mathbb {R}})^0\) and so we get an identification of \({\mathfrak {H}}_n^{\pm }\) (resp. \({\mathfrak {H}}_n^+\)) with \(G_n({\mathbb {R}})/U_{n,\infty }^0A_n({\mathbb {R}})^0\) (resp. \(G_n({\mathbb {R}})^+/U_{n,\infty }^0A_n({\mathbb {R}})^0\)). The natural map
$$\begin{aligned} {\mathfrak {H}}_n^\pm =G_n({\mathbb {R}})\Big /U_{n,\infty }^0A_n({\mathbb {R}})^0 \hookrightarrow G_n({\mathbb {C}})/{\mathfrak {Q}}_n({\mathbb {C}}) \end{aligned}$$
is an open embedding and gives \({\mathfrak {H}}_n^\pm \) the structure of a complex manifold. The action of \(G_n({\mathbb {R}})\) is holomorphic, and the complex structure induced on the tangent space \(T_{J_n} {\mathfrak {H}}_n^\pm \cong {\mathfrak {p}}_n\) is the complex structure described in the previous paragraph.
If \(\rho \) is a finite dimensional algebraic representation of \({\mathfrak {Q}}_n\) on a \({\mathbb {C}}\)-vector space \(W_\rho \), then there is a holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^\pm \) together with a holomorphic action of \(G_n({\mathbb {R}})\), defined as the pull-back to \({\mathfrak {H}}^\pm \) of \((G_n({\mathbb {C}}) \times W_\rho )/{\mathfrak {Q}}_n({\mathbb {C}})\), where
  • \(h \in {\mathfrak {Q}}_n({\mathbb {C}})\) sends (gw) to \((gh,h^{-1}w)\),

  • and where \(h \in G_n({\mathbb {R}})\) sends [(gw)] to [(hgw)].

If \(N_2 \ge N_1 \ge 0\) are integers we will write \(U_p(N_1,N_2)_n\) for the subgroup of \(G_n({\mathbb {Z}}_p)\) consisting of elements whose reduction modulo \(p^{N_2}\) preserves
$$\varLambda _{n,(n)} \otimes _{\mathbb {Z}}\left( {\mathbb {Z}}/p^{N_2}{\mathbb {Z}}\right) \subset \varLambda _n \otimes _{\mathbb {Z}}\left( {\mathbb {Z}}/p^{N_2}{\mathbb {Z}}\right) $$
and acts trivially on \(\varLambda _n/(\varLambda _{n,(n)} + p^{N_1} \varLambda _n)\). If \(N_2 \ge N_1 \ge N_1' \ge 0\) then \(U_p(N_1,N_2)_n\) is a normal subgroup of \(U_p(N_1',N_2)_n\) and
$$U_p(N_1',N_2)_n/U_p(N_1,N_2)_n \cong \ker \left( GL_n\left( {\mathcal {O}}_F/p^{N_1}\right) \rightarrow GL_n \left( {\mathcal {O}}_F/p^{N_1'}\right) \right) .$$
We will also set
$$\begin{array}{rcl} U_p(N_1,N_2)^{(m)}_n&{}=&{}U_p(N_1,N_2)_n \ltimes {{\text {Hom}}}_{{\mathcal {O}}_{F,p}}\left( {\mathcal {O}}_{F,p}^m, \varLambda _{n,(n)} +p^{N_1}\varLambda _n\right) \\ &{}\subset &{}G_n^{(m)}({\mathbb {Z}}_p) \end{array}$$
and set \(\widetilde{{U}}_p(N_1,N_2)^{(m)}_n\) to be the pre-image of \(U_p(N_1,N_2)^{(m)}_n\) in \(\widetilde{{G}}_n^{(m)}({\mathbb {Z}}_p)\). Pictorially we can think of \(U_p(N_1,N_2)_n\) as
$$\left( \begin{array}{c|c} \mu 1_n \bmod p^{N_1} \,\,&{}\,\, * \\ \hline 0 \bmod p^{N_2} \,\,&{}\,\, 1_n \bmod p^{N_1} \end{array} \right) $$
and of \(U_p(N_1,N_2)_n^{(m)}\) as
If \(U^p\) is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) (resp. of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\), resp. of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\)) we will set \(U^p(N_1,N_2)\) to be \(U^p \times U_p(N_1,N_2)_n\) (resp. \(U^p \times U_p(N_1,N_2)_n^{(m)}\), resp. \(U^p \times \widetilde{{U}}_p(N_1,N_2)_n^{(m)}\)), a compact open subgroup of \(G_n({\mathbb {A}}^{\infty })\) (resp. \(G_n^{(m)}({\mathbb {A}}^{\infty })\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })\)).

2.2 Maximal parabolic subgroups

We will write \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) for the subgroup of \(G_n\) (resp. \(G_n^{(m)}\), resp. \(\widetilde{{G}}_n^{(m)}\)) consisting of elements which (after projection to \(G_n\)) take \(\varLambda _{n,(i)}\) to itself. We will also write \(N_{n,(i)}^+\) (resp. \(N^{(m),+}_{n,(i)}\), resp. \(\widetilde{{N}}^{(m),+}_{n,(i)}\)) for the subgroups of \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) consisting of elements which act trivially on \(\varLambda _{n,(i)}\) and \(\varLambda _{n,(i)}^\perp /\varLambda _{n,(i)}\) and \(\varLambda _n/\varLambda _{n,(i)}^\perp \). Over \({\mathbb {Q}}\) the groups \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) are maximal parabolic subgroups of \(G_n\) (resp. \(G_n^{(m)}\), resp. \(\widetilde{{G}}_n^{(m)}\)) containing the pre-image of \(B_n\). The groups \(N_{n,(i)}^+\) (resp. \(N^{(m),+}_{n,(i)}\), resp. \(\widetilde{{N}}^{(m),+}_{n,(i)}\)) are their unipotent radicals.

In some instances it will be useful to replace these groups by their ‘hermitian part’. We will write \(P_{n,(i)}\) for the normal subgroup of \(P_{n,(i)}^+\) consisting of elements which act trivially on \(\varLambda _n/\varLambda _{n,(i)}^\perp \). We will also write \(P^{(m)}_{n,(i)}\) for the normal subgroup
$$\begin{aligned} P_{n,(i)} \ltimes {{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^m,\varLambda _{n,(i)}^\perp \right) \end{aligned}$$
of \(P^{(m),+}_{n,(i)}\), and \(\widetilde{{P}}^{(m)}_{n,(i)}\) for the pre-image of \(P^{(m)}_{n,(i)}\) in \(\widetilde{{P}}^{(m),+}_{n,(i)}\). We will let
$$\begin{aligned} N_{n,(i)}=N_{n,(i)}^+ \end{aligned}$$
and
$$\begin{aligned} N^{(m)}_{n,(i)}=N^{(m),+}_{n,(i)} \cap P^{(m)}_{n,(i)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{N}}^{(m)}_{n,(i)}=\widetilde{{N}}^{(m),+}_{n,(i)} \cap \widetilde{{P}}^{(m)}_{n,(i)}. \end{aligned}$$
Over \({\mathbb {Q}}\) these are the unipotent radicals of \(P_{n,(i)}\) (resp. \(P^{(m)}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m)}_{n,(i)}\)).
Pictorially one can think of \(P_{n,(i)}^+\) and \(P_{n,(i)}\) as matrices of the following shapes
and
respectively. If we picture an element of \(G^{(m)}_n\) as a pair of matrices
(the first \(2n \times 2n\) and the second \(2n \times m\)) then we can picture \(P_{n,(i)}^{(m),+}\) and \(P_{n,(i)}^{(m)}\) as consisting of matrices of the shape
and
respectively.
We have an isomorphism
$$\begin{aligned} P_{n,(i)} \cong \widetilde{{G}}_{n-i}^{(i)}. \end{aligned}$$
To describe it let \(\varLambda _{n,(i)}'\) denote the subspace of \(\varLambda _n\) consisting of vectors with their first \(2n-i\) entries 0, so that
$$\begin{aligned} \varLambda _{n,(i)}' \cong {\mathcal {O}}_F^i \end{aligned}$$
and
$$\varLambda _{n-i} \cong \varLambda _{n,(i)}^\perp \cap \left( \varLambda _{n,(i)}'\right) ^\perp \mathop {\longrightarrow }\limits ^{\sim }\varLambda _{n,(i)}^\perp \Big /\varLambda _{n,(i)}.$$
We define
$$\begin{aligned} G_{n-i} \hookrightarrow P_{n,(i)} \end{aligned}$$
by letting \(g \in G_{n-i}\) act as \(\nu (g)\) on \(\varLambda _{n,(i)}\), as g on \(\varLambda _{n-i} \cong \varLambda _{n,(i)}^\perp \cap (\varLambda _{n,(i)}')^\perp \) and as 1 on \(\varLambda _{n,(i)}'\), i.e.
$$g \longmapsto \left( \begin{array}{ccc} \nu (g) 1_i &{}\quad 0&{}\quad 0 \\ 0 &{}\quad g &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 1_i \end{array} \right) \in P_{n,(i)}.$$
We define
$$\begin{aligned} N_{n,(i)} \longrightarrow {{\text {Hom}}}_{n-i}^{(i)} \end{aligned}$$
by sending h to the map
$$\begin{aligned} {\mathcal {O}}_F^i \cong \varLambda _{n,(i)}' \mathop {\longrightarrow }\limits ^{h-1_{2n}} \varLambda _{n,(i)}^\perp \twoheadrightarrow \varLambda _{n-i}. \end{aligned}$$
We also define
$$\begin{aligned} Z(N_{n,(i)}) \mathop {\longrightarrow }\limits ^{\sim }{{\text {Herm}}}_{\varLambda _{n,(i)}'} \cong {{\text {Herm}}}^{(i)} \end{aligned}$$
by sending z to the pairing
$$\begin{aligned} (x,y)_z=\left\langle (z-1_{2n})x,y\right\rangle _n \end{aligned}$$
on \(\varLambda _{n,(i)}'\). In the other direction \((f,z) \in N_{n-i}^{(i)}\) is mapped to
$$\left( \begin{array}{ccc} 1_i &{}\quad \varPsi _i\,{}^{c,t}f\,J_{n-i} &{}\quad \varPsi _i {}^t\left( z-\frac{1}{2} {}^tf\,J_{n-i}\,{}^cf\right) \\ 0 &{}\quad 1_{2(n-i)} &{}\quad f \\ 0 &{}\quad 0 &{}\quad 1_i \end{array} \right) \in N_{n,(i)},$$
where we think of \(f \in M_{2(n-i) \times i}(F)\) with first \(n-i\) rows in \(({\mathcal {D}}_F^{-1})^i\) and second \((n-i)\) rows in \({\mathcal {O}}_F^i\), and we think of \(z \in M_{i \times i}(F)^{t=c}\).
We also have isomorphisms
$$\begin{aligned} P_{n,(i)}^{(m)} \cong \widetilde{{G}}_{n-i}^{(i+m)}\Big /{{\text {Herm}}}^{(m)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{P}}_{n,(i)}^{(m)} \cong \widetilde{{G}}_{n-i}^{(i+m)}. \end{aligned}$$
We will describe the second of these isomorphisms. Suppose \(f \in {{\text {Hom}}}_{n-i}^{(i)}\) and \(g \in {{\text {Hom}}}_{n-i}^{(m)}\). Also suppose that \(z \in \frac{1}{2}{{\text {Herm}}}^{(i)}\) and \(w \in \frac{1}{2}{{\text {Herm}}}^{(m)}\) and
$$\begin{aligned} u \in \frac{1}{2}{{\text {Hom}}}\left( {\mathcal {O}}_F^i \otimes _{{\mathcal {O}}_F,c} {\mathcal {O}}_F^m,{\mathbb {Z}}\right) , \end{aligned}$$
so that
$$\begin{aligned} \left( (f,g),(z,u,w) \right) \in N_{n-i}^{(i+m)}. \end{aligned}$$
Let h(fz) denote the element of \(P_{n,(i)}\) corresponding to \((f,z) \in N_{n,(i)}\). Think of g as a map
$$\begin{aligned} g: {\mathcal {O}}_F^m \longrightarrow \varLambda _{n-i} \cong \varLambda _{n,(i)}^\perp \cap \left( \varLambda _{n,(i)}'\right) ^\perp \subset \varLambda _n. \end{aligned}$$
Define \(j(f,g,u) \in {{\text {Hom}}}({\mathcal {O}}_F^m, \varLambda _{n,(i)})\) by
$$\begin{aligned} \langle y, j(f,g,u)(x)\rangle _n=1/2 \langle f(y),g(x)\rangle _{n-i} -u(y \otimes x) \end{aligned}$$
for all \(x \in {\mathcal {O}}_F^m\) and \(y \in \varLambda _{n,(i)}' \cong {\mathcal {O}}_F^i\). Then
$$\begin{aligned} ((f,g),(z,u,w)) \longmapsto h(f,z) (g+j(f,g,u),w) \in N_{n,(i)} \ltimes \widetilde{{N}}_n^{(m)}. \end{aligned}$$
Note that
$$\begin{aligned} Z\left( \widetilde{{N}}^{(m)}_{n,(i)}\right) \cong {{\text {Herm}}}^{(i+m)} \end{aligned}$$
and that
$$\begin{aligned} Z\left( N^{(m)}_{n,(i)}\right) \cong {{\text {Herm}}}^{(i+m)}/{{\text {Herm}}}^{(m)}. \end{aligned}$$
Write \(L_{n,(i),{{\text {lin}}}}\) for the subgroup of \(P_{n,(i)}^+\) consisting of elements with \(\nu =1\) which preserve \(\varLambda _{n,(i)}'\subset \varLambda _n\) and act trivially on \(\varLambda _{n,(i)}^\perp /\varLambda _{n,(i)}\). We set \(N(L_{n,(i),{{\text {lin}}}}^{(m)})\) to be the additive group scheme over \({\mathbb {Z}}\) associated to
$$\begin{aligned} {{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^m, \varLambda _{n,(i)}'\right) , \end{aligned}$$
and write \(L_{n,(i),{{\text {lin}}}}^{(m)}\) for
$$\begin{aligned} L_{n,(i),{{\text {lin}}}} \ltimes N\left( L_{n,(i),{{\text {lin}}}}^{(m)}\right) \subset P_{n,(i)}^{(m),+} \end{aligned}$$
and \(\widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)}\) for
$$\begin{aligned} L_{n,(i),{{\text {lin}}}} \ltimes N\left( L_{n,(i),{{\text {lin}}}}^{(m)}\right) \subset \widetilde{{P}}_{n,(i)}^{(m),+}. \end{aligned}$$
Note that
$$\begin{aligned} P_{n,(i)}^+=L_{n,(i),{{\text {lin}}}} \ltimes P_{n,(i)} \end{aligned}$$
and
$$\begin{aligned} P_{n,(i)}^{(m),+}=L_{n,(i),{{\text {lin}}}}^{(m)} \ltimes P_{n,(i)}^{(m)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{P}}_{n,(i)}^{(m),+}=\widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)} \ltimes \widetilde{{P}}_{n,(i)}^{(m)}. \end{aligned}$$
Also note that
$$\begin{aligned} L_{n,(i),{{\text {lin}}}} \cong {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_i \end{aligned}$$
via its action on \(\varLambda _{n,(i)}'\cong {\mathcal {O}}_F^i\), and that
$$\begin{aligned} \widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)} \mathop {\longrightarrow }\limits ^{\sim }L_{n,(i),{{\text {lin}}}}^{(m)} \cong \left( {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_i\right) \ltimes {{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^m,{\mathcal {O}}_F^i\right) . \end{aligned}$$
Pictorially we can think of \(L_{n,(i),{{\text {lin}}}}\) as consisting of matrices of the form
$$\left( \begin{array}{c|c|c} \varPsi _i\,{}^{c,t}h^{-1}\,\varPsi _n \,\,&{}\,\, 0 \,\,&{}\,\, 0 \\ \hline 0 \,\,&{}\,\, 1_{2(n-i)} \,\,&{}\,\, 0 \\ \hline 0 \,\,&{}\,\, 0 \,\,&{}\,\, h \end{array} \right) $$
and \(L_{n,(i),{{\text {lin}}}}^{(m)}\) as consisting of matrices of the form
We let \(L_{n,(i),{{\text {herm}}}}\) denote the subgroup of \(P_{n,(i)}\) consisting of elements which preserve \(\varLambda _{n,(i)}'\). Thus
$$\begin{aligned} L_{n,(i),{{\text {herm}}}} \cong G_{n-i}. \end{aligned}$$
In particular
$$\begin{aligned} \nu : L_{n,(n),{{\text {herm}}}} \mathop {\longrightarrow }\limits ^{\sim }{\mathbb {G}}_m. \end{aligned}$$
Pictorially we can think of \(L_{n,(i),{{\text {herm}}}}\) as consisting of matrices of the form
$$\left( \begin{array}{c|c|c} \nu (g) 1_i \,\,&{}\,\, 0\,\,&{}\,\,0\\ \hline 0 \,\,&{}\,\, g \,\,&{}\,\, 0 \\ \hline 0 \,\,&{}\,\, 0 \,\,&{}\,\, 1_i \end{array} \right) . $$
Over \({\mathbb {Q}}\) it is a Levi component for \(P_{n,(i)}\) and \(P^{(m)}_{n,(i)}\) and \(\widetilde{{P}}^{(m)}_{n,(i)}\), so in particular
$$\begin{aligned} P_{n,(i)}=L_{n,(i),{{\text {herm}}}} \ltimes N_{n,(i)} \end{aligned}$$
and
$$\begin{aligned} P_{n,(i)}^{(m)}=L_{n,(i),{{\text {herm}}}} \ltimes N_{n,(i)}^{(m)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{P}}_{n,(i)}^{(m)}=L_{n,(i),{{\text {herm}}}} \ltimes \widetilde{{N}}_{n,(i)}^{(m)}. \end{aligned}$$
We also set
$$\begin{aligned} L_{n,(i)}=L_{n,(i),{{\text {herm}}}} \times L_{n,(i),{{\text {lin}}}} \end{aligned}$$
and
$$\begin{aligned} L_{n,(i)}^{(m)}=L_{n,(i),{{\text {herm}}}} \times L_{n,(i),{{\text {lin}}}}^{(m)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{L}}_{n,(i)}^{(m)}=L_{n,(i),{{\text {herm}}}} \times \widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)}. \end{aligned}$$
Over \({\mathbb {Q}}\) we see that \(L_{n,(i)}\) is a Levi component for each of \(P_{n,(i)}^+\) and \(P^{(m),+}_{n,(i)}\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}\). Moreover
$$\begin{aligned} P_{n,(i)}^+=L_{n,(i)} \ltimes N_{n,(i)} \end{aligned}$$
and
$$\begin{aligned} P_{n,(i)}^{(m),+}=L_{n,(i)}^{(m)} \ltimes N_{n,(i)}^{(m)} \end{aligned}$$
and
$$\begin{aligned} \widetilde{{P}}_{n,(i)}^{(m),+}=\widetilde{{L}}_{n,(i)}^{(m)} \ltimes \widetilde{{N}}_{n,(i)}^{(m)}. \end{aligned}$$
We will occasionally write \(P_{n,(i)}^{(m),-}\) (resp. \(L_{n,(i),{{\text {herm}}}}^-\)) for the kernel of the map \(P_{n,(i)}^{(m)}\rightarrow C_{n-i}\) (resp. \(L_{n,(i),{{\text {herm}}}} \rightarrow C_{n-i}\)).

We will write \(R_{n,(n),(i)}\) for the subgroup of \(L_{n,(n)}\) mapping \(\varLambda _{n,(i)}\) to itself. We will write \(N(R_{n,(n),(i)})\) for the subgroup of \(R_{n,(n),(i)}\) which acts trivially on \(\varLambda _{n,(i)}\) and \(\varLambda ^\perp _{n,(i)}/\varLambda _{n,(i)}\) and \(\varLambda _n/\varLambda ^\perp _{n,(i)}\).

We will also write \(R_{n,(n)}^{(m)}\) for the semi-direct product
$$\begin{aligned} L_{n,(n)} \ltimes {{\text {Hom}}}_{{\mathcal {O}}_F}\left( {\mathcal {O}}_F^{m},\varLambda _{n,(n)}\right) . \end{aligned}$$
If \(m'\le m\) we will fix \({\mathbb {Z}}^m \twoheadrightarrow {\mathbb {Z}}^{m-m'}\) to be projection onto the last \(m-m'\) coordinates and define \(Q_{m,(m')}\) for the subgroup of \(GL_m\) consisting of elements preserving the kernel of this map. We also define \(Q_{m,(m')}'\) to be the subgroup of \(Q_{m,(m')}\) consisting of elements which induce \(1_{{\mathbb {Z}}^{m-m'}}\) on \({\mathbb {Z}}^{m-m'}\). Thus there is an exact sequence
$$(0) \longrightarrow {{\text {Hom}}}\left( {\mathbb {Z}}^{m-m'},{\mathbb {Z}}^{m'}\right) \longrightarrow Q_{m,(m')}' \longrightarrow GL_{m'} \longrightarrow \{1\}.$$
Moreover
$$\widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)} \cong L_{n,(i),{{\text {lin}}}}^{(m)} \cong {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}Q'_{m+i,(i)}.$$
We will also write \(A_{n,(i),{{\text {lin}}}}\) (resp. \(A_{n,(i),{{\text {herm}}}}\)) for the image of the map from \({\mathbb {G}}_m\) to \(L_{n,(i),{{\text {lin}}}}\) (resp. \(L_{n,(i),{{\text {herm}}}}\)) sending t to \(t 1_i\) (resp. \((t^2,t 1_{2(n-i)})\)). Moreover write \(A_{n,(i)}\) for \(A_{n,(i),{{\text {lin}}}} \times A_{n,(i),{{\text {herm}}}}\). Over \({\mathbb {Q}}\) the group \(A_{n,(i)}\) (resp. \(A_{n,(i),{{\text {lin}}}}\), resp. \(A_{n,(i),{{\text {herm}}}}\)) is the maximal split torus in the centre of \(L_{n,(i)}\) (resp. \(L_{n,(i),{{\text {lin}}}}\), resp. \(L_{n,(i),{{\text {herm}}}}\)).

Again suppose that \(\varOmega \) is an algebraically closed field of characteristic 0. Let \(\varPhi _{(n)} \subset \varPhi _n\) denote the set of roots of \(T_n\) on \({{\text {Lie}}\,}L_{n,(n)}\), and set \(\varPhi _{(n)}^+=\varPhi _n^+ \cap \varPhi _{(n)}\) and \(\varDelta _{(n)}=\varDelta _n \cap \varPhi _{(n)}\). We will write \(\varrho _{n,(n)}\) for half the sum of the elements of \(\varPhi _{(n)}^+\). If \(R \subset {\mathbb {R}}\) then \(X^*(T_{n,/\varOmega })^+_{(n),R}\) will denote the subset of \(X^*(T_{n,/\varOmega })_R\) consisting of elements which pair non-negatively with the coroot \(\check{\alpha } \in X_*(T_{n,/\varOmega })\) corresponding to each \(\alpha \in \varDelta _{(n)}\). We write \(X^*(T_{n,/\varOmega })_{(n)}^+\) for \(X^*(T_{n,/\varOmega })_{(n),{\mathbb {Z}}}^+\). If \(\lambda \in X^*(T_{n,/\varOmega })_{(n)}^+\) we will let \(\rho _{(n),\lambda }\) denote the irreducible representation of \(L_{n,(n)}\) with highest weight \(\lambda \). When \(\rho _{(n),\lambda }\) is used as a subscript we will sometimes replace it by just \((n),\lambda \).

Note that \({{\text {Lie}}\,}P_{n,(n)}({\mathbb {C}})\) and \({\mathfrak {q}}_n\) are conjugate under \(G_n({\mathbb {C}})\) and hence we obtain an identification (‘Cayley transform’) of \(({{\text {Lie}}\,}U_{n,\infty } A_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\) and \({{\text {Lie}}\,}L_{n,(n)}({\mathbb {C}})\), which is well defined up to conjugation by \(L_{n,(n)}({\mathbb {C}})\). Similarly \({\mathfrak {Q}}_n\) and \(P_{n,(n)}({\mathbb {C}})\) are conjugate in \(G_n \times _{\mathbb {Q}}{\mathbb {C}}\). Thus \(L_{n,(n)}({\mathbb {C}})\) can be identified with \({\mathfrak {Q}}_n\) modulo its unipotent radical, canonically up to \(L_{n,(n)}({\mathbb {C}})\)-conjugation. Thus if \(\rho \) is a finite dimensional algebraic representation of \(L_{n,(n)}\) over \({\mathbb {C}}\), we can associate to it a representation of \({\mathfrak {Q}}_n\) and of \({\mathfrak {q}}_n\), and hence a holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^\pm \) with \(G_n({\mathbb {R}})\)-action.

The isomorphism \(L_{n,(n)} \cong GL_1 \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_n\) gives rise to a natural identification
$$\begin{aligned} L_{n,(n)} \times {{\text {Spec}}\,}\varOmega \cong GL_1 \times GL_n^{{{\text {Hom}}}(F,\varOmega )}, \end{aligned}$$
and hence to identifications
$$\begin{aligned} T_n \times {{\text {Spec}}\,}\varOmega \cong GL_1 \times \left( GL_1^{n}\right) ^{{{\text {Hom}}}(F,\varOmega )} \end{aligned}$$
and
$$\begin{aligned} X^*(T_{n,/\varOmega }) \cong {\mathbb {Z}}\oplus ({\mathbb {Z}}^{n})^{{{\text {Hom}}}(F,\varOmega )}. \end{aligned}$$
Under this identification \(X^*(T_{n,/\varOmega })_{(n)}^+\) is identified to the set of
$$\begin{aligned} (b_0,(b_{\tau ,i})) \in {\mathbb {Z}}\oplus ({\mathbb {Z}}^{n})^{{{\text {Hom}}}(F,\varOmega )} \end{aligned}$$
with
$$\begin{aligned} b_{\tau ,1} \ge b_{\tau ,2} \ge \cdots \ge b_{\tau ,n} \end{aligned}$$
for all \(\tau \).
To compare this parametrization of \(X^*(T_{n,/\varOmega })\) with the one introduced in Sect. 1.1 note that the map
$$\begin{aligned} GL_1 \times GL_n^{{{\text {Hom}}}(F,\varOmega )} \hookrightarrow \left\{ (\mu ,g_\tau ) \in {\mathbb {G}}_m \times GL_{2n}^{{{\text {Hom}}}(F,\varOmega )}: g_{\tau c} =\mu J_n {}^tg_\tau ^{-1} J_n \,\, \forall \tau \right\} \end{aligned}$$
coming from \(L_{n,(n)} \hookrightarrow G_n\) sends
$$\begin{aligned} (\mu ,(g_\tau )_{\tau \in {{\text {Hom}}}(F,\varOmega )}) \longmapsto \left( \mu , \left( \left( \begin{array}{cc} {\mu \varPsi _n {}^t g_{\tau c}^{-1} \varPsi _n} &{} {\quad 0} \\ {0} &{} {\quad g_\tau } \end{array} \right) \right) _{\tau \in {{\text {Hom}}}(F,\varOmega )} \right) . \end{aligned}$$
Thus the map
$$\begin{aligned} {\mathbb {Z}}\oplus ({\mathbb {Z}}^{2n})^{{{\text {Hom}}}(F,\varOmega )} \twoheadrightarrow X^*(T_{n,/\varOmega }) \cong {\mathbb {Z}}\oplus ({\mathbb {Z}}^{n})^{{{\text {Hom}}}(F,\varOmega )} \end{aligned}$$
sends
$$\begin{aligned} \left( a_0, (a_{\tau ,i})_{\tau \in {{\text {Hom}}}(F,\varOmega ); i=1,\ldots ,2n}\right) \longmapsto \left( a_0+\sum _\tau \sum _{j=1}^n a_{\tau ,j}, \,\, \left( a_{\tau ,n+i}-a_{\tau c, n+1-i}\right) _{\tau ,i}\right) . \end{aligned}$$
A section is provided by the map
$$\begin{aligned} (b_0,(b_{\tau ,i})) \longmapsto \left( b_0,\left( 0,\ldots ,0,b_{\tau ,1},\ldots ,b_{\tau ,n}\right) _\tau \right) . \end{aligned}$$
In particular we see that \(X^*(T_{n,/\varOmega })^+\subset X^*(T_{n,/\varOmega })_{(n)}^+\) is identified with the set of
$$\begin{aligned} (b_0,(b_{\tau ,i})) \in {\mathbb {Z}}\oplus ({\mathbb {Z}}^{n})^{{{\text {Hom}}}(F,\varOmega )} \end{aligned}$$
with
$$\begin{aligned} b_{\tau ,1} \ge b_{\tau ,2} \ge \cdots \ge b_{\tau ,n} \end{aligned}$$
and
$$\begin{aligned} b_{\tau ,1}+b_{\tau c,1} \le 0 \end{aligned}$$
for all \(\tau \).
Note that
$$\begin{aligned} 2(\varrho _n-\varrho _{n,(n)})= \left( n^2[F^+: {\mathbb {Q}}],(-n)_{\tau ,i}\right) . \end{aligned}$$
We write \({{\text {Std}}}\) for the representation of \(L_{n,(n)}\) on \(\varLambda _n/\varLambda _{n,(n)}\) over \({\mathbb {Z}}\), and if \(\tau : F \hookrightarrow \overline{{{\mathbb {Q}}}}\) we write \({{\text {Std}}}_\tau \) for the representation of \(L_{n,(n)}\) on \((\varLambda _n/\varLambda _{n,(n)}) \otimes _{{\mathcal {O}}_F,\tau }{\mathcal {O}}_{\overline{{{\mathbb {Q}}}}}\). If \(\varOmega \) is an algebraically closed field of characteristic 0 and if \(\tau : F \hookrightarrow \varOmega \) we will sometimes write \({{\text {Std}}}_\tau \) for the representation of \(L_{n,(n)}\) on \((\varLambda _n/\varLambda _{n,(n)}) \otimes _{{\mathcal {O}}_F,\tau }\varOmega \). We hope that context will make clear the distinction between these two slightly different meaning of \({{\text {Std}}}_\tau \). We also let \({{\text {KS}}}\) denote the representation
$$\begin{aligned} S({{\text {Std}}}^\vee )\otimes \nu \end{aligned}$$
of \(L_{n,(n)}\) over \({\mathbb {Z}}\). (See Sect. 1.1.) Note that over \(\overline{{{\mathbb {Q}}}}\) the representation \({{\text {Std}}}_\tau ^\vee \) is irreducible and in our normalizations has highest weight \((0,b_{\tau '})\) where
$$\begin{aligned} b_\tau =(0,\ldots ,0,-1) \end{aligned}$$
but \(b_{\tau '}=0\) for \(\tau ' \ne \tau \). Similarly the representation \(\wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee \) is irreducible with highest weight
$$\begin{aligned} (0,(-1,\ldots ,-1)_\tau ). \end{aligned}$$
Finally \({{\text {KS}}}\) is the direct sum of the \([F^+: {\mathbb {Q}}]\) irreducible representations indexed by \(\tau \in {{\text {Hom}}}(F^+, \overline{{{\mathbb {Q}}}})\) with highest weights \((1,b_{\tau '})\), where
$$\begin{aligned} b_{\tau '}=(0,\ldots ,0,-1) \end{aligned}$$
if \(\tau '\) extends \(\tau \), and \(b_{\tau '}=0\) otherwise.

We will let \(\varsigma _p \in L_{n,(n),{{\text {herm}}}}({\mathbb {Q}}_p)\cong {\mathbb {Q}}_p^\times \) denote the unique element with multiplier \(p^{-1}\).

Set
$$\begin{aligned} U_{p}(N)_{n,(i)}=\ker \left( L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_p) \rightarrow L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}/p^N{\mathbb {Z}})\right) \end{aligned}$$
and
$$U_{p}(N)_{n,(i)}^{(m)}=\ker \left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_p) \rightarrow L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}/p^N{\mathbb {Z}})\right) .$$
Also set
$$\begin{aligned} U_p(N_1,N_2)_{n,(i)}^{(m)}=U_p(N_1,N_2)_{n-i} \times U_p(N_1)_{n,(i)}^{(m)} \subset L_{n,(i)}^{(m)}({\mathbb {Z}}_p) \end{aligned}$$
and
$$\widetilde{{U}}_p(N_1,N_2)_{n,(i)}^{(m),+}= U_p(N_1)^{(m)}_{n,(i)} \ltimes \widetilde{{U}}_p(N_1,N_2)_{n-i}^{(m+i)} \subset \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {Z}}_p)$$
and
$$\begin{aligned} U_p(N_1,N_2)_{n,(i)}^{(m),+}=\widetilde{{U}}_p(N_1,N_2)_{n,(i)}^{(m),+} \Big /{{\text {Herm}}}_{{\mathcal {O}}_{F,p}^m} \subset P^{(m),+}_{n,(i)}({\mathbb {Z}}_p). \end{aligned}$$
Let \(U^p\) be an open compact subgroup of \(L_{n,(i)}({\mathbb {A}}^{\infty ,p})\) (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\), resp. \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty ,p})\), resp. \((P_{n,(i)}^{(m),+}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^{\infty ,p})\), resp. \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\), resp. \(\widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\)). Then set
$$\begin{aligned} U^p(N_1,N_2)=U^p \times U_p(N_1,N_2)_{n,(i)} \subset L_{n,(i)}({\mathbb {A}}^\infty ) \end{aligned}$$
(resp.
$$\begin{aligned} U^p(N)= U^p \times U_{p}(N)_{n,(i)}, \end{aligned}$$
resp.
$$\begin{aligned} U^p(N_1,N_2)=U^p \times U_p(N_1,N_2)_{n,(i)}^{(m)} \subset L_{n,(i)}^{(m)}({\mathbb {A}}^\infty ), \end{aligned}$$
resp.
$$U^p(N_1,N_2)=U^p \times \left( U_p(N_1,N_2)_{n,(i)}^{(m),+}\Big /Z\left( N_{n,(i)}^{(m)}\right) ({\mathbb {Z}}_p)\right) \subset P_{n,(i)}^{(m),+}\Big /Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {A}}^\infty ),$$
resp.
$$\begin{aligned} U^p(N_1,N_2) =U^p \times U_p(N_1,N_2)_{n,(i)}^{(m),+} \subset P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty ), \end{aligned}$$
resp.
$$\begin{aligned} \left. U^p(N_1,N_2) =U^p \times \widetilde{{U}}_p(N_1,N_2)_{n,(i)}^{(m),+} \subset \widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\right) . \end{aligned}$$
In the case \(i=n\) these groups do not depend on \(N_2\), so we will write simply \(U^p(N_1)\).
For the study of the ordinary locus we will need a variant of \(G_n({\mathbb {A}}^\infty )\) and \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\). More specifically define a semigroup
$$\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}=\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p}) \times \left( \varsigma _p^{{\mathbb {Z}}_{\ge 0}}\widetilde{{P}}^{(m),+}_{n,(n)}({\mathbb {Z}}_p)\right) .$$
Its maximal sub-semigroup that is also a group is
$$\begin{aligned} \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }=\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p}) \times \widetilde{{P}}^{(m),+}_{n,(n)}({\mathbb {Z}}_p). \end{aligned}$$
If H is an algebraic subgroup of \(\widetilde{{G}}_n^{(m)}\) (over \({{\text {Spec}}\,}{\mathbb {Q}}\)) we set
$$\begin{aligned} H({\mathbb {A}}^\infty )^{{\text {ord}}}=H({\mathbb {A}}^{\infty }) \cap \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}. \end{aligned}$$
Its maximal sub-semigroup that is also a group is
$$\begin{aligned} H({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }=H({\mathbb {A}}^{\infty }) \cap \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }. \end{aligned}$$
Thus
$$\begin{aligned} G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }=G_n({\mathbb {A}}^{\infty ,p}) \times P_{n,(n)}^+({\mathbb {Z}}_p) \end{aligned}$$
and
$$\begin{aligned} G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }=G^{(m)}_n({\mathbb {A}}^{\infty ,p}) \times P^{(m),+}_{n,(n)}({\mathbb {Z}}_p). \end{aligned}$$
If \(U^p\) is an open compact subgroup of \(H({\mathbb {A}}^{\infty ,p})\), we set
$$\begin{aligned} U^p(N)=H({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } \cap \left( U^p \times \widetilde{{U}}_p(N,N')_{n,(n)}^{(m),+}\right) \end{aligned}$$
for any \(N' \ge N\). The group does not depend on the choice of \(N'\).

2.3 Base change

We will write \(B_{GL_m}\) for the subgroup of upper triangular elements of \(GL_m\) and \(T_{GL_m}\) for the subgroup of diagonal elements of \(B_{GL_m}\).

We will also let \(G_n^1\) denote the group scheme over \({\mathcal {O}}_{F^+}\) defined by
$$G_n^1(R)=\left\{ g \in {{\text {Aut}}}\left( \left( \varLambda _n \otimes _{{\mathcal {O}}_{F^+}} R\right) \Big / \left( {\mathcal {O}}_F \otimes _{{\mathcal {O}}_{F^+}} R\right) \right) : {}^tgJ_n {}^cg=J_n\right\} .$$
Thus
$$\begin{aligned} \ker \nu \cong {{\text {RS}}}^{{\mathcal {O}}_{F^+}}_{\mathbb {Z}}G_n^1. \end{aligned}$$
We will write \(B_n^1\) for the subgroup of \(G_n^1\) consisting of upper triangular matrices and \(T_n^1\) for the subgroup of \(B_n^1\) consisting of diagonal matrices. There is a natural projection \(B_n^1 \twoheadrightarrow T_n^1\) obtained by setting the off diagonal entries of an element of \(B_n^1\) to 0.
Suppose that q is a rational prime. Let \(u_1,\ldots ,u_r\) denote the primes of \(F^+\) above q which split \(u_i=w_i{}^cw_i\) in F and let \(v_1,\ldots ,v_s\) denote the primes of \(F^+\) above q which do not split in F. Then
$$\begin{aligned} G_n({\mathbb {Q}}_q) \cong \prod _{i=1}^r GL_{2n}(F_{w_i}) \times H \end{aligned}$$
where
$$\begin{aligned} H= \left\{ (\mu ,g_i) \in {\mathbb {Q}}_q^\times \times \prod _{i=1}^s GL_{2n}(F_{v_i}): {}^tg_i J_n{}^cg_i=\mu J_n \,\, \forall i\right\} \supset \prod _{i=1}^s G_n^1\left( F^+_{v_i}\right) . \end{aligned}$$
If \(\varPi \) is an irreducible smooth representation of \(G_n({\mathbb {Q}}_q)\) then
$$\begin{aligned} \varPi =\left( \bigotimes _{i=1}^r \varPi _{w_i} \right) \otimes \varPi _H. \end{aligned}$$
We define \({{\text {BC}}}(\varPi )_{w_i}=\varPi _{w_i}\) and \({{\text {BC}}}(\varPi )_{c w_i}= \varPi _{w_i}^{\vee ,c}\). Note that this does not depend on the choice of primes \(w_i|u_i\). We will say that \(\varPi \) is unramified at \(w_i\) if \({{\text {BC}}}(\varPi )_{w_i}^{GL_{2n}({\mathcal {O}}_{F,w_i})} \ne (0)\). We will say that \(\varPi \) is unramified at \(v_i\) if \(v_i\) is unramified in F and
$$\begin{aligned} \varPi ^{G_n^1({\mathcal {O}}_{F^+,v_i})} \ne (0). \end{aligned}$$
We will say that \(\varPi \) is unramified at q if \(\varPi \) is unramified at all primes above q and either q splits in \(F_0\) or q is unramified in F.
Suppose that \(\varPi \) is unramified at \(v_i\). Then there is a character \(\chi \) of the quotient \(T_n^1(F^+_{v_i})/T_n^1({\mathcal {O}}_{F^+,v_i})\) such that \(\varPi |_{G_n^1(F^+_{v_i})}\) and \({{\text {n-Ind}}}_{B_n^1(F^+_{v_i})}^{G_n^1(F^+_{v_i})} \chi \) share an irreducible subquotient with a \(G_n^1({\mathcal {O}}_{F^+,v_i})\)-fixed vector. Moreover this character \(\chi \) is unique modulo the action of the normalizer \(N_{G_n^1(F_{v_i}^+)}(T_n^1(F_{v_i}^+))/T_n^1(F_{v_i}^+)\). (If \(\pi \) and \(\pi '\) are two irreducible subquotients of \(\varPi _H|_{G_n^1(F_{v_i}^+)}\) then we must have \(\pi ' \cong \pi ^{\varsigma _{v_i}^{-1}}\) where
$$\begin{aligned} \varsigma _{v_i}=\left( \begin{array}{cc} {\varpi _{v_i}^{-1} 1_n} &{} {\quad 0} \\ {0} &{} {\quad 1_n} \end{array} \right) \in GL_{2n}(F_{v_i}). \end{aligned}$$
However
$$\begin{aligned} \left. \left( {{\text {n-Ind}}}_{B_n^1(F^+_{v_i})}^{G_n^1(F^+_{v_i})} \chi \right) ^{\varsigma _v^{-1}} \cong {{\text {n-Ind}}}_{B_n^1(F^+_{v_i})}^{G_n^1(F^+_{v_i})} \chi .\right) \end{aligned}$$
Let
$$\begin{aligned} \begin{array}{rcl} {\text{ N }}: T_{GL_{2n}}(F_{v_i}) &{}\longrightarrow &{}T_n^1(F_{v_i}^+) \\ {{\text {diag}}}(t_1,\ldots ,t_{2n})&{} \longmapsto &{} {{\text {diag}}}(t_1/{}^ct_{2n},\ldots ,t_{2n}/{}^ct_1). \end{array} \end{aligned}$$
Then we define \({{\text {BC}}}(\varPi )_{v_i}\) to be the unique subquotient of
$$\begin{aligned} {{\text {n-Ind}}}_{B_{GL_{2n}}(F_{v_i})}^{GL_{2n}(F_{v_i})} (\chi \circ {\text{ N }}) \end{aligned}$$
with a \(GL_{2n}({\mathcal {O}}_{F,v_i})\)-fixed vector. The next lemma is easy to prove.

Lemma 1.1

Suppose that \(\psi \otimes \pi \) is an irreducible smooth representation of
$$\begin{aligned} L_{n,(n)}({\mathbb {Q}}_q) \cong L_{n,(n),{{\text {herm}}}}({\mathbb {Q}}_q) \times L_{n,(n),{{\text {lin}}}}({\mathbb {Q}}_q)={\mathbb {Q}}_q^\times \times GL_n(F_q). \end{aligned}$$
  1. (1)

    If v is unramified over \(F^+\) and \(\pi _v\) is unramified then \({{\text {n-Ind}}}_{P_{n,(n)}({\mathbb {Q}}_q)}^{G_n({\mathbb {Q}}_q)} (\psi \otimes \pi )\) has a subquotient \(\varPi \) which is unramified at v. Moreover \({{\text {BC}}}(\varPi )_v\) is the unramified irreducible subquotient of \({{\text {n-Ind}}}_{Q_{2n,(n)}(F_v)}^{GL_{2n}(F_v)} (\pi ^{\vee ,c}_v \otimes \pi _v)\).

     
  2. (2)

    If v is split over \(F^+\) and \(\varPi \) is an irreducible subquotient of the normalized induction \({{\text {n-Ind}}}_{P_{n,(n)}({\mathbb {Q}}_q)}^{G_n({\mathbb {Q}}_q)} (\psi \otimes \pi )\), then \({{\text {BC}}}(\varPi )_v\) is an irreducible subquotient of \({{\text {n-Ind}}}_{Q_{2n,(n)}(F_v)}^{GL_{2n}(F_v)} ((\pi _{{}^cv})^{\vee ,c} \otimes \pi _v)\).

     
Note that in both cases \({{\text {BC}}}(\varPi _v)\) does not depend on \(\psi \).
In this paragraph let K be a number field, \(m\in {\mathbb {Z}}_{>0}\), and write \(U_{K,\infty }\) for a maximal compact subgroup of \(GL_m(K_\infty )\). We shall (slightly abusively) refer to an admissible
$$\begin{aligned} G_n({\mathbb {A}}^\infty ) \times (({{\text {Lie}}\,}G_n({\mathbb {R}}))_{\mathbb {C}}, U_{n,\infty }) \end{aligned}$$
(resp.
$$L_{n,(i)}({\mathbb {A}}^\infty ) \times \left( \left( {{\text {Lie}}\,}L_{n,(i)}({\mathbb {R}})\right) _{\mathbb {C}},U_{n,\infty }\cap L_{n,(i)}({\mathbb {R}})\right) ,$$
resp.
$$GL_m\left( {\mathbb {A}}^\infty _K\right) \times \left( ({{\text {Lie}}\,}GL_m(K_\infty ))_{\mathbb {C}}, U_{K,\infty } )\right) $$
module as an admissible \(G_n({\mathbb {A}})\)-module (resp. \(L_{n,(i)}({\mathbb {A}})\)-module, resp. \(GL_m({\mathbb {A}}_K)\)-module). By a square integrable automorphic representation of \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)) we shall mean the twist by a character of an irreducible admissible \(G_n({\mathbb {A}})\)-module (resp. \(L_{n,(i)}({\mathbb {A}})\)-module, resp. \(GL_m({\mathbb {A}}_K)\)-module) that occurs discretely in the space of square integrable automorphic forms on the double coset space \(G_n({\mathbb {Q}}) \backslash G_n({\mathbb {A}})/A_n({\mathbb {R}})^0\) (respectively \(L_{n,(i)}({\mathbb {Q}})\backslash L_{n,(i)}({\mathbb {A}})/A_{n,(i)}({\mathbb {R}})^0\) or \(GL_m(K)\backslash GL_m({\mathbb {A}}_K)/{\mathbb {R}}^\times _{>0}\)). By a cuspidal automorphic representation of \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)) we shall mean an irreducible admissible \(G_n({\mathbb {A}})\)-submodule (resp. \(L_{n,(i)}({\mathbb {A}})\)-submodule, resp. \(GL_m({\mathbb {A}}_K)\)-submodule) of the space of cuspidal automorphic forms on \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)).

Proposition 1.2

Suppose that \(\varPi \) is a square integrable automorphic representation of \(G_n({\mathbb {A}})\) and that \(\varPi _\infty \) is cohomological. Then there is an expression
$$\begin{aligned} 2n=m_1n_1+\cdots +m_rn_r \end{aligned}$$
with \(m_i,n_i \in {\mathbb {Z}}_{>0}\) and cuspidal automorphic representations \(\widetilde{\varPi }_i\) of \(GL_{m_i}({\mathbb {A}}_F)\) such that
  • \(\widetilde{\varPi }_i^\vee \cong \widetilde{\varPi }_i^c\);

  • \(\widetilde{\varPi }_i ||\det ||^{(m_i+n_i-1)/2}\) is cohomological;

  • if v is a prime of F above a rational prime q such that
    • either q splits in \(F_0\),

    • or F and \(\varPi \) are unramified above q,

    then
    $$\begin{aligned} {{\text {BC}}}(\varPi _q)_v=\boxplus _{i=1}^r \boxplus _{j=0}^{n_i-1} \widetilde{\varPi }_{i,v} |\det |_v^{(n_i-1)/2-j}. \end{aligned}$$

Proof

This follows from the main theorem of [52] and the classification of square integrable automorphic representations of \(GL_m({\mathbb {A}}_F)\) in [48]. (Here we are using our assumption that F contains an imaginary quadratic field.) \(\square \)

Corollary 1.3

Keep the assumptions of the proposition. Then there is a continuous, semi-simple, algebraic (i.e. unramified almost everywhere and de Rham above p) representation
$$\begin{aligned} r_{p,\imath }(\varPi ): G_F \longrightarrow GL_{2n}(\overline{{{\mathbb {Q}}}}_p) \end{aligned}$$
with the following property: If v is a prime of F above a rational prime \(q \ne p\) such that
  • either q splits in \(F_0\),

  • or F and \(\varPi \) are unramified above q,

then
$$\imath {{\text {WD}}}\left( r_{p,\imath } (\varPi )|_{G_{F_v}}\right) ^{{\text {ss}}}\cong {\text {rec}}_{F_v} \left( {{\text {BC}}}(\varPi _q)_v|\det |_v^{(1-2n)/2}\right) .$$

Proof

Combine the proposition with, for instance, theorem 1.2 of [11] and theorem A of [10]. (These results are due to many people and we simply choose these particular references for convenience.) \(\square \)

2.4 Spaces of hermitian forms

If \(R \subset {\mathbb {R}}\) then we will denote by \({{\text {Herm}}}_X^{>0}\) (resp. \({{\text {Herm}}}_X^{\ge 0}\)) the set of pairings \((\,\,\,,\,\,\,)\) in \({{\text {Herm}}}_X\) such that
$$\begin{aligned} (x,x) > 0 \end{aligned}$$
(resp. \(\ge 0\)) for all \(x \in X-\{0\}\). We will denote by \(S(F^m)^{>0}\) (resp. \(S(F^m)^{\ge 0}\)) the set of elements \(a \in S(F^m)\) such that for each \(\tau : F \hookrightarrow {\mathbb {C}}\) the image of a under the map
$$\begin{aligned} \begin{array}{rcl} S(F^m) &{} \longrightarrow &{} M_m(F)^{t=c} \\ x \otimes y &{} \longmapsto &{} x\,{}^{c,t}y+y\,{}^{c,t}x \end{array} \end{aligned}$$
is positive definite (resp. positive semi-definite), i.e. all the roots of its characteristic polynomial are strictly positive (resp. non-negative) real numbers. Then \(S(F^m)^{> 0}\) is the set of elements of \(S(F^m)\) whose pairing with every nonzero element of \({{\text {Herm}}}_{F^m}^{\ge 0}\) is strictly positive, and \({{\text {Herm}}}_{F^m}^{>0}\) is the set of elements of \({{\text {Herm}}}_{F^m}\) whose pairing with every nonzero element of \(S(F^m)^{\ge 0}\) is strictly positive. We will also write
$$\begin{aligned} S\left( {\mathcal {O}}_{F,(p)}^m\right) ^{>0}= S\left( {\mathcal {O}}_{F,(p)}^m\right) \cap S(F^m)^{>0}. \end{aligned}$$
We will next turn to the study of certain spaces which play a key role in the definition of the auxiliary data controlling toroidal compactifications.
Suppose that \(W \subset V_n\) is an isotropic F-direct summand. We set
$$\begin{aligned} {\mathfrak {C}}^{(m)}(W)= \left( {{\text {Herm}}}_{V_n/W^\perp } \oplus {{\text {Hom}}}_F(F^m,W)\right) \otimes _{\mathbb {Q}}{\mathbb {R}}. \end{aligned}$$
If \(m=0\) we will drop it from the notation. Note that we have a natural identification
$$\begin{aligned} {\mathfrak {C}}^{(m)}(V_{n,(i)}) \cong Z\left( N_{n,(i)}^{(m)}\right) ({\mathbb {R}}). \end{aligned}$$
There is also a natural map
$$\begin{aligned} {\mathfrak {C}}^{(m)}(W) \longrightarrow {\mathfrak {C}}(W). \end{aligned}$$
Note that if \(f \in {{\text {Hom}}}_F(F^m,W)\) we can define \(f' \in {{\text {Hom}}}(F^m \otimes _{F,c} (V_n/W^\perp ),{\mathbb {Q}})\) by
$$\begin{aligned} f'(x \otimes y)=\langle f(x),y\rangle _n. \end{aligned}$$
This establishes an isomorphism
$$\begin{aligned} {{\text {Hom}}}_F(F^m,W) \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}\left( F^m \otimes _{F,c} (V_n/W^\perp ),{\mathbb {Q}}\right) \end{aligned}$$
and hence an isomorphism
$$\begin{aligned} {\mathfrak {C}}^{(m)}(W) \mathop {\longrightarrow }\limits ^{\sim }({{\text {Herm}}}_{V_n/W^\perp \oplus F^m}/{{\text {Herm}}}_{F^m}) \otimes _{\mathbb {Q}}{\mathbb {R}}. \end{aligned}$$
Thus
$$\begin{aligned} {\mathfrak {C}}^{(m)}(V_{n,(i)}) \cong Z\left( N_{n,(i)}^{(m)}\right) ({\mathbb {R}}). \end{aligned}$$
If \(g \in G_n({\mathbb {Q}})\) we define
$$\begin{aligned} \begin{array}{rcl} g: {\mathfrak {C}}^{(m)}(W) &{}\longrightarrow &{}{\mathfrak {C}}^{(m)}(gW) \\ (z,f) &{} \longmapsto &{} (gz,g \circ f), \end{array} \end{aligned}$$
where
$$\begin{aligned} (x,y)_{gz}=|\nu (g)| \left( g^{-1}x,g^{-1}y\right) _z. \end{aligned}$$
We extend this to an action of \(G_n^{(m)}({\mathbb {Q}})\) as follows: If \(g \in {{\text {Hom}}}_F(F^m,V_n)\) then we set
$$\begin{aligned} g(z,f)=(z,f - \theta _z \circ g) \end{aligned}$$
where \(\theta _z: V_n \rightarrow W\) satisfies
$$\begin{aligned} \left( x \bmod W^\perp , y \bmod W^\perp \right) _z=\langle \theta _z(x),y \rangle _n \end{aligned}$$
for all \(x,y \in V_n\). If \(W' \subset W\) there is a natural embedding
$$\begin{aligned} {\mathfrak {C}}^{(m)}(W') \hookrightarrow {\mathfrak {C}}^{(m)}(W). \end{aligned}$$
We will write \({\mathfrak {C}}^{>0}(W)={{\text {Herm}}}_{V_n/W^\perp \otimes _{\mathbb {Q}}{\mathbb {R}}}^{>0}\) and \({\mathfrak {C}}^{\ge 0}(W)={{\text {Herm}}}_{V_n/W^\perp \otimes _{\mathbb {Q}}{\mathbb {R}}}^{\ge 0}\). We will also write \({\mathfrak {C}}^{(m),>0}(W)\) (resp. \({\mathfrak {C}}^{(m),\ge 0}(W)\)) for the pre-image of \({\mathfrak {C}}^{>0}(W)\) (resp. \({\mathfrak {C}}^{\ge 0}(W)\)) in \({\mathfrak {C}}^{(m)}(W)\). Moreover we will set
$$\begin{aligned} {\mathfrak {C}}^{(m),\succ 0}(W)=\bigcup _{W' \subset W} {\mathfrak {C}}^{(m),>0}(W'), \end{aligned}$$
and \({\mathfrak {C}}^{\succ 0}(W)={\mathfrak {C}}^{(0),\succ 0}(W)\). Thus
$$\begin{aligned} {\mathfrak {C}}^{(m),>0}(W) \subset {\mathfrak {C}}^{(m),\succ 0}(W) \subset {\mathfrak {C}}^{(m),\ge 0}(W). \end{aligned}$$
Note that the natural map \({\mathfrak {C}}^{(m)}(W) \twoheadrightarrow {\mathfrak {C}}(W)\) gives rise to a surjection
$$\begin{aligned} {\mathfrak {C}}^{(m),\succ 0}(W) \twoheadrightarrow {\mathfrak {C}}^{\succ 0}(W) \end{aligned}$$
and that the pre-image of a point in \({\mathfrak {C}}^{>0}(W')\) is \({{\text {Hom}}}_F(F^m,W')\) (and in particular the pre-image of (0) is (0)). Also note that if \(W' \subset W\) then there is a closed embedding
$$\begin{aligned} {\mathfrak {C}}^{(m),\succ 0}(W') \hookrightarrow {\mathfrak {C}}^{(m),\succ 0}(W). \end{aligned}$$
Finally note that the action of \(G_n^{(m)}({\mathbb {Q}})\) takes \({\mathfrak {C}}^{(m),\succ 0}(W)\) (resp. \({\mathfrak {C}}^{(m),> 0}(W)\), resp. \({\mathfrak {C}}^{(m),\ge 0}(W)\)) to \({\mathfrak {C}}^{(m),\succ 0}(gW)\) (resp. \({\mathfrak {C}}^{(m),> 0}(gW)\), resp. \({\mathfrak {C}}^{(m),\ge 0}(gW)\)).
Note that \(L_{n,(i)}^{(m)}({\mathbb {R}})\) acts on
$$\begin{aligned} \pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}(V_{n,(i)}) \end{aligned}$$
and preserves
$$\begin{aligned} \pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)}). \end{aligned}$$
Moreover \(L_{n,(i)}^{(m)}({\mathbb {Q}})\) preserves
$$\begin{aligned} \pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)}). \end{aligned}$$
In fact, \(L_{n,(i)}^{(m)}({\mathbb {R}})\) acts transitively on \(\pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\). For this paragraph let \((\,\,\, ,\,\,\,)_0\in {\mathfrak {C}}^{>0}(V_{n,(i)})\) denote the pairing on \((V_n/V_{n,(i)}^\perp \otimes _{\mathbb {Q}}{\mathbb {R}})^2\) induced by \(\langle J_n\,\,\, ,\,\,\,\rangle _n\). Then the stabilizer of \(1 \times ((\,\,\, ,\,\,\,)_0,0)\) in \(L_{n,(i)}^{(m)}({\mathbb {R}})\) is
$$L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^{\nu =1} \left( U_{n,\infty } \cap L_{n,(i), {{\text {lin}}}}^{(m)}({\mathbb {R}})\right) A_n({\mathbb {R}})^0.$$
Thus we get an \(L_{n,(i)}^{(m)}({\mathbb {R}})\)-equivariant identification
$$\begin{array}{l} \pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\Big /{\mathbb {R}}^\times _{>0} \\ \quad \cong L_{n,(i)}^{(m)}({\mathbb {R}}) \Big / L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^+ \left( U_{n,\infty } \cap L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {R}})\right) ^0 A_{n,(i)}({\mathbb {R}})^0. \end{array}$$
We define \({\mathfrak {C}}^{(m)}\) to be the topological space
$$\begin{aligned} \Big (\bigcup _W {\mathfrak {C}}^{(m),\succ 0}(W) \Big )\Big / \sim , \end{aligned}$$
where \(\sim \) is the equivalence relation generated by the identification of \({\mathfrak {C}}^{(m),\succ 0}(W')\) with its image in \({\mathfrak {C}}^{(m),\succ 0}(W)\) whenever \(W' \subset W\). (This is sometimes referred to as the ‘conical complex’.) Thus as a set
$$\begin{aligned} {\mathfrak {C}}^{(m)}=\coprod _W {\mathfrak {C}}^{(m),>0}(W). \end{aligned}$$
We will let \({\mathfrak {C}}^{(m)}_{=i}\) denote
$$\begin{aligned} \coprod _{\dim _F W=i} {\mathfrak {C}}^{(m),>0}(W). \end{aligned}$$
Note that \({\mathfrak {C}}^{(m)}_{=n}\) is a dense open subset of \({\mathfrak {C}}^{(m)}\). If \(m=0\) we drop it from the notation.

The space \({\mathfrak {C}}^{(m)}\) has a natural, continuous, left action of \(G^{(m)}_n({\mathbb {Q}}) \times {\mathbb {R}}^\times _{>0}\). (The second factor acts on each \({\mathfrak {C}}^{(m),\succ 0}(W)\) by scalar multiplication.)

We have homeomorphisms
$$\begin{array}{ll} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\Big /{\mathbb {R}}_{>0}^\times \right) \\ \quad \cong P_{n,(i)}^{(m),+}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U \times \pi _0(G_n({\mathbb {R}})) \times \left( {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\Big /{\mathbb {R}}^\times _{>0}\right) \right) \\ \quad \cong \coprod _{h \in P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty ) \backslash G_n^{(m)}({\mathbb {A}}^\infty )/U} L_{n,(i)}^{(m)}({\mathbb {Q}}) \Big \backslash L_{n,(i)}^{(m)}({\mathbb {A}}) \Big / \left( hUh^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\right) \\ \quad \quad L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^+ \left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(i)}({\mathbb {R}})^0 . \end{array}$$
(Use the fact, strong approximation for unipotent groups, that
$$\begin{aligned} N_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )=V+N_{n,(i)}^{(m),+}({\mathbb {Q}}) \end{aligned}$$
for any open compact subgroup V of \(N_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\).) If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(g^{-1}Ug \subset U'\) then the right translation map
$$\begin{array}{l} g: G_n^{(m)}({\mathbb {Q}}) \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}/{\mathbb {R}}_{>0}^\times \right) . \\ \quad \longrightarrow G_n^{(m)}({\mathbb {Q}}) \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U' \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}/{\mathbb {R}}_{>0}^\times \right) . \end{array} $$
corresponds to the coproduct of the right translation maps
$$\begin{array}{l} g': L_{n,(i)}^{(m)}({\mathbb {Q}}) \Big \backslash L_{n,(i)}^{(m)}({\mathbb {A}}) \Big /\left( hUh^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\right) L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^+\\ \quad \left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(i)}({\mathbb {R}})^0 \\ \quad \longrightarrow L_{n,(i)}^{(m)}({\mathbb {Q}}) \Big \backslash L_{n,(i)}^{(m)}({\mathbb {A}}) \Big / \left( h'U'(h')^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\right) L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^+\\ \quad \quad \left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(i)}({\mathbb {R}})^0 \end{array}$$
where \(hg=g'h'u'\) with \(g' \in P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and \(u' \in U'\).

When considering compactifications of just the ordinary locus we will need a variant of the above discussion.

We set
$$\begin{aligned} \left( G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}\end{aligned}$$
to be the subset of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) consisting of elements \((g,\delta ,x)\) such that for some W we have
$$\begin{aligned} x \in {\mathfrak {C}}^{(m),\succ 0}(W) \end{aligned}$$
and
$$\begin{aligned} W \otimes _{\mathbb {Q}}{\mathbb {Q}}_p=g_p (V_{n,(n)} \otimes _{\mathbb {Q}}{\mathbb {Q}}_p). \end{aligned}$$
It has a left action of \(G_n^{(m)}({\mathbb {Q}})\) and a right action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\times {\mathbb {R}}^\times _{>0}\). We define
$$\begin{aligned} \left( G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\end{aligned}$$
similarly. We also set
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}$$
(resp.
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left. \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i} \right) ^{{\text {ord}}}\right) $$
to be the image of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) in
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) $$
(resp.
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left. \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) \right) .$$
Then as a set
$$\begin{array}{l} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}\\ \quad =\coprod _i G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}. \end{array}$$
In the case \(i=n\) we have a simpler description of
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}.$$

Lemma 1.4

$$\begin{array}{l} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=n}\right) ^{{\text {ord}}}\Big /U^p(N_1) \\ \quad \mathop {\longrightarrow }\limits ^{\sim }G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=n}\right) ^{{\text {ord}}}.\end{array}$$

Proof

There is a natural surjection. We must check that it is also injective. The right hand side equals
$$\begin{array}{l} P_{n,(n)}^{(m),+}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p \times \left( P_{n,(n)}^{(m),+}({\mathbb {Q}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)} \right. \\ \left. \quad \times \, \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(n)})\right) \\ \quad \quad \cong P_{n,(n)}^{(m),+}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\Big /U^p(N_1) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(n)})\right) , \end{array}$$
which is clearly isomorphic to the left hand side. \(\square \)
There does not seem to be such a simple description of
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}$$
for \(i \ne n\). However the interested reader can see the end of this section for a partial result, with a very unpleasant proof.
We set
$${\mathfrak {T}}_{U^p(N_1),=n}^{(m),{{\text {ord}}}} = G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=n}\right) ^{{\text {ord}}}\Big /{\mathbb {R}}_{>0}^\times .$$
If U is a neat open compact subgroup of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\), set
$${\mathfrak {T}}^{(m)}_{(i),U}=L_{n,(i)}^{(m)}({\mathbb {Q}}) \Big \backslash L_{n,(i)}^{(m)}({\mathbb {A}}) \Big / U L_{n,(i),{{\text {herm}}}}({\mathbb {R}})^0 \left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0.$$

Corollary 1.5

$${\mathfrak {T}}^{(m),{{\text {ord}}}}_{U^p(N_1),=n} \cong \coprod _{h \in P_{n,(n)}^{(m),+}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } \backslash G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/U^p(N_1)} {\mathfrak {T}}_{(n),\left( hU^ph^{-1} \cap P_{n,(n)}^{(m),+}({\mathbb {A}}^{\infty ,p})\right) {\mathbb {Z}}_p^\times U_p(N_1)^{(m)}_{n,(n)}}^{(m)}.$$
If Y is a locally compact, Hausdorff topological space then we write \(H^i_{{\text {Int}}}(Y,{\mathbb {C}})\) for the image of
$$\begin{aligned} H^i_c(Y,{\mathbb {C}}) \longrightarrow H^i(Y,{\mathbb {C}}). \end{aligned}$$
We define
$$H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{=n}^{(m),{{\text {ord}}}},\overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U^p,N}H^i_{{\text {Int}}}\left( {\mathfrak {T}}^{(m),{{\text {ord}}}}_{U^p(N),=n},\overline{{{\mathbb {Q}}}}_p\right) $$
a smooth \(G^{(m)}_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-module, and
$$H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p\right) =\lim _{\rightarrow U}H^i_{{\text {Int}}}\left( {\mathfrak {T}}^{(m)}_{(n),U},\overline{{{\mathbb {Q}}}}_p\right) $$
a smooth \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-module. Note that
$$H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p\right) ^{{\mathbb {Z}}_p^\times }=\lim _{\rightarrow U^p,N}H^i_{{\text {Int}}}\left( {\mathfrak {T}}^{(m)}_{(n),U^pU_p(N)^{(m)}_{n,(n)} {\mathbb {Z}}_p^\times },\overline{{{\mathbb {Q}}}}_p\right) $$
as N runs over positive integers and \(U^p\) runs over neat open compact subgroups of \(L_{n,(n)}^{(m)}({\mathbb {A}}^{\infty ,p})\). With these definitions we have the following corollary.

Corollary 1.6

There is a \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant isomorphism
$${{\text {Ind}}}^{G_n^{(m)}({\mathbb {A}}^{\infty ,p})}_{P_{n,(n)}^{(m),+}({\mathbb {A}}^{\infty ,p})} H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p\right) ^{{\mathbb {Z}}_p^\times } \cong H^i_{{\text {Int}}}\left( {\mathfrak {T}}^{(m),{{\text {ord}}}}_{=n},\overline{{{\mathbb {Q}}}}_p\right) .$$

Interior cohomology has the following property which will be key for us.

Lemma 1.7

Suppose that G is a locally compact, totally disconnected topological group. Suppose that for any sufficiently small open compact subgroup \(U \subset G\) we are given a compact Hausdorff space \(Z_U\) and an open subset \(Y_U \subset Z_U\). Suppose moreover that whenever U, \(U'\) are sufficiently small open compact subgroups of G and \(g \in G\) with \(g^{-1} U g \subset U'\), then there is a proper continuous map
$$\begin{aligned} g: Z_U \longrightarrow Z_{U'} \end{aligned}$$
with \(g Y_U \subset Y_{U'}\). Also suppose that \(g \circ h=hg\) whenever these maps are all defined and that if \(g\in U\) then the map \(g: Z_U \rightarrow Z_U\) is the identity.
If \(\varOmega \) is a field, set
$$\begin{aligned} H^i(Z,\varOmega )=\lim _{\rightarrow U} H^i(Z_U,\varOmega ) \end{aligned}$$
and
$$\begin{aligned} H^i_{{\text {Int}}}(Y,\varOmega )=\lim _{\rightarrow U} H^i_{{\text {Int}}}(Y_U,\varOmega ). \end{aligned}$$
These are both smooth G-modules. Moreover \(H^i_{{\text {Int}}}(Y,\varOmega )\) is a subquotient of \(H^i(Z,\varOmega )\) as G-modules.

Proof

Note that the diagram
$$\begin{array}{rcl} H^i_c(Y_U,\varOmega ) &{} \longrightarrow &{} H^i(Y_U,\varOmega ) \\ \downarrow &{}&{} \uparrow \\ H^i_c(Z_U,\varOmega ) &{}=&{} H^i(Z_U,\varOmega ) \end{array}$$
is commutative. Set
$$A=\lim _{\rightarrow U} {\text {Im}}\left( H^i_c(Y_U,\varOmega ) \longrightarrow H^i_c(Z_U,\varOmega )=H^i(Z_U,\varOmega )\right) $$
and
$$B= \lim _{\rightarrow U} {\text {Im}}\left( \ker \left( H^i_c(Y_U,\varOmega ) \longrightarrow H^i(Y_U,\varOmega )\right) \longrightarrow H^i(Z_U,\varOmega )\right) .$$
Then
$$\begin{aligned} B \subset A \subset H^i(Z,\varOmega ) \end{aligned}$$
are G-invariant subspaces with
$$\begin{aligned} A/B \mathop {\longrightarrow }\limits ^{\sim }H^i_{{\text {Int}}}(Y,\varOmega ). \end{aligned}$$
\(\square \)

We finish this section with our promised generalization of Lemma 1.4. This generalization is not needed for the proofs of the main results of this paper, but we include it for completeness sake. The reader may wish to skip the proof.

Lemma 1.8

There is a natural homeomorphism
where \(U^p(N_1) \subset G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\).
In particular
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}$$
and
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}$$
are independent of \(N_2 \ge N_1\).

Proof

Firstly we have that
$$\begin{array}{ll} &{} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\\ &{}\quad \cong P_{n,(i)}^{(m),+}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p\right. \\ &{}\quad \quad \times \left( P_{n,(i)}^{(m),+}({\mathbb {Q}}) P_{n,(n)}^{(m),+}({\mathbb {Q}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)} \\ &{}\quad \quad \left. \times \,\pi _0 (G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) . \end{array}$$
We can replace the second \(P_{n,(i)}^{(m),+}({\mathbb {Q}})\) by \(P_{n,(i)}^{(m),+}({\mathbb {Q}}_p)\), and then, using in particular the Iwasawa decomposition for \(L_{n,(n)}({\mathbb {Q}}_p)\), replace \(P_{n,(n)}^{(m),+}({\mathbb {Q}}_p)\) by \(P_{n,(n)}^{(m),+}({\mathbb {Z}}_p)\). Next we can replace \(P_{n,(i)}^{(m),+}({\mathbb {Q}}_p)\) by \(P_{n,(i)}^{(m),+}({\mathbb {Z}}_p)\) as long as we also replace \(P_{n,(i)}^{(m),+}({\mathbb {Q}})\) by \(P_{n,(i)}^{(m),+}({\mathbb {Z}}_{(p)})\). This gives
$$\begin{array}{ll} &{} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\\ &{}\quad \cong P_{n,(i)}^{(m),+}({\mathbb {Z}}_{(p)}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p \right. \\ &{}\quad \quad \times \,\left( P_{n,(i)}^{(m),+}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)} \\ &{}\quad \quad \left. \times \,\pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) . \end{array}$$
Note that
$$\begin{aligned} P_{n-i,(n-i)}^{+}({\mathbb {Z}}_p) \twoheadrightarrow C_{n-i}({\mathbb {Z}}_p). \end{aligned}$$
[This follows from the fact that primes above p of \(F^+\) are unramified in F, which implies that
$$\left. \ker \left( {\text{ N }}_{F/F^+}: {\mathcal {O}}_{F,p}^\times \rightarrow {\mathcal {O}}_{F^+,p}^\times \right) =\left\{ {}^cx x^{-1}: x \in {\mathcal {O}}_{F,p}^\times \right\} . \right] $$
Thus
$$\begin{aligned} L_{n,(i),{{\text {herm}}}}^-({\mathbb {Z}}_p) P_{n-i,(n-i)}^+({\mathbb {Z}}_p)=L_{n,(i),{{\text {herm}}}}({\mathbb {Z}}_p) \end{aligned}$$
and
$$\begin{aligned} P_{n,(i)}^{(m),+}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p)=P_{n,(i)}^{(m),-}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p). \end{aligned}$$
Moreover, by strong approximation, \(P_{n,(i)}^{(m),-}({\mathbb {Z}}_{(p)})\) (resp. \(L_{n,(i),{{\text {herm}}}}^-({\mathbb {Z}}_{(p)})\)) is dense in \(P_{n,(i)}^{(m),-}({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_{p})\) (resp. \(L_{n,(i),{{\text {herm}}}}^-({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_{p})\)). Thus
$$\begin{array}{ll} &{} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\\ &{}\quad \cong P_{n,(i)}^{(m),+}({\mathbb {Z}}_{(p)}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p \right. \\ &{}\quad \quad \times \left( P_{n,(i)}^{(m),-}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)} \\ &{}\left. \quad \quad \times \,\pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) \\ &{}\quad \cong L_{n,(i)}^{(m)}({\mathbb {Z}}_{(p)}) \Big \backslash \left( \left( P_{n,(i)}^{(m), -}({\mathbb {A}}^{\infty ,p})\Big \backslash G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p\right) \right. \\ &{}\quad \quad \times \left( P_{n,(i)}^{(m),-}({\mathbb {Z}}_{p}) \Big \backslash \left( P_{n,(i)}^{(m),-}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)}\right) \\ &{}\quad \quad \left. \times \, \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) . \end{array}$$
Next we claim that the natural map
$$\begin{array}{rl} &{} \left( P_{n,(i)}^{(m),-} \cap P_{n,(n)}^{(m),+}\right) ({\mathbb {Z}}_{p}) \Big \backslash P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) \Big /\left( U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p)\right) \\ &{}\quad \longrightarrow P_{n,(i)}^{(m),-}({\mathbb {Z}}_{p}) \Big \backslash \left( P_{n,(i)}^{(m),-}({\mathbb {Z}}_p) P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) U_p(N_1,N_2)_n^{(m)}\right) \Big /U_p(N_1,N_2)_n^{(m)} \end{array}$$
is an isomorphism. It suffices to check this modulo \(p^{N_2}\), where the map becomes
$$\begin{array}{l} \left( P_{n,(i)}^{(m),-} \cap P_{n,(n)}^{(m),+}\right) ({\mathbb {Z}}/p^{N_2}{\mathbb {Z}}) \Big \backslash P_{n,(n)}^{(m),+}({\mathbb {Z}}/p^{N_2}{\mathbb {Z}}) \Big /\left( U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p)\right) \\ \quad \longrightarrow P_{n,(i)}^{(m),-}({\mathbb {Z}}/p^{N_2}{\mathbb {Z}}) \Big \backslash \left( P_{n,(i)}^{(m),-}({\mathbb {Z}}/p^{N_2}{\mathbb {Z}}) P_{n,(n)}^{(m),+}({\mathbb {Z}}/p^{N_2}{\mathbb {Z}}) \Big /\left( U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p)\right) \right. , \end{array}$$
which is clearly an isomorphism. Thus we have
$$\begin{array}{ll} &{} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\\ &{}\quad \cong L_{n,(i)}^{(m)}({\mathbb {Z}}_{(p)}) \Big \backslash \left( \left( P_{n,(i)}^{(m), -}({\mathbb {A}}^{\infty ,p})\Big \backslash G_n^{(m)}({\mathbb {A}}^{\infty ,p})\Big /U^p\right) \right. \\ &{} \quad \quad \times \left( \left( P_{n,(i)}^{(m),-} \cap P_{n,(n)}^{(m),+}\right) ({\mathbb {Z}}_{p}) \Big \backslash P_{n,(n)}^{(m),+}({\mathbb {Z}}_p) \Big /\left( U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p)\right) \right) \\ &{}\left. \quad \quad \times \, \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) , \end{array}$$
where \(\gamma \in L_{n,(i)}^{(m)}({\mathbb {Z}}_{(p)})\) acts on \((P_{n,(i)}^{(m),-} \cap P_{n,(n)}^{(m),+}) ({\mathbb {Z}}_{p}) \backslash P_{n,(n)}^{(m),+}({\mathbb {Z}}_p)\) via an element of \(P_{n-i,(n-i)}^+({\mathbb {Z}}_p) \times L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_p)\) with the same image in \(C_{n-i}({\mathbb {Z}}_p) \times L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_p)\).
Note that
Also note that, if we set \(U_p=(U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p))\), then
However as the primes above p split in \(F^+\) split in F we see that
$${\text {Im}}\left( P_{n-i,(n-i)}({\mathbb {Z}}_p) \rightarrow C_{n-i}({\mathbb {Z}}_p)\right) =L_{n,(i),{{\text {herm}}}} ({\mathbb {Z}}_p)/L_{n,(i),{{\text {herm}}}}^-({\mathbb {Z}}_p),$$
and so
Thus we see that
$$\begin{array}{ll} &{} G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )\Big /U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}\\ &{}\quad \cong \coprod _{h \in P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } \backslash G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/U^p(N_1)} L_{n,(i)}^{(m)}({\mathbb {Z}}_{(p)}) \\ &{}\quad \quad \Big \backslash \left( L^{(m)}_{n,(i)}({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\Big / L_{n,(i),{{\text {herm}}}}^-({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\right. \\ &{}\quad \quad \left. \left( hU^p(N_1)h^{-1} \cap P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\right) \times \pi _0(G_n^{(m)}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\right) . \end{array}$$
As \(L_{n,(i),{{\text {herm}}}}^{-}({\mathbb {Z}}_{(p)})\) acts trivially on
$$\left( L^{(m)}_{n,(i)}({\mathbb {Z}}_p)/ L_{n,(i),{{\text {herm}}}}^-({\mathbb {Z}}_p)\right) \times \pi _0(G_n^{(m)}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})$$
and is dense in \(L_{n,(i),{{\text {herm}}}}^{-}({\mathbb {A}}^{\infty ,p})\), we further see that
as desired. \(\square \)
Abusing notation slightly, we will write
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}}$$
for
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i}\right) ^{{\text {ord}}},$$
and
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}$$
for
$$G_n^{(m)}({\mathbb {Q}}) \Big \backslash \left( G_n^{(m)}({\mathbb {A}}^\infty )/U^p(N_1,N_2) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\right) ^{{\text {ord}}}.$$

2.5 Locally symmetric spaces

In this section we will calculate \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p)\) in terms of automorphic forms on \(L_{n,(n)}({\mathbb {A}})\). Our main result will be the following, which will be an immediate consequence of Corollary 1.12 and Lemma 1.13 below.

Corollary 1.9

Suppose that \(n>1\) and that \(\rho \) is an irreducible algebraic representation of \(L_{n,(n),{{\text {lin}}}}\) on a finite dimensional \({\mathbb {C}}\)-vector space. Suppose also that \(\pi \) is a cuspidal automorphic representation of \(L_{n,(n),{{\text {lin}}}}({\mathbb {A}})\) such that \(\pi _\infty \) has the same infinitesimal character as \(\rho ^\vee \) and that \(\psi \) is a continuous character of \({\mathbb {Q}}^\times \backslash {\mathbb {A}}^\times / {\mathbb {R}}^\times _{>0}\). Then for all sufficiently large integers N there are integers \(m(N) \in {\mathbb {Z}}_{\ge 0}\) and \(i(N) \in {\mathbb {Z}}_{>0}\), and an \(L_{n,(n)}({\mathbb {A}}^\infty )\)-equivariant embedding
$$\begin{aligned} (\pi ^\infty ||\det ||^N)\times \psi ^\infty \hookrightarrow H^{i(N)}_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m(N))},{\mathbb {C}}\right) . \end{aligned}$$
If \(m=0\) we will write \({\mathfrak {T}}_{(n)}\) for \({\mathfrak {T}}^{(0)}_{(n)}\). Let \(\varOmega \) denote an algebraically closed field of characteristic 0. If \(\rho \) is a finite dimensional algebraic representation of \(L_{n,(n)}\) on a \(\varOmega \)-vector space \(W_\rho \) then we define a locally constant sheaf \({\mathcal {L}}_{\rho ,U}/{\mathfrak {T}}_{(n),U}\) as
$$\begin{array}{c} L_{n,(n)}({\mathbb {Q}}) \Big \backslash \left( W_\rho \times L_{n,(n)}({\mathbb {A}}) \Big /U \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0\right) \\ \downarrow \\ L_{n,(n)}({\mathbb {Q}}) \Big \backslash L_{n,(n)}({\mathbb {A}})\Big /U \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0. \end{array}$$
The system of sheaves \({\mathcal {L}}_{\rho ,U}\) has a right action of \(L_{n,(n)}({\mathbb {A}}^\infty )\). We define
$$\begin{aligned} H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)},{\mathcal {L}}_\rho )= \lim _{\rightarrow U} H^i_{{\text {Int}}}({\mathfrak {T}}_{(n),U},{\mathcal {L}}_{\rho ,U}), \end{aligned}$$
smooth \(L_{n,(n)}({\mathbb {A}}^\infty )\)-module. Note that if \(\rho \) has a central character \(\chi _\rho \) then,
$$\begin{aligned} \alpha \in Z(L_{n,(n)})({\mathbb {Q}})^+ \subset L_{n,(n)}({\mathbb {A}}^\infty ) \end{aligned}$$
acts on \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)},{\mathcal {L}}_\rho )\) via \(\chi _\rho (\alpha )^{-1}\). (Use the fact that \(Z(L_{n,(n)})({\mathbb {Q}})^+ \subset (L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0)A_{n,(n)}({\mathbb {R}})^0\).)
The natural map \(L_{n,(n)}^{(m)} \rightarrow L_{n,(n)}\) gives rise to continuous maps
$$\begin{aligned} \pi ^{(m)}: {\mathfrak {T}}^{(m)}_{(n),U} \longrightarrow {\mathfrak {T}}_{(n), U} \end{aligned}$$
compatible with the action of \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\).

Lemma 1.10

  1. (1)

    The maps \(\pi ^{(m)}\) are real-torus bundles (i.e. \((S^1)^r\)-bundles for some r), and in particular are proper maps.

     
  2. (2)
    There are \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\)-equivariant identifications
    $$\begin{aligned} R^i\pi _{*}^{(m)} \varOmega \cong {\mathcal {L}}_{\wedge ^i \left( \bigoplus _{\tau : F \hookrightarrow \varOmega } {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }. \end{aligned}$$
    In particular the action of \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) on the relative cohomology sheaf \(R^i\pi _{*}^{(m)} \varOmega \) factors through \(L_{n,(n)}({\mathbb {A}}^\infty )\).
     

Proof

Recall that
$$\begin{aligned} N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) = \ker \left( L_{n,(n)}^{(m)} \rightarrow L_{n,(n)}\right) . \end{aligned}$$
Suppose that U is a neat open compact subgroup of \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(L_{n,(n)}({\mathbb {A}}^\infty )\). Then \(L_{n,(n)}({\mathbb {Q}}) \times U'\) acts freely on
$$\begin{aligned} L_{n,(n)}({\mathbb {A}})\Big / \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0. \end{aligned}$$
Thus it suffices to prove that the map \({\widetilde{\pi }} ^{(m)}\)
$$\begin{array}{c} N \left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {Q}}) \Big \backslash L^{(m)}_{n,(n)}({\mathbb {A}})\Big /\left( U \cap N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {A}}^\infty )\right) \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0 \\ \downarrow \\ L_{n,(n)}({\mathbb {A}})/\left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0 \end{array}$$
is a real-torus bundle and that there are \(L_{n,(n)}({\mathbb {Q}}) \times L^{(m)}_{n,(n)}({\mathbb {A}}^\infty ) \)-equivariant isomorphisms
$$R^i{\widetilde{\pi }} _{*}^{(m)} \varOmega \cong {\mathcal {L}}_{\wedge ^i \left( \bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }.$$
Using the identification of spaces (but not of groups) that comes from the group product
$$\begin{aligned} L_{n,(n)}^{(m)}({\mathbb {A}})=N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {A}}) \times L_{n,(n)}({\mathbb {A}}), \end{aligned}$$
we see that \({\widetilde{\pi }} ^{(m)}\) can be identified with the map
$$\begin{array}{c} \left( N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {Q}}) \Big \backslash N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {A}})\Big / \left( U \cap N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {A}}^\infty )\right) \right) \\ \times \left( L_{n,(n)}({\mathbb {A}})\Big / \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0\right) \\ \downarrow \\ L_{n,(n)}({\mathbb {A}})\Big /\left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \end{array}$$
or, using the equality \(N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {A}}^\infty )=N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {Q}}) (U \cap N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {A}}^\infty ))\), even with
$$\begin{array}{c} \!\!\! \left( \!\left( N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {Q}})\! \cap \! U\right) \Big \backslash N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) ({\mathbb {R}}) \! \right) \!\! \times \!\! \left( L_{n,(n)}({\mathbb {A}})\Big / \left( L_{n,(n)}({\mathbb {R}})\! \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0\right) \\ \downarrow \\ L_{n,(n)}({\mathbb {A}})\Big /\left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \end{array}$$
The right \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-action is by right translation on the second factor. The left action of \(L_{n,(n)}({\mathbb {Q}})\) is via conjugation on the first factor and left translation on the second.
The first part of the lemma follows, and we see that
$$\begin{aligned} R^i{\widetilde{\pi }} ^{(m)}_* \varOmega \end{aligned}$$
is \(L_{n,(n)}({\mathbb {Q}}) \times L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) equivariantly identified with the locally constant sheaf
$$\begin{array}{c} \left( \wedge ^i N\left( L_{n,(n),{{\text {lin}}}}^{(m)}\right) (\varOmega )^\vee \right) \times \left( L_{n,(n)}({\mathbb {A}})\Big / \left( L_{n,(n)}({\mathbb {R}}) \cap \! U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0\right) \\ \downarrow \\ L_{n,(n)}({\mathbb {A}})\Big /\left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0. \end{array}$$
The lemma follows. \(\square \)

Lemma 1.11

There is an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant isomorphism
$$H^{k}_{{\text {Int}}}\left( {\mathfrak {T}}^{(m)}_{(n)},\varOmega \right) \cong \bigoplus _{i+j=k} H^i_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j\left( \bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }\right) .$$

Proof

There is an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant spectral sequence
$$E_{2}^{i,j}=H^i\left( {\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j \left( \bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }\right) \Rightarrow H^{i+j}\left( {\mathfrak {T}}^{(m)}_{(n)},\varOmega \right) .$$
If \(\alpha \in {\mathbb {Q}}^\times _{>0} \subset Z(L_{n,(n),{{\text {lin}}}})({\mathbb {A}}^\infty )\), then \(\alpha \) acts on \(E_{2}^{i,j}\) via \(\alpha ^j\). We deduce that all the differentials (on the second and any later page) vanish, and so the spectral sequence degenerates on the second page. Moreover the \(\alpha \mapsto \alpha ^j\) eigenspace in \(H^{i+j}({\mathfrak {T}}^{(m)}_{(n)},\varOmega )\) is naturally identified with \(H^i({\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j(\bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m})^\vee })\). (This standard argument is sometimes referred to as ‘Lieberman’s trick’.)
As the maps \(\pi ^{(m)}\) are proper, there is also an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant spectral sequence
$$E_{c,2}^{i,j}=H^i_c\left( {\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j\left( \bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }\right) \Rightarrow H^{i+j}_c\left( {\mathfrak {T}}^{(m)}_{(n)},\varOmega \right) $$
and \(\alpha \in {\mathbb {Q}}^\times _{>0} \subset Z(L_{n,(n),{{\text {lin}}}})({\mathbb {A}}^\infty )\) acts on \(E_{c,2}^{i,j}\) via \(\alpha ^j\). Again we see that the spectral sequence degenerates on the second page and that the \(\alpha \mapsto \alpha ^j\) eigenspace in \(H^{i+j}_c({\mathfrak {T}}^{(m)}_{(n)},\varOmega )\) is naturally identified with \(H^i_c({\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j(\bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m})^\vee })\).

The lemma follows. \(\square \)

Corollary 1.12

Suppose that \(\rho \) is an irreducible representation of \(L_{n,(n),{{\text {lin}}}}\) over \(\varOmega \), which we extend to a representation of \(L_{n,(n)}\) by letting it be trivial on \(L_{n,(n),{{\text {herm}}}}\). Let \(d={\text{ N }}_{F/{\mathbb {Q}}}\circ \det : L_{n,(n),{{\text {lin}}}} \rightarrow {\mathbb {G}}_m\). Then for all N sufficiently large there are \(j(N),\, m(N) \in {\mathbb {Z}}_{\ge 0}\) such that, for all i,
$$\begin{aligned} H^i_{{\text {Int}}}( {\mathfrak {T}}_{(n)},{\mathcal {L}}_{\rho \otimes d^{-N}}) \end{aligned}$$
is an \(L_{n,(n)}({\mathbb {A}}^\infty )\)-direct summand of
$$\begin{aligned} H^{i+j(N)}_{{\text {Int}}}\left( {\mathfrak {T}}_{(n)}^{(m(N))},\varOmega \right) . \end{aligned}$$

Proof

It follows from Weyl’s construction of the irreducible representations of \(GL_n\) that, for N sufficiently large, \(\rho \otimes d^{-N}\) is a direct summand of
$$\begin{aligned} \bigotimes _\tau \left( {{\text {Std}}}_\tau ^{\vee }\right) ^{\otimes m_\tau (N)} \end{aligned}$$
for certain non-negative integers \(m_\tau (N)\). Hence for N sufficiently large and \(m(N)=\max \{m_\tau (N)\}\) the representation \(\rho \otimes d^{-N}\) is also a direct summand of
$$\wedge ^{\sum _\tau m_\tau (N)}\left( \bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m(N)}\right) ^\vee .$$
\(\square \)

Lemma 1.13

Suppose that \(\rho \) is an irreducible algebraic representation of \(L_{n,(n)}\) on a finite dimensional \({\mathbb {C}}\)-vector space.
  1. (1)
    Then
    $$\bigoplus _\varPi \varPi ^\infty \otimes H^i \left( {{\text {Lie}}\,}L_{n,(n)}, \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \varPi _\infty \otimes \rho \right) \hookrightarrow H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}, {\mathcal {L}}_\rho ),$$
    where \(\varPi \) runs over cuspidal automorphic representations of \(L_{n,(n)}({\mathbb {A}})\).
     
  2. (2)
    If \(n>1\) and if \(\varPi \) is a cuspidal automorphic representation of \(L_{n,(n)}({\mathbb {A}})\) such that \(\varPi _\infty \) has the same infinitesimal character as \(\rho ^\vee \), then
    $$H^i\left( {{\text {Lie}}\,}L_{n,(n)}, \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \varPi _\infty \otimes \rho \right) \ne (0)$$
    for some \(i>0\).
     

Proof

The first part results from [13], more precisely from combining theorem 5.2, the discussion in section 5.4 and corollary 5.5 of that paper. The second part results from [20], see the proof of theorem 3.13, and in particular lemma 3.14, of that paper. \(\square \)

We are now in a position to deduce Corollary 1.9, which we stated at the start of this section.

3 Tori, torsors and torus embeddings

The main aim of this section is to recall some basic facts about relative torus embeddings of tori torsors, which will provide local models for the boundary of toroidal compactifications of Shimura and Kuga–Sato varieties. Much of this material is in some sense standard, but we need to work with infinite cone decompositions, which are not treated in much of the literature. It will also be convenient to use a notation which emphasizes the boundary of the torus embedding and the completion of the torus embedding along the boundary. These seem to be more naturally parameterized by certain partial fans rather than fans. In Sect. 2.4 we compute certain cohomology groups. For finite fans (or partial fans) such results are fairly standard, but we found it quite tricky to formulate and prove the results we need in the presence of infinitely many cones. Maybe this is just our incompetence.

Throughout this section let \(R_0\) denote an irreducible noetherian ring (i.e. a noetherian ring with a unique minimal prime ideal). In the applications of this section elsewhere in this paper it will be either \({\mathbb {Q}}\) or \({\mathbb {Z}}_{(p)}\) or \({\mathbb {Z}}/p^r{\mathbb {Z}}\) for some r. We will consider \(R_0\) endowed with the discrete topology so that \({{\text {Spf}}\,}R_0 \cong {{\text {Spec}}\,}R_0\).

3.1 Tori and torsors

If S / Y is a torus (i.e. a group scheme etale locally on Y isomorphic to \({\mathbb {G}}_m^N\) for some N) then we can define its sheaf of characters \(X^*(S)={{\text {Hom}}}(S,{\mathbb {G}}_m)\) and its sheaf of cocharacters \(X_*(S)={{\text {Hom}}}({\mathbb {G}}_m,S)\). These are locally constant sheaves of free \({\mathbb {Z}}\)-modules in the etale topology on Y. They are naturally \({\mathbb {Z}}\)-dual to each other. More generally if \(S_1/Y\) and \(S_2/Y\) are two tori then \({{\text {Hom}}}(S_1,S_2)\) is a locally constant sheaf of free \({\mathbb {Z}}\)-modules in the etale topology on Y. In fact,
$${{\text {Hom}}}(S_1,S_2)={{\text {Hom}}}\left( X_*(S_1),X_*(S_2)\right) = {{\text {Hom}}}\left( X^*(S_2),X^*(S_1)\right) .$$
By a quasi-isogeny (resp. isogeny) from \(S_1\) to \(S_2\) we shall mean a global section of the sheaf \({{\text {Hom}}}(S_1,S_2)_{\mathbb {Q}}\) (resp. \({{\text {Hom}}}(S_1,S_2)\)) with an inverse in \({{\text {Hom}}}(S_2,S_1)_{\mathbb {Q}}\). We will write \([S]_{{\text {isog}}}\) for the category whose objects are tori over Y quasi-isogenous to S and whose morphisms are isogenies. The sheaves \(X_*(S)_{\mathbb {Q}}\) and \(X^*(S)_{\mathbb {Q}}\) only depend on the quasi-isogeny class of S so we will write \(X_*([S]_{{\text {isog}}})_{\mathbb {Q}}\) and \(X^*([S]_{{\text {isog}}})_{\mathbb {Q}}\).
If \({\overline{{y}}}\) is a geometric point of Y then we define
and
with the transition map from MN to N being multiplication by M. (The Tate modules of S.) Also define
$$\begin{aligned} VS_{\overline{{y}}}=TS_{\overline{{y}}}\otimes _{\mathbb {Z}}{\mathbb {Q}}\end{aligned}$$
and
$$\begin{aligned} V^pS_{\overline{{y}}}=T^pS_{\overline{{y}}}\otimes _{\mathbb {Z}}{\mathbb {Q}}. \end{aligned}$$
If Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Q}}\) then
$$\begin{aligned} TS_{\overline{{y}}}\cong X_*(S)_{\overline{{y}}}\otimes _{\mathbb {Z}}{\widehat{{\mathbb {Z}}}}(1). \end{aligned}$$
If Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) then
$$\begin{aligned} T^pS_{\overline{{y}}}\cong X_*(S)_{\overline{{y}}}\otimes _{\mathbb {Z}}{\widehat{{\mathbb {Z}}}}^p(1). \end{aligned}$$
Now suppose that S is split, i.e. isomorphic to \({\mathbb {G}}_m^N\) for some N. By an S-torsor T / Y we mean a scheme T / Y with an action of S, which locally in the Zariski topology on Y is isomorphic to S. By a rigidification of T along \(e: Y' \rightarrow Y\) we mean an isomorphism of S-torsors \(e^* T \cong S\) over \(Y'\). If U is a connected open subset of Y then
$$\begin{aligned} T|_U=\underline{{{\text {Spec}}\,}} \bigoplus _{\chi \in X^*(S)(U)} {\mathcal {L}}_T(\chi ), \end{aligned}$$
where \({\mathcal {L}}_T(\chi )\) is a line bundle on U. If Z is any open subset of Y and if \(\chi \in X^*(S)(Z)\) then there is a unique line bundle \({\mathcal {L}}_T(\chi )\) on Z whose restriction to any connected open subset \(U \subset Z\) is \({\mathcal {L}}_T(\chi |_U)\). Multiplication gives isomorphisms
$$\begin{aligned} {\mathcal {L}}_{T}(\chi _1) \otimes {\mathcal {L}}_{T}(\chi _2) \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {L}}_{T}(\chi _1+\chi _2). \end{aligned}$$
The map
$$\begin{aligned} T \longmapsto {\mathcal {L}}_{T,1}^\vee \end{aligned}$$
gives a bijection between isomorphism classes of \({\mathbb {G}}_m\)-torsors and isomorphism classes of line bundles on Y. The inverse map sends \({\mathcal {L}}\) to
$$\begin{aligned} \underline{{{\text {Spec}}\,}} \bigoplus _{N \in {\mathbb {Z}}} {\mathcal {L}}^{\vee , \otimes N}. \end{aligned}$$
If \(\alpha : S \rightarrow S'\) is a morphism of split tori and if T / Y is an S-torsor we can form a push-out \(\alpha _*T\), an \(S'\) torsor on Y defined as the quotient
$$\begin{aligned} (S' \times _Y T)/S \end{aligned}$$
where S acts by
$$\begin{aligned} s: (s',t) \longmapsto (s's,s^{-1}t). \end{aligned}$$
There is a natural map \(T \rightarrow \alpha _* T\) compatible with \(\alpha : S \rightarrow S'\). If \(\alpha \) is an isogeny then
$$\begin{aligned} \alpha _* T=(\ker \alpha ) \backslash T. \end{aligned}$$
If \(T_1\) and \(T_2\) are S-torsors over Y we define
$$\begin{aligned} (T_1 \otimes _S T_2)/Y \end{aligned}$$
to be the S-torsor
$$\begin{aligned} (T_1 \times _Y T_2) /S \end{aligned}$$
where S acts by
$$\begin{aligned} s: (t_1,t_2) \longmapsto (st_1,s^{-1}t_2). \end{aligned}$$
If T is an S-torsor on Y we define an S-torsor \(T^\vee /Y\) by taking \(T^\vee =T\) as schemes but defining an S action . on \(T^\vee \) by
$$\begin{aligned} s.t=s^{-1}t, \end{aligned}$$
i.e. \(T^\vee =[-1]_{S,*} T\). Then
$$\begin{aligned} T^\vee \otimes _S T \cong S \end{aligned}$$
via the map that sends \((t_1,t_2)\) to the unique section s of S with \(st_1=t_2\).

3.2 Log structures

We will call a formal scheme
$$\begin{aligned} {\mathfrak {X}}\longrightarrow {{\text {Spf}}\,}R_0 \end{aligned}$$
suitable if it has a cover by affine opens \({\mathfrak {U}}_i={{\text {Spf}}\,}(A_i)^\wedge _{I_i}\), where \(A_i\) is a finitely generated \(R_0\)-algebra and \(I_i\) is an ideal of \(A_i\) whose inverse image in \(R_0\) is (0).
By a log structure on a scheme X (resp. formal scheme \({\mathfrak {X}}\)) we mean a sheaf of monoids \({\mathcal {M}}\) on X (resp. \({\mathfrak {X}}\)) together with a morphism
$$\begin{aligned} \alpha : {\mathcal {M}}\longrightarrow ({\mathcal {O}}_X,\times ) \end{aligned}$$
(resp.
$$\begin{aligned} \alpha : {\mathcal {M}}\longrightarrow ({\mathcal {O}}_{\mathfrak {X}},\times )) \end{aligned}$$
such that the induced map
$$\begin{aligned} \alpha ^{-1} {\mathcal {O}}_X^\times \longrightarrow {\mathcal {O}}_X^\times \end{aligned}$$
(resp.
$$\begin{aligned} \alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \longrightarrow {\mathcal {O}}_{\mathfrak {X}}^\times ) \end{aligned}$$
is an isomorphism. We will refer to a scheme (resp. formal scheme) endowed with a log structure as a log scheme (resp. log formal scheme). By a morphism of log schemes (resp. morphism of log formal schemes)
$$\begin{aligned} (\phi ,\psi ): (X,{\mathcal {M}},\alpha ) \longrightarrow (Y,{\mathcal {N}},\beta ) \end{aligned}$$
(resp.
$$\begin{aligned} (\phi ,\psi ): ({\mathfrak {X}},{\mathcal {M}},\alpha ) \longrightarrow ({\mathfrak {Y}},{\mathcal {N}},\beta )\,) \end{aligned}$$
we shall mean a morphism \(\phi : X \rightarrow Y\) (resp. \(\phi : {\mathfrak {X}}\rightarrow {\mathfrak {Y}}\)) and a map
$$\begin{aligned} \psi : \phi ^{-1} {\mathcal {N}}\longrightarrow {\mathcal {M}}\end{aligned}$$
such that \(\phi ^* \circ \phi ^{-1}(\beta )=\alpha \circ \psi \). We will consider \(R_0\) endowed with the trivial log structure \(({\mathcal {O}}_{{{\text {Spec}}\,}R_0}^\times , 1)\) (resp. \(({\mathcal {O}}_{{{\text {Spf}}\,}R_0}^\times , 1)\)). We will call a log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\) suitable if \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\) is suitable and if, locally in the Zariski topology, \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) is finitely generated. (In the case of schemes these definitions are well known: See, for example, [34]. We have not attempted to optimize the definition in the case of formal schemes. We are simply making a definition which works for the limited purposes of this article. The reader might like to compare our definitions with those in [5].)
If D is a closed subscheme of X we define a log structure \({\mathcal {M}}(D)\) on X by setting
$$\begin{aligned} {\mathcal {M}}(D)(U)={\mathcal {O}}_X(U) \cap {\mathcal {O}}_X(U-D)^\times . \end{aligned}$$
If \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type and if \(Z \subset X\) is a closed subscheme which is flat over \({{\text {Spec}}\,}R_0\), then the formal completion \(X_Z^\wedge \) is a suitable formal scheme. Let \(i^\wedge \) denote the map of ringed spaces \(X_Z^\wedge \rightarrow X\). If \(({\mathcal {M}},\alpha )\) is a log structure on X, then we get a map
$$\begin{aligned} (i^\wedge )^{-1}(\alpha ): (i^\wedge )^{-1} {\mathcal {M}}\longrightarrow {\mathcal {O}}_{X_Z^\wedge }. \end{aligned}$$
It induces a log structure \(({\mathcal {M}}^\wedge ,\alpha ^\wedge )\) on \(X_Z^\wedge \), where \({\mathcal {M}}^\wedge \) denotes the push-out
$$\begin{array}{rcl} ((i^\wedge )^{-1}(\alpha ))^{-1} {\mathcal {O}}_{X_Z^\wedge }^\times &{} \hookrightarrow &{} (i^\wedge )^{-1} {\mathcal {M}}\\ \downarrow &{}&{}\downarrow \\ {\mathcal {O}}_{X_Z^\wedge }^\times &{} \longrightarrow &{} {\mathcal {M}}^\wedge . \end{array}$$
If
$$\begin{aligned} (\phi ,\psi ): (X,{\mathcal {M}},\alpha ) \longrightarrow (Y,{\mathcal {N}},\beta ) \end{aligned}$$
is a morphism of schemes with log structures over \({{\text {Spec}}\,}R_0\) then there is a right exact sequence
$$\begin{aligned} \phi ^* \varOmega ^1_Y(\log {\mathcal {N}}) \longrightarrow \varOmega ^1_X(\log {\mathcal {M}}) \longrightarrow \varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}}) \longrightarrow (0) \end{aligned}$$
of sheaves of log differentials. If the map \((\phi ,\psi )\) is log smooth then this sequence is also left exact and the sheaf \(\varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})\) is locally free. (See, e.g., [34].) As usual, we write \(\varOmega ^i_X(\log {\mathcal {M}})=\wedge ^i \varOmega ^1_X(\log {\mathcal {M}})\) and \(\varOmega ^i_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})=\wedge ^i\varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})\).
By a coherent sheaf of differentials on a formal scheme \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\) we will mean a coherent sheaf \(\varOmega /{\mathfrak {X}}\) together with a differential \(d: {\mathcal {O}}_{\mathfrak {X}}\rightarrow \varOmega \) which vanishes on \(R_0\). By a coherent sheaf of log differentials on a log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\) we shall mean a coherent sheaf \(\varOmega /{\mathfrak {X}}\) together with a differential, which vanishes on \(R_0\),
$$\begin{aligned} d: {\mathcal {O}}_{\mathfrak {X}}\longrightarrow \varOmega , \end{aligned}$$
and a homomorphism
$$\begin{aligned} {{\text {dlog}}}: {\mathcal {M}}\longrightarrow \varOmega \end{aligned}$$
such that
$$\begin{aligned} \alpha (m) {{\text {dlog}}}\,m=d( \alpha (m)) . \end{aligned}$$
By a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) we shall mean a coherent sheaf of differentials \((\varOmega ,d)\) (resp. a coherent sheaf of log differentials \((\varOmega ,d,{{\text {dlog}}})\)) such that for any other coherent sheaf of differentials \((\varOmega ',d')\) (resp. a coherent sheaf of log differentials \((\varOmega ',d',{{\text {dlog}}}')\)) there is a unique map \(f: \varOmega \rightarrow \varOmega '\) such that \(f \circ d=d'\) (resp. \(f \circ d=d'\) and \(f \circ {{\text {dlog}}}={{\text {dlog}}}'\)).

Note that if a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) exists, it is unique up to unique isomorphism.

Lemma 2.1

Suppose that \(R_0\) is a discrete, noetherian topological ring.
  1. (1)

    A universal sheaf of coherent differentials \(\varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}\) exists for any suitable formal scheme \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\).

     
  2. (2)
    If \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type and if \(Z \subset X\) is flat over \(R_0\) then
    $$\varOmega _{X_Z^\wedge /{{\text {Spf}}\,}R_0}^1 \cong \left( \varOmega ^1_{X/{{\text {Spec}}\,}R_0}\right) ^\wedge .$$
     
  3. (3)

    A universal sheaf of coherent log differentials \(\varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}})\) exists for any suitable log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\).

     
  4. (4)
    Suppose that \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type, that \(Z \subset X\) is flat over \(R_0\) and that \(({\mathcal {M}},\alpha )\) is a log structure on X such that Zariski locally \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_X^\times \) is finitely generated. Then
    $$\varOmega _{X_Z^\wedge /{{\text {Spf}}\,}R_0}^1(\log {\mathcal {M}}^\wedge ) \cong \left( \varOmega ^1_{X/{{\text {Spec}}\,}R_0}(\log {\mathcal {M}})\right) ^\wedge .$$
     

Proof

Consider the first part. Suppose that \({\mathfrak {U}}={{\text {Spf}}\,}A_I^\wedge \) is an affine open in \({\mathfrak {X}}\), where A is a finitely generated \(R_0\)-algebra and I is an ideal of A with inverse image (0) in \(R_0\). Then there exists a universal finite module of differentials \(\varOmega ^1_{{\mathfrak {U}}}\) for \({\mathfrak {U}}\), namely the coherent sheaf of \({\mathcal {O}}_{\mathfrak {U}}\)-modules associated to \((\varOmega ^1_{A/R_0})_I^\wedge \). (See sections 11.5 and 12.6 of [40].) We must show that if \({\mathfrak {U}}' \subset {\mathfrak {U}}\) is open then \(\varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'}\) is a universal finite module of differentials for \({\mathfrak {U}}'\). For then uniqueness will allow us to glue the coherent sheaves \(\varOmega ^1_{{\mathfrak {U}}}\) to form \(\varOmega ^1_{{\mathfrak {X}}}\).

So suppose that \((\varOmega ',d')\) is a finite module of differentials for \({\mathfrak {U}}'\). We must show that there is a unique map of \({\mathcal {O}}_{{\mathfrak {U}}'}\)-modules
$$\begin{aligned} f: \varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'} \longrightarrow \varOmega ' \end{aligned}$$
such that \(d'=f \circ d\). We may cover \({\mathfrak {U}}'\) by affine opens of the form \({{\text {Spf}}\,}(A_g)^\wedge _I\) and it will suffice to find, for each g, a unique
$$\begin{aligned} f_g: \varOmega ^1_{{\mathfrak {U}}}\big |_{{{\text {Spf}}\,}(A_g)^\wedge _I} \longrightarrow \varOmega '|{{\text {Spf}}\,}(A_g)^\wedge _I \end{aligned}$$
with \(d'=f_g \circ d\). Thus we may assume that \({\mathfrak {U}}'={{\text {Spf}}\,}(A_g)^\wedge _I\). But in this case we know \(\varOmega ^1_{{\mathfrak {U}}'}\) exists, and is the coherent sheaf associated to
$$\left( \varOmega ^1_{A_g/R_0}\right) ^\wedge _I \cong \left( \varOmega ^1_{A/R_0} \otimes _A A_g\right) ^\wedge _I.$$
On the other hand \(\varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'}\) is the coherent sheaf associated to
$$\begin{aligned} \left( \varOmega ^1_{A/R_0}\right) ^\wedge _I \otimes _{A_I^\wedge } (A_g)_I^\wedge . \end{aligned}$$
Thus
$$\begin{aligned} \varOmega ^1_{{\mathfrak {U}}'} \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'} \end{aligned}$$
and the first part follows. The second part also follows from the proof of the first part.
For the third part, because of uniqueness, it suffices to work locally. Thus we may assume that there are finitely many sections \(m_1,\ldots ,m_r \in {\mathcal {M}}({\mathfrak {X}})\), which together with \(\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) generate \({\mathcal {M}}\). Then we define \(\varOmega ^1_{({\mathfrak {X}},{\mathcal {M}},\alpha )}\) to be the cokernel of the map
$$\begin{array}{rcl} {\mathcal {O}}_{\mathfrak {X}}^{\oplus r} &{} \longrightarrow &{} \varOmega ^1_{\mathfrak {X}}\oplus {\mathcal {O}}_{{\mathfrak {X}}}^{\oplus r} \\ (f_i)_i &{} \longmapsto &{} \left( -\sum _i f_i d \alpha (m_i),(f_i \alpha (m_i))_i\right) . \end{array}$$
It is elementary to check that this has the desired universal property. The fourth part is also elementary to check. \(\square \)
If
$$\begin{aligned} (\phi ,\psi ): ({\mathfrak {X}},{\mathcal {M}},\alpha ) \longrightarrow ({\mathfrak {Y}},{\mathcal {N}},\beta ) \end{aligned}$$
is a map of suitable log formal schemes over \({{\text {Spf}}\,}R_0\) then we set
$$\varOmega ^1_{{\mathfrak {X}}/{\mathfrak {Y}}}(\log {\mathcal {M}}/{\mathcal {N}})=\varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}})\big /\phi ^* \varOmega ^1_{{\mathfrak {Y}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {N}}).$$
We also set
$$\begin{aligned} \varOmega ^i_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}=\wedge ^i \varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0} \end{aligned}$$
and
$$\begin{aligned} \varOmega ^i_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}})=\wedge ^i \varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}}) \end{aligned}$$
and
$$\begin{aligned} \varOmega ^i_{{\mathfrak {X}}/{\mathfrak {Y}}}(\log {\mathcal {M}}/{\mathcal {N}})= \wedge ^i \varOmega ^1_{{\mathfrak {X}}/{\mathfrak {Y}}}(\log {\mathcal {M}}/{\mathcal {N}}). \end{aligned}$$

Corollary 2.2

Suppose that \(R_0\) is a discrete, noetherian topological ring; that
$$\begin{aligned} (X,{\mathcal {M}},\alpha ) \rightarrow (Y,{\mathcal {N}},\beta ) \end{aligned}$$
is a map of log schemes over \({{\text {Spec}}\,}R_0\); and that \(Z \subset X\) and \(W \subset Y\) are closed subschemes flat over \({{\text {Spec}}\,}R_0\) which map to each other under \(X \rightarrow Y\). Suppose moreover that X and Y have finite type over \({{\text {Spec}}\,}R_0\) and that \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) and \({\mathcal {N}}/\beta ^{-1} {\mathcal {O}}_{\mathfrak {Y}}^\times \) are locally (in the Zariski topology) finitely generated. Then
$$\varOmega ^1_{(X_Z^\wedge ,{\mathcal {M}}^\wedge ,\alpha ^\wedge )/(Y_W^\wedge ,{\mathcal {N}}^\wedge ,\beta ^\wedge )} \cong ( \varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}}))_Z^\wedge .$$

Proof

This follows from the lemma and from the exactness of completion. \(\square \)

If Y is a scheme we will let
$$\begin{aligned} {{\text {Aff}}}^n_Y=\underline{{{\text {Spec}}\,}} {\mathcal {O}}_Y[T_1,\ldots ,T_n] \end{aligned}$$
denote affine n-space over Y and
$$\begin{aligned} {{\text {Coord}}}^n_Y=\underline{{{\text {Spec}}\,}} {\mathcal {O}}_Y[T_1,\ldots ,T_n]/(T_1\ldots T_n) \subset {{\text {Aff}}}^n_Y \end{aligned}$$
denote the union of the coordinate hyperplanes in \({{\text {Aff}}}^n_Y\). Now suppose that \(X \rightarrow Y\) is a smooth map of schemes of relative dimension n. By a simple normal crossings divisor in X relative to Y we shall mean a closed subscheme \(D \subset X\) such that X has an affine Zariski-open cover \(\{ U_i\}\) such that each \(U_i\) admits an etale map \(f_i: U_i \rightarrow {{\text {Aff}}}^n_Y\) so that \(D|_{U_i}\) is the (scheme-theoretic) pre-image of \({{\text {Coord}}}^n_Y\). In the case that Y is just the spectrum of a field we will refer simply to a simple normal crossings divisor in X.
Suppose that Y is locally noetherian and separated and that the connected components of Y are irreducible. If S is a finite set of irreducible components of D we will set
$$\begin{aligned} D_S=\bigcap _{E \in S} E. \end{aligned}$$
It is smooth over Y. We will also set
$$\begin{aligned} D^{(s)}=\coprod _{\# S=s} D_S. \end{aligned}$$
If E is an irreducible component of \(D^{(s)}\) then the set S(E) of irreducible components of D containing E has cardinality s. If \(\ge \) is a total order on the set of irreducible components of D, we can define a delta set \({\mathcal {S}}(D, \ge )\), or simply \({\mathcal {S}}(D)\), as follows. (For the definition of ‘delta set’, see, for instance, [25]. We can, if we prefer to be more abstract, replace \({\mathcal {S}}(D,\ge )\) by the associated simplicial set.) The n cells consist of all irreducible components of \(D^{(n+1)}\). If E is such an irreducible component and if \(i\in \{0,\ldots ,n\}\) then the image of E under the face map \(d_i\) is the unique irreducible component of
$$\begin{aligned} \bigcap _{F \in S(E)_i} F \end{aligned}$$
which contains E. Here \(S(E)_i\) equals S(E) with its \((i+1)\)th smallest element removed. The topological realization \(|{\mathcal {S}}(D,\ge )|\) does not depend on the total order \(\ge \), so we will often write \(|{\mathcal {S}}(D)|\).

We record a general observation about log de Rham complexes and divisors with simple normal crossings, which is probably well known. We include a proof because it is of crucial importance for our argument.

Lemma 2.3

Suppose that Y is a smooth scheme of finite type over a field k and that \(Z \subset Y\) is a divisor with simple normal crossings. Let \(Z_1,\ldots ,Z_m\) denote the distinct irreducible components of Z and set
$$\begin{aligned} Z_S=\bigcap _{j \in S} Z_j \subset Y \end{aligned}$$
(in particular \(Z_\emptyset =Y\)), and
$$\begin{aligned} Z^{(s)}=\coprod _{\# S=s} Z_S. \end{aligned}$$
Let \(i_S\) (resp. \(i^{(s)}\)) denote the natural maps \(Z_S \rightarrow Y\) (resp. \(Z^{(s)} \rightarrow Y\)). Also let \({\mathcal {I}}_Z\) denote the ideal of definition of Z.
There is a double complex
$$\begin{aligned} i^{(s)}_* \varOmega ^r_{Z^{(s)}} \end{aligned}$$
with maps
$$\begin{aligned} d: i^{(s)}_* \varOmega ^r_{Z^{(s)}} \longrightarrow i^{(s)}_* \varOmega ^{r+1}_{Z^{(s)}} \end{aligned}$$
and
$$\begin{aligned} i^{(s)}_* \varOmega ^r_{Z^{(s)}} \longrightarrow i^{(s+1)}_* \varOmega ^{r}_{Z^{(s+1)}} \end{aligned}$$
being the sum of the maps
$$\begin{aligned} i_{S,*}\varOmega ^r_{Z_S} \longrightarrow i_{S',*} \varOmega ^{r}_{Z_{S'}}, \end{aligned}$$
which are
  • 0 if \( S \not \subset S'\),

  • and \((-1)^{\# \{ i \in S: i<j\} }\) times the natural pull-back if \(S \cup \{ j\} =S'\).

The natural inclusions
$$\begin{aligned} \varOmega ^r_Y(\log {\mathcal {M}}(Z)) \otimes {\mathcal {I}}_Z \longrightarrow \varOmega ^r_Y \end{aligned}$$
give rise to a map of complexes
$$\begin{aligned} \varOmega ^\bullet _Y(\log {\mathcal {M}}(Z)) \otimes {\mathcal {I}}_Z \longrightarrow \varOmega ^\bullet _Y=i^{(0)}_* \varOmega ^\bullet _{Z^{(0)}}. \end{aligned}$$
For fixed r the simple complexes
$$\begin{aligned} (0) \longrightarrow \varOmega ^r_Y(\log {\mathcal {M}}(Z)) \otimes {\mathcal {I}}_Z \longrightarrow i^{(0)}_* \varOmega ^r_{Z^{(0)}} \longrightarrow i^{(1)}_* \varOmega ^r_{Z^{(1)}} \longrightarrow \cdots \end{aligned}$$
are exact.

Proof

Only the last assertion is not immediate. So consider the last assertion. We can work Zariski locally, so we may assume that the complex is pulled back from the corresponding complex for the case \(Y={{\text {Spec}}\,}k[X_1,\ldots ,X_d]\) and Z is given by \(X_1X_2\ldots X_m=0\). In this case we take \(Z_j\) to be the scheme \(X_j=0\), for \(j=1,\ldots ,m\). In this case
$$\begin{aligned} \varOmega ^r_Y(\log {\mathcal {M}}(Z)) \otimes {\mathcal {I}}_Z=\bigoplus _{T} k[X_1,\ldots ,X_d] \left( \prod _{j=1, \,\, j \not \in T}^m X_j \right) \bigwedge _{j \in T} dX_j \end{aligned}$$
where T runs over r element subsets of \(\{1,\ldots ,d\}\). On the other hand
$$\begin{aligned} i_{S,*} \varOmega ^r_{Z_S}=\bigoplus _T k[X_1,\ldots ,X_d]/(X_j)_{j \in S} \bigwedge _{j \in T} dX_j \end{aligned}$$
where T runs over r element subsets of \(\{ 1,\ldots ,d\}-S\). Thus it suffices to show that, for each subset \(T \subset \{ 1,\ldots ,d\}\) the sequence
$$\begin{array}{l} (0) \longrightarrow \left( \prod _{j=1, \,\, j \not \in T}^m X_j \right) k[X_1,\ldots ,X_d] \longrightarrow k[X_1,\ldots ,X_d] \longrightarrow \cdots \\ \quad \cdots \longrightarrow \bigoplus _{\# S=s, \,\, S\cap T=\emptyset } k[X_1,\ldots ,X_d]/(X_j)_{j \in S} \longrightarrow \cdots \end{array}$$
is exact, where \(S \subset \{ 1,\ldots ,m\}\). The sequence for \(T \subset \{1,\ldots ,d\}\) is obtained from the sequence for \(\emptyset \subset \{1,\ldots ,m\}-T\) by tensoring over k with \(k[X_j]_{j \in T \cup \{ m+1,\ldots ,d\} }\), and so we only need treat the case \(m=d\) and \(T=\emptyset \).
If \(\mu \) is a monomial in the variables \(X_1,\ldots ,X_m\), let \(R(\mu )\) denote the subset of \(\{ 1,\ldots ,m\}\) consisting of the indices j for which \(X_j\) does not occur in \(\mu \). Then our complex is the direct sum over \(\mu \) of the complexes
$$\begin{aligned} (0) \longrightarrow A_\mu \longrightarrow k \longrightarrow \cdots \longrightarrow \bigoplus _{S \subset R(\mu ),\,\, \# S=s} k \longrightarrow \cdots \end{aligned}$$
where \(A_\mu =k\) if \(R(\mu )=\emptyset \) and \(=(0)\) otherwise. So it suffices to prove this latter complex exact for all \(\mu \). If \(R(\mu )=\emptyset \) then it becomes
$$\begin{aligned} (0) \longrightarrow k \longrightarrow k \longrightarrow (0) \longrightarrow (0) \longrightarrow \cdots , \end{aligned}$$
which is clearly exact. If \(R(\mu ) \ne \emptyset \), our complex becomes
$$\begin{aligned} (0) \longrightarrow k \longrightarrow \bigoplus _{S \subset R(\mu ),\,\, \# S=1} k \longrightarrow \cdots \longrightarrow \bigoplus _{S \subset R(\mu ),\,\, \# S=s} k \longrightarrow \cdots . \end{aligned}$$
If we suppress the first k, this is the complex that computes the simplicial cohomology with k-coefficients of the simplex with \(\# R(\mu )\) vertices. Thus it is exact everywhere except \(\bigoplus _{S \subset R(\mu ),\,\, \# S=1} k\) and the kernel of
$$\begin{aligned} \bigoplus _{S \subset R(\mu ),\,\, \# S=1} k \longrightarrow \bigoplus _{S \subset R(\mu ),\,\, \# S=2} k \end{aligned}$$
is just k. The desired exactness follows. \(\square \)

3.3 Torus embeddings

We will now discuss relative torus embeddings, crucially in the context of infinite fans.

If W is a real vector space with dual \(W^\vee \) and if \(A \subset W\) is a subspace we set
$$\begin{aligned} A^\vee =\left\{ \chi \in W^\vee : \chi (A) \subset {\mathbb {R}}_{\ge 0}\right\} \end{aligned}$$
and
$$\begin{aligned} A^{\vee ,0}=\left\{ \chi \in W^\vee : \chi (A-\{0\}) \subset {\mathbb {R}}_{> 0}\right\} \end{aligned}$$
and
$$\begin{aligned} A^\perp =\left\{ \chi \in W^\vee : \chi (A) =\{0\} \right\} . \end{aligned}$$
We will suppose that \(Y/{{\text {Spec}}\,}R_0\) is flat and locally of finite type. To simplify the notation, for now we will restrict to the case of a split torus S / Y with Y connected. We will record the (trivial) generalization to the case of a disconnected base below. Thus we can think of \(X^*(S)\) and \(X_*(S)\) as abelian groups, rather than as locally constant sheaves on Y, i.e. we replace the sheaf by its global sections over Y. We will let T / Y denote an S-torsor.
By a rational polyhedral cone \(\sigma \subset X_*(S)_{\mathbb {R}}\) we mean a non-empty subset consisting of all \({\mathbb {R}}_{\ge 0}\)-linear combinations of a finite set of elements of \(X_*(S)\), but which contains no complete line through 0. (We include the case \(\sigma =\{0\}\). The notion we define here is sometimes called a ‘non-degenerate rational polyhedral cone’.) By the interior \(\sigma ^0\) of \(\sigma \) we shall mean the complement in \(\sigma \) of all its proper faces. (We consider \(\sigma \) as a face of \(\sigma \), but not a proper face.) We call \(\sigma \) smooth if it consists of all \({\mathbb {R}}_{\ge 0}\)-linear combinations of a subset of a \({\mathbb {Z}}\)-basis of \(X_*(S)\). Note that any face of a smooth cone is smooth. Moreover we set
$$\begin{aligned} T_\sigma =\underline{{{\text {Spec}}\,}} \bigoplus _{\chi \in X^*(S) \cap \sigma ^\vee } {\mathcal {L}}_T(\chi ). \end{aligned}$$
Then \(T_\sigma \) is a scheme over Y with an action of S and there is a natural S-equivariant dense open embedding \(T \hookrightarrow T_\sigma \). If \(\sigma ' \subset \sigma \) there is a natural map \(T_{\sigma '} \rightarrow T_\sigma \) compatible with the embeddings of T. If \(f: Y' \rightarrow Y\) then \(T_\sigma /Y\) pulls back under f to \((f^*T)_\sigma /Y'\) compatibly with the maps \(T_{\sigma '}\hookrightarrow T_\sigma \) for \(\sigma '\subset \sigma \).
Suppose that \(\varSigma _0\) is a set of faces of \(\sigma \) such that
  • \(\{0\} \not \in \varSigma _0\),

  • and, if \(\tau ' \supset \tau \in \varSigma _0\), then \(\tau ' \in \varSigma _0\).

In this case define
$$\begin{aligned} |\varSigma _0|^0=\sigma -\bigcup _{\tau \not \in \varSigma _0} \tau . \end{aligned}$$
Thus
$$\begin{aligned} |\varSigma _0|^{0,\vee ,0} \cap \sigma ^\vee =\sigma ^\vee - \bigcup _{\tau \in \varSigma _0} \tau ^\perp . \end{aligned}$$
Then we define \(\partial _{\varSigma _0} T_\sigma \subset T_\sigma \) to be the closed subscheme defined by the sheaf of ideals
$$\begin{aligned} \bigoplus _{\chi \in X^*(S) \cap |\varSigma _0|^{0,\vee ,0} \cap \sigma ^\vee } {\mathcal {L}}_T(\chi ) \subset \bigoplus _{\chi \in X^*(S) \cap \sigma ^\vee } {\mathcal {L}}_T(\chi ). \end{aligned}$$
If \(\varSigma _0\) contains all the faces of \(\sigma \) other than \(\{0\}\) we will write \(\partial T_\sigma \) for \(\partial _{\varSigma _0} T_\sigma \). Note that \(\partial _\emptyset T_\sigma =\emptyset \). If \(\sigma '\) is a face of \(\sigma \) then under the open embedding
$$\begin{aligned} T_{\sigma '} \hookrightarrow T_\sigma \end{aligned}$$
\(\partial _{\varSigma _0}T_\sigma \) pulls back to \(\partial _{\{ \tau \in \varSigma _0: \tau \subset \sigma '\}} T_{\sigma '}\).
By a fan in \(X_*(S)_{\mathbb {R}}\) we shall mean a non-empty collection \(\varSigma \) of rational polyhedral cones \(\sigma \subset X_*(S)_{\mathbb {R}}\) which satisfy
  • if \(\sigma \in \varSigma \), so is each face of \(\sigma \),

  • if \(\sigma , \sigma ' \in \varSigma \) then \(\sigma \cap \sigma '\) is a face of \(\sigma \) and of \(\sigma '\).

Note that unless otherwise stated we will not assume that \(\varSigma \) is finite. We call \(\varSigma \) smooth if each \(\sigma \in \varSigma \) is smooth. We will call \(\varSigma \) full if every element of \(\varSigma \) is contained in an element of \(\varSigma \) with the same dimension as \(X_*(S)_{\mathbb {R}}\). Define
$$\begin{aligned} |\varSigma |= \bigcup _{\sigma \in \varSigma } \sigma . \end{aligned}$$
We call \(\varSigma '\) a refinement of \(\varSigma \) if each \(\sigma ' \in \varSigma '\) is a subset of some element of \(\varSigma \) and each element \(\sigma \in \varSigma \) is a finite union of elements of \(\varSigma '\).

Lemma 2.4

  1. (1)
    If \(\varSigma \) is a fan and \(\varSigma ' \subset \varSigma \) is a finite cardinality subfan then there is a refinement \(\widetilde{\varSigma }\) of \(\varSigma \) with the following properties:
    • any element of \(\varSigma \) which is smooth also lies in \(\widetilde{\varSigma }\);

    • any element of \(\widetilde{\varSigma }\) contained in an element of \(\varSigma '\) is smooth;

    • and if \(\sigma ' \in \varSigma -\widetilde{\varSigma }\) then \(\sigma '\) has a non-smooth face lying in \(\varSigma '\).

     
  2. (2)

    Any fan \(\varSigma \) has a smooth refinement \(\varSigma '\) such that every smooth cone \(\sigma \in \varSigma \) also lies in \(\varSigma '\).

     

Proof

The first part is proved just as for finite fans by making a finite series of elementary subdivisions by 1 cones that lie in some element \(\sigma ' \in \varSigma '\) but not in any of its smooth faces. See, for instance, section 2.6 of [26].

For the second part, consider the set \({\mathcal {S}}\) of pairs \((\widetilde{\varSigma },\varDelta )\) where \(\widetilde{\varSigma }\) is a refinement of \(\varSigma \) and \(\varDelta \) is a subfan of \(\varSigma \) such that
  • every smooth element of \(\varSigma \) lies in \(\widetilde{\varSigma }\);

  • and if \(\sigma \in \widetilde{\varSigma }\) is contained in an element of \(\varDelta \) then \(\sigma \) is smooth.

It suffices to show that \({\mathcal {S}}\) contains an element \((\widetilde{\varSigma },\varDelta )\) with \(\varDelta =\varSigma \).
If \((\widetilde{\varSigma },\varDelta ) \in {\mathcal {S}}\) and \(\sigma \in \varSigma \) we define \(\widetilde{\varSigma }(\sigma )\) to be the set of elements of \(\widetilde{\varSigma }\) contained in \(\sigma \). We define a partial order on \({\mathcal {S}}\) by decreeing that \((\widetilde{\varSigma },\varDelta ) \ge (\widetilde{\varSigma }',\varDelta ')\) if and only if the following conditions are satisfied:
  • \(\widetilde{\varSigma }\) refines \(\widetilde{\varSigma }'\);

  • \(\varDelta \supset \varDelta '\);

  • \(\widetilde{\varSigma }'(\sigma ) =\widetilde{\varSigma }(\sigma )\) unless \(\sigma \) has a face that is contained in an element of \(\varDelta \) but in no element of \(\varDelta '\).

Suppose that \({\mathcal {S}}' \subset {\mathcal {S}}\) is totally ordered. Set
$$\begin{aligned} \varDelta =\bigcup _{(\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}'} \varDelta ', \end{aligned}$$
and let \(\widetilde{\varSigma }\) denote the set of cones \(\sigma '\) which lie in \(\widetilde{\varSigma }'\) for all sufficiently large elements of \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}'\). If \(\sigma \in \varSigma \) then we can choose \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}'\) so that the number of faces of \(\sigma \) in \(\varDelta '\) is maximal. If \((\widetilde{\varSigma }',\varDelta ') \le (\widetilde{\varSigma }'',\varDelta '') \in {\mathcal {S}}'\) then \(\widetilde{\varSigma }'(\sigma )=\widetilde{\varSigma }''(\sigma )\). Thus \(\widetilde{\varSigma }(\sigma )=\widetilde{\varSigma }'(\sigma )\). We conclude that \(\widetilde{\varSigma }\) is a refinement of \(\varSigma \). Thus \((\widetilde{\varSigma },\varDelta ) \in {\mathcal {S}}\) and it is an upper bound for \({\mathcal {S}}'\).
By Zorn’s lemma \({\mathcal {S}}\) has a maximal element \((\widetilde{\varSigma },\varDelta )\). We will show that \(\varDelta =\varSigma \), which will complete the proof of the lemma. Suppose not. Choose \(\sigma \in \varSigma -\varDelta \). Set \(\varDelta '\) to be the union of \(\varDelta \) and the faces of \(\sigma \). Let \(\widetilde{\varSigma }'\) be a refinement of \(\widetilde{\varSigma }\) such that
  • any element of \(\widetilde{\varSigma }\) which is smooth also lies in \(\widetilde{\varSigma }'\);

  • any element of \(\widetilde{\varSigma }'\) contained in \(\sigma \) is smooth;

  • and if \(\sigma ' \in \widetilde{\varSigma }-\widetilde{\varSigma }'\) then \(\sigma '\) has a non-smooth face contained in \(\sigma \).

Then \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}\) and \((\widetilde{\varSigma }',\varDelta ') > (\widetilde{\varSigma },\varDelta )\), a contradiction. \(\square \)
To a fan \(\varSigma \) one can attach a connected scheme \(T_\varSigma \) that is separated, locally (on \(T_\varSigma \)) of finite type and flat over Y of relative dimension \(\dim _{\mathbb {R}}X_*(S)_{\mathbb {R}}\), together with an action of S and an S-equivariant dense open embedding \(T \hookrightarrow T_\varSigma \) over Y. The scheme \(T_\varSigma \) has an open cover by the \(T_\sigma \) for \(\sigma \in \varSigma \) such that \(T_{\sigma '} \subset T_{\sigma }\) if and only if \(\sigma ' \subset \sigma \). We write \({\mathcal {O}}_{T_\varSigma }\) for the structure sheaf of \(T_\varSigma \). If \(\varSigma \) is smooth then \(T_\varSigma /Y\) is smooth. If \(\varSigma \) is finite and \(|\varSigma |=X_*(S)_{\mathbb {R}}\), then \(T_\varSigma /Y\) is proper. If \(\varSigma ' \subset \varSigma \) then \(T_{\varSigma '}\) can be identified with an open subscheme of \(T_\varSigma \). If \(\varSigma '\) refines \(\varSigma \) then there is an S-equivariant proper map
$$\begin{aligned} T_{\varSigma '} \rightarrow T_\varSigma \end{aligned}$$
which restricts to the identity on T: Its restriction to \(T_{\sigma '}\) equals the map
$$\begin{aligned} T_{\sigma '} \longrightarrow T_\sigma \hookrightarrow T_\varSigma \end{aligned}$$
where \(\sigma ' \subset \sigma \in \varSigma \).
By boundary data for \(\varSigma \) we shall mean a proper subset \(\varSigma _0 \subset \varSigma \) such that \(\varSigma - \varSigma _0\) is a fan. (Note that \(\varSigma _0\) may not be closed under taking faces.) If \(\varSigma _0\) is boundary data we define \(\partial _{\varSigma _0}T_\varSigma \) to be the closed subscheme of \(T_\varSigma \) with
$$\begin{aligned} (\partial _{\varSigma _0} T_\varSigma ) \cap T_\sigma =\partial _{\{ \tau \in \varSigma _0: \tau \subset \sigma \}} T_\sigma . \end{aligned}$$
Note that
$$\begin{aligned} \partial _{\varSigma _0} T_\varSigma \subset \bigcup _{\sigma \in \varSigma _0} T_\sigma . \end{aligned}$$
Thus \(\partial _{\varSigma _0} T_\varSigma \) has an open cover by the sets
$$\begin{aligned} (\partial _{\varSigma _0}T_\varSigma )_\sigma =T_\sigma \cap \partial _{\varSigma _0}T_\varSigma \end{aligned}$$
as \(\sigma \) runs over \(\varSigma _0\). We write \({\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma }\) for the ideal sheaf in \({\mathcal {O}}_{T_\varSigma }\) defining \(\partial _{\varSigma _0} T_\varSigma \). If \(\varSigma _0 \subset \varSigma ' \subset \varSigma \) then
$$\begin{aligned} \partial _{\varSigma _0} T_{\varSigma '} \mathop {\longrightarrow }\limits ^{\sim }\partial _{\varSigma _0} T_\varSigma . \end{aligned}$$
Note that \({\mathcal {I}}_{\partial _{\varSigma _0}T_\varSigma }|_{T_\sigma }\) corresponds to the ideal
$$\begin{aligned} \bigoplus _{\chi \in {\mathfrak {X}}_{\varSigma _0,\sigma ,1}} {\mathcal {L}}_T(\chi ) \end{aligned}$$
of
$$\begin{aligned} \bigoplus _{\chi \in X^*(S) \cap \sigma ^\vee } {\mathcal {L}}_T(\chi ), \end{aligned}$$
where
$$\begin{aligned} {\mathfrak {X}}_{\varSigma _0,\sigma ,1}= X^*(S) \cap \sigma ^\vee - \bigcup _{\tau \in \varSigma _0, \tau \subset \sigma } \tau ^\perp \end{aligned}$$
and \(\tau ^\perp \) denotes the annihilator of \(\tau \) in \(X^*(S)_{\mathbb {R}}\). If we let \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\) denote the set of sums of m elements of \({\mathfrak {X}}_{\varSigma _0,\sigma ,1}\), then \({\mathcal {I}}_{\partial _{\varSigma _0}T_\varSigma }^m|_{T_\sigma }\) corresponds to the ideal
$$\begin{aligned} \bigoplus _{\chi \in {\mathfrak {X}}_{\varSigma _0,\sigma ,m}} {\mathcal {L}}_T(\chi ). \end{aligned}$$
If \(\sigma \not \in \varSigma _0\) then
$$\begin{aligned} {\mathfrak {X}}_{\varSigma _0,\sigma ,m}= X^*(S) \cap \sigma ^\vee \end{aligned}$$
for all m. If on the other hand \(\sigma \in \varSigma _0\) then
$$\begin{aligned} \bigcap _m {\mathfrak {X}}_{\varSigma _0,\sigma ,m}=\emptyset . \end{aligned}$$
(For if \(\chi \in \sigma ^0 \cap X_*(S)\) then \(\chi \ge m\) on \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\).)
In the special case \(\varSigma _0=\varSigma -\{ \{0\}\}\) we will write \(\partial T_\varSigma \) for \(\partial _{\varSigma _0} T_\varSigma \) and \({\mathcal {I}}_{\partial T_\varSigma }\) for \({\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma }\). Then
$$\begin{aligned} T=T_\varSigma - \partial T_\varSigma . \end{aligned}$$
We will write \({\mathcal {M}}_\varSigma \rightarrow {\mathcal {O}}_{T_\varSigma }\) for the log structure corresponding to the closed embedding \(\partial T_\varSigma \hookrightarrow T_\varSigma \). We will write \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log \infty )\) for the log differentials previously denoted \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log {\mathcal {M}}_\varSigma )\).

If \(\varSigma \) is smooth then \(\partial T_\varSigma \) is a simple normal crossings divisor on \(T_\varSigma \) relative to Y.

If \(\varSigma _0\) is boundary data for \(\varSigma \) we will set
$$\begin{aligned} |\varSigma _0|=\{ 0\} \cup \bigcup _{\sigma \in \varSigma _0} \sigma . \end{aligned}$$
and
$$\begin{aligned} |\varSigma _0|^0=|\varSigma _0|- \bigcup _{ \sigma \in \varSigma -\varSigma _0} \sigma . \end{aligned}$$
We will call \(\varSigma _0\)
  • open if \(|\varSigma _0|^0\) is open in \(X_*(S)_{\mathbb {R}}\);

  • finite if it has finite cardinality;

  • locally finite if for every rational polyhedral cone \(\tau \subset |\varSigma _0|\) (not necessarily in \(\varSigma _0\)) the intersection \(\tau \cap |\varSigma _0|^0\) meets only finitely many elements of \(\varSigma _0\). (We remark that although this condition may be intuitive in the case \(|\varSigma _0|^0=|\varSigma _0|\), in other cases it may be less so.)

Let \(\varSigma \) continue to denote a fan and \(\varSigma _0\) boundary data for \(\varSigma \). If \(\sigma \in \varSigma \) we write
$$\begin{aligned} \varSigma (\sigma )=\left\{ \tau \in \varSigma : \tau \supset \sigma \right\} . \end{aligned}$$
If \(\sigma \ne \{0\}\), then this is an example of boundary data for \(\varSigma \). If \(\sigma \in \varSigma _0\) then
$$\begin{aligned} \varSigma (\sigma )=\left\{ \tau \in \varSigma _0: \tau \supset \sigma \right\} \end{aligned}$$
and we will sometimes denote it \(\varSigma _0(\sigma )\). If \(\varSigma _0\) is locally finite then \(\varSigma _0(\sigma )\) is finite for all \(\sigma \in \varSigma _0\). If \(\{0\} \ne \sigma \in \varSigma \) we write
$$\begin{aligned} \partial _\sigma T_\varSigma =\partial _{\varSigma (\sigma )} T_\varSigma \end{aligned}$$
and
$$\begin{aligned} \partial ^0_\sigma T_\varSigma =\partial _\sigma T_\varSigma - \bigcup _{\sigma ' \supsetneq \sigma } \partial _{\sigma '} T_\varSigma \end{aligned}$$
Sometimes we also write
$$\begin{aligned} \partial _{\{0\}}^0T_\varSigma =T. \end{aligned}$$
If \(\varSigma _0\) is locally finite then the \(\partial _\sigma T_\varSigma \) for \(\sigma \in \varSigma _0\) form a locally finite closed cover of \(\partial _{\varSigma _0} T_\varSigma \). Set theoretically we have
$$\begin{aligned} \partial _\sigma T_\varSigma =\coprod _{\sigma ' \in \varSigma (\sigma )} \partial _{\sigma '}^0T_\varSigma \end{aligned}$$
and
$$\begin{aligned} (\partial _{\varSigma _0}T_\varSigma )_\sigma =\coprod _{\begin{array}{c} \sigma ' \in \varSigma _0 \\ \sigma ' \subset \sigma \end{array}} \partial _{\sigma '}^0 T_\varSigma \end{aligned}$$
and
$$\begin{aligned} T_\sigma =\coprod _{\sigma ' \subset \sigma } \partial _{\sigma '}^0 T_\varSigma \end{aligned}$$
and
$$\begin{aligned} \partial _{\varSigma _0} T_\varSigma =\coprod _{\sigma ' \in \varSigma _0} \partial _{\sigma '}^0T_\varSigma . \end{aligned}$$
If \(\dim \sigma =1\) then \(\partial _\sigma ^0 T_\varSigma = \partial T_\sigma \).
Keep the notation of the previous paragraph. We define \(S(\sigma )\) to be the split torus with cocharacter group \(X_*(S)\) divided by the subgroup generated by \(\sigma \cap X_*(S)\), and \(T(\sigma )\) to be the push-out of T to \(S(\sigma )\). We also define \(\overline{\varSigma }(\sigma )\) to be the set of images in \(X_*(S(\sigma ))_{\mathbb {R}}\) of elements of \(\varSigma (\sigma )\). It is a fan for \(X_*(S)_{\mathbb {R}}/\langle \sigma \rangle _{\mathbb {R}}\). [The main point to check is that if \(\tau , \tau ' \in \varSigma (\sigma )\) then \((\tau \cap \tau ') +\langle \sigma \rangle _{\mathbb {R}}= (\tau +\langle \sigma \rangle _{\mathbb {R}}) \cap (\tau ' + \langle \sigma \rangle _{\mathbb {R}})\). To verify this suppose that \(x \in \tau \) and \(y \in \tau '\) with \(x-y \in \langle \sigma \rangle _{\mathbb {R}}\). Then \(x-y=z-w\) with \(z ,w \in \sigma \). Thus \(x+w=y+z \in \tau \cap \tau '\) and \(x+\langle \sigma \rangle _{\mathbb {R}}=(x+w) +\langle \sigma \rangle _{\mathbb {R}}\).] If \(\sigma \in \varSigma _0\) we will sometimes write \(\overline{\varSigma }_0(\sigma )\) for \(\overline{\varSigma }(\sigma )\), as it depends only on \(\varSigma _0\) and not on \(\varSigma \). Then
$$\begin{aligned} \partial ^0_\sigma T_\varSigma \cong T(\sigma ) \subset T(\sigma )_{\overline{\varSigma }(\sigma )} \cong \partial _\sigma T_\varSigma . \end{aligned}$$
Thus \(\partial _\sigma T_\varSigma \) is separated, locally (on the source) of finite type and flat over Y. The closed subscheme \(\partial _\sigma T_\varSigma \) has codimension in \(T_\varSigma \) equal to the dimension of \(\sigma \). If \(\varSigma (\sigma )\) is smooth then \(\partial _\sigma T_\varSigma \) is smooth over Y.

If \(\varSigma (\sigma )\) is open then \(\partial _\sigma T_\varSigma \) satisfies the valuative criterion of properness over Y. If in addition \(\varSigma (\sigma )\) is finite then \(\partial _\sigma T_\varSigma \) is proper over Y. If \(\varSigma _0\) is open, then \(\partial _{\varSigma _0} T_\varSigma \) satisfies the valuative criterion of properness over Y. If in addition \(\varSigma _0\) is finite then \(\partial _{\varSigma _0}T_\varSigma \) is proper over Y.

The schemes \(\partial _{\sigma _1} T_\varSigma ,\ldots ,\partial _{\sigma _s} T_\varSigma \) intersect if and only if \(\sigma _1, \ldots , \sigma _s\) are all contained in some \(\sigma \in \varSigma \). In this case the intersection equals \(\partial _\sigma T_\varSigma \) for the smallest such \(\sigma \). We set
$$\begin{aligned} \partial _i T_\varSigma =\coprod _{\dim \sigma =i} \partial _\sigma T_\varSigma . \end{aligned}$$
If Y is irreducible then \(T_\varSigma \) and each \(\partial _\sigma T_\varSigma \) is irreducible. Moreover the irreducible components of \(\partial T_\varSigma \) are the \(\partial _\sigma T_\varSigma \) as \(\sigma \) runs over one-dimensional elements of \(\varSigma \). If \(\varSigma \) is smooth then we see that \({\mathcal {S}}(\partial T_\varSigma )\) is the delta set with cells in bijection with the elements of \(\varSigma -\{ \{0\}\}\) and with the same ‘face relations’. In particular it is in fact a simplicial complex and
$$\begin{aligned} |{\mathcal {S}}(\partial T_\varSigma )|=(|\varSigma |-\{0\})/{\mathbb {R}}^{\times }_{>0}. \end{aligned}$$
We say that \((\varSigma ',\varSigma _0')\) refines \((\varSigma ,\varSigma _0)\) if \(\varSigma '\) refines \(\varSigma \) and \(\varSigma '-\varSigma _0'\) is the set of elements of \(\varSigma '\) contained in some element of \(\varSigma -\varSigma _0\). In this case \(\partial _{\varSigma _0'}T_{\varSigma '}\) maps to \(\partial _{\varSigma _0} T_{\varSigma }\), and in fact set theoretically \(\partial _{\varSigma _0'}T_{\varSigma '}\) is the pre-image of \(\partial _{\varSigma _0} T_{\varSigma }\) in \(T_{\varSigma '}\).
If \(\varSigma \) is a fan, then by line bundle data for \(\varSigma \) we mean a continuous function \(\psi : |\varSigma | \rightarrow {\mathbb {R}}\), such that for each cone \(\sigma \in \varSigma \), the restriction \(\psi |_{\sigma }\) equals some \(\psi _{\sigma } \in X^*(S)\). To \(\psi \) we can attach a line bundle \({\mathcal {L}}_\psi \) on \(T_\varSigma \): On \(T_{\sigma }\) (with \(\sigma \in \varSigma \)) it corresponds to the \(\bigoplus _{\chi \in \sigma ^\vee \cap X^*(S)} {\mathcal {L}}_T(\chi )\)-module
$$\begin{aligned} \bigoplus _{\begin{array}{c} \chi \in X^*(S) \\ \chi -\psi \ge 0 \,\,\mathrm{on}\,\, \sigma \end{array}} {\mathcal {L}}_T(\chi ). \end{aligned}$$
Note that there are natural isomorphisms
$$\begin{aligned} {\mathcal {L}}_\psi \otimes {\mathcal {L}}_{\psi '}\cong {\mathcal {L}}_{\psi +\psi '}, \end{aligned}$$
and that
$$\begin{aligned} {\mathcal {L}}_\psi ^{\otimes -1} \cong {\mathcal {L}}_{-\psi }. \end{aligned}$$
We have the following examples of line bundle data.
  1. (1)

    \({\mathcal {O}}_{T_\varSigma }\) is the line bundle associated to \(\psi \equiv 0\).

     
  2. (2)
    If \(\varSigma \) is smooth then \({\mathcal {I}}_{\partial T_\varSigma }\) is the line bundle associated to the unique such function \(\psi _\varSigma \) which for every one-dimensional cone \(\sigma \in \varSigma \) satisfies
    $$\begin{aligned} \psi _{\varSigma }(X_*(S) \cap \sigma )={\mathbb {Z}}_{\ge 0}. \end{aligned}$$
     
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori over Y. Then \(X^*(\alpha ): X^*(S') \hookrightarrow X^*(S)\) and \(X_*(\alpha ): X_*(S) \rightarrow X_*(S')\), the latter with finite cokernel. We call fans \(\varSigma \) for \(X_*(S)\) and \(\varSigma '\) for \(X_*(S')\) compatible if for all \(\sigma \in \varSigma \) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma '\). In this case the map \(\alpha : T \rightarrow \alpha _* T\) extends to an S-equivariant map
$$\begin{aligned} \alpha : T_\varSigma \longrightarrow (\alpha _*T)_{\varSigma '}. \end{aligned}$$
We will write
$$\begin{aligned} \varOmega _{T_\varSigma /(\alpha _* T)_{\varSigma '}}^1(\log \infty )=\varOmega _{T_\varSigma /(\alpha _* T)_{\varSigma '}}^1(\log {\mathcal {M}}_\varSigma /{\mathcal {M}}_{\varSigma '}). \end{aligned}$$
If for all \(\sigma ' \in \varSigma '\) the pre-image \(X_*(\alpha )^{-1}(\sigma ')\) is a finite union of elements of \(\varSigma \), then \(\alpha : T_\varSigma \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.

If \(\alpha \) is an isogeny, if \(\varSigma \) and \(\varSigma '\) are compatible, and if every element of \(\varSigma '\) is a finite union of elements of \(\varSigma \), then we call \(\varSigma \) a quasi-refinement of \(\varSigma '\). In that case the map \(\alpha : T_{\varSigma } \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.

Lemma 2.5

If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then
$$\begin{aligned} \alpha : (T_\varSigma ,{\mathcal {M}}_\varSigma ) \rightarrow ((\alpha _*T)_{\varSigma '},{\mathcal {M}}_{\varSigma '}) \end{aligned}$$
is log smooth, and there is a natural isomorphism
$$\begin{aligned} (X^*(S)/X^*(\alpha )X^*(S')) \otimes _{\mathbb {Z}}{\mathcal {O}}_{T_\varSigma } \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{T_\varSigma /(\alpha _*T)_{\varSigma '}}(\log \infty ). \end{aligned}$$

Proof

We can work Zariski locally on \(T_\varSigma \). Thus we may replace \(T_\varSigma \) by \(T_\sigma \) and \((\alpha _* T)_{\varSigma '}\) by \((\alpha _*T)_{\sigma '}\) for cones \(\sigma \) and \(\sigma '\) with \(X_*(\alpha )\sigma \subset \sigma '\). We may also replace Y by an affine open subset U such that \(T|_U\) is trivial, i.e. each \({\mathcal {L}}_T(\chi ) \cong {\mathcal {O}}_Y\) compatibly with \({\mathcal {L}}_T(\chi ) \otimes {\mathcal {L}}_T(\chi ') \mathop {\rightarrow }\limits ^{\sim }{\mathcal {L}}_T(\chi +\chi ')\). Then the log structure on \(T_\sigma \) has a chart \({\mathbb {Z}}[\sigma ^\vee \cap X^*(S)] \rightarrow {\mathcal {O}}_{T_{\sigma }}\) sending \(\chi \) to
$$\begin{aligned} 1 \in {\mathcal {O}}_Y(Y) \cong {\mathcal {L}}_T(\chi ). \end{aligned}$$
Similarly the log structure on \((\alpha _*T)_{\sigma '}\) has a chart \({\mathbb {Z}}[(\sigma ')^\vee \cap X^*(S')] \rightarrow {\mathcal {O}}_{(\alpha _*T)_{\sigma '}}\) sending \(\chi \) to
$$\begin{aligned} 1 \in {\mathcal {O}}_Y(Y) \cong {\mathcal {L}}_{\alpha _*T}(\chi ). \end{aligned}$$
The lemma follows because
$$\begin{aligned} X^*(\alpha ): X^*(S') \longrightarrow X^*(S) \end{aligned}$$
is injective and the torsion subgroup of the cokernel is finite with order invertible on Y. \(\square \)
We will call pairs \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) of fans and boundary data for S and \(S'\), respectively, compatible if \(\varSigma \) and \(\varSigma '\) are compatible and if no cone of \(\varSigma _0\) maps into any cone of \(\varSigma '-\varSigma _0'\). In this case
$$\begin{aligned} \partial _{\varSigma _0} T_\varSigma \longrightarrow \partial _{\varSigma _0'} (\alpha _*T)_{\varSigma '}. \end{aligned}$$
We will call them strictly compatible if they are compatible and \(\varSigma -\varSigma _0\) is the set of cones in \(\varSigma \) mapping into some element of \(\varSigma '-\varSigma _0'\).

Lemma 2.6

Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori, that T / Y is an S-torsor and that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible fans with boundary data for S and \(S'\), respectively. Then locally on \(T_\varSigma \) there is a strictly positive integer m such that
$$\begin{aligned} \alpha ^* {\mathcal {I}}_{\partial _{\varSigma _0'}(\alpha _* T)_{\varSigma '}} \supset {\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma }^m \end{aligned}$$
and
$$\begin{aligned} {\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma } \supset \alpha ^* {\mathcal {I}}_{\partial _{\varSigma _0'}(\alpha _* T)_{\varSigma '}}. \end{aligned}$$

Proof

We may work locally on Y and so we may suppose that \(Y={{\text {Spec}}\,}A\) is affine and that each \({\mathcal {L}}_T(\chi )\) is trivial. It also suffices to check the lemma locally on \(T_{\varSigma }\). Thus we may suppose that \(\varSigma \) consists of a cone \(\sigma \) and all its faces. Let \(\sigma '\) denote the smallest element of \(\varSigma '\) containing the image of \(\sigma \). Then we may further suppose that \(\varSigma '\) consists of \(\sigma '\) and all its faces. We may further suppose that \(\sigma \in \varSigma _0\) and \(\sigma ' \in \varSigma _0'\), else there is nothing to prove.

Then
$$\begin{aligned} T_\varSigma ={{\text {Spec}}\,}\bigoplus _{\chi \in X^*(S) \cap \sigma ^\vee } {\mathcal {L}}_T(\chi ) \end{aligned}$$
and \(\partial _{\varSigma _0} T_\varSigma \) is defined by
$$\begin{aligned} \bigoplus _{\chi \in X^*(S) \cap |\varSigma _0|^{0,\vee ,0}} {\mathcal {L}}_T(\chi ). \end{aligned}$$
Moreover \(T_\varSigma \times _{(\alpha _* T)_{\varSigma '}} \partial _{\varSigma _0'} (\alpha _*T)_{\varSigma '}\) is defined by
$$\begin{aligned} \bigoplus _{\begin{array}{c} \chi _1 \in X^*(S') \cap |\varSigma _0'|^{0,\vee ,0} \\ \chi _2 \in X^*(S) \cap \sigma ^\vee \end{array}} {\mathcal {L}}_T(X^*(\alpha )\chi _1 + \chi _2). \end{aligned}$$
Thus it suffices to show that for some positive integer m we have
$$\begin{array}{rcl} X^*(S) \cap |\varSigma _0|^{0,\vee ,0} &{}\supset &{}X^* (\alpha )(X^*(S') \cap |\varSigma _0'|^{0,\vee ,0}) + \left( X^*(S) \cap \sigma ^\vee \right) \\ &{} \supset &{} m (X^*(S) \cap |\varSigma _0|^{0,\vee ,0}). \end{array}$$
This is equivalent to
$$\begin{aligned} |\varSigma _0|^{0,\vee ,0}= X^*(\alpha )|\varSigma _0'|^{0,\vee ,0} + \sigma ^\vee . \end{aligned}$$
Suppose that \(\chi _1 \in |\varSigma _0'|^{0,\vee ,0}\) and \(\chi _2 \in \sigma ^\vee \). Then
$$\begin{aligned} X^*(\alpha )(\chi _1) (\sigma - |\varSigma -\varSigma _0|)=\chi _1(X_*(\alpha )(\sigma - |\varSigma -\varSigma _0|))\subset \chi _1(\sigma '-|\varSigma '-\varSigma _0'|) \subset {\mathbb {R}}_{>0} \end{aligned}$$
and so
$$\begin{aligned} (X^*(\alpha )(\chi _1)+\chi _2) (\sigma - |\varSigma -\varSigma _0|) \subset {\mathbb {R}}_{>0}. \end{aligned}$$
Thus
$$\begin{aligned} |\varSigma _0|^{0,\vee ,0}\supset X^*(\alpha )|\varSigma _0'|^{0,\vee ,0} + \sigma ^\vee . \end{aligned}$$
Conversely suppose that \(\chi \in |\varSigma _0|^{0,\vee ,0}\). Let \(\tau \) denote the face of \(\sigma \), where \(\chi =0\). Then \(\tau \in \varSigma -\varSigma _0\). Let \(\tau '\) denote the smallest face of \(\sigma '\) containing \(X_*(\alpha ) \tau \). Then \(\tau ' \in \varSigma '-\varSigma _0'\). We can find \(\chi _1 \in |\varSigma _0'|^{0,\vee ,0}\) with \(\chi _1(\tau ')=\{0\}\). Note that if \(a \in \sigma \) and \(\chi (a)=0\) then \((X^*(\alpha )(\chi _1))(a)=0\). Thus we can find \(\epsilon >0\) such that
$$\begin{aligned} \chi - X^*(\alpha )(\epsilon \chi _1) \in \sigma ^\vee . \end{aligned}$$
It follows that
$$\begin{aligned} |\varSigma _0|^{0,\vee ,0}\subset X^*(\alpha )|\varSigma _0'|^{0,\vee ,0} + \sigma ^\vee . \end{aligned}$$
The lemma follows. \(\square \)
Suppose that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible. We will say that
  • \(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0\);

  • and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).

If \(\alpha \) is an isogeny, if \(\varSigma \) is a quasi-refinement of \(\varSigma '\) and if \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible, then we call \((\varSigma ,\varSigma _0)\) a quasi-refinement of \((\varSigma ',\varSigma _0')\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\).

Lemma 2.7

Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori, that T / Y is an S-torsor, and that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible fans with boundary data for S and \(S'\), respectively. If \(\varSigma _0\) is locally finite and \(\varSigma _0\) is open over \(\varSigma _0'\) then
$$\begin{aligned} \partial _{\varSigma _0} T_\varSigma \longrightarrow \partial _{\varSigma _0'} (\alpha _*T)_{\varSigma '} \end{aligned}$$
satisfies the valuative criterion of properness. If in addition \(\varSigma _0\) is finite over \(\varSigma _0'\) then this morphism is proper.

Proof

It suffices to show that if \(\sigma \in \varSigma _0\) and if \(\sigma '\) is the smallest element of \(\varSigma '_0\) containing \(X_*(\alpha ) \sigma \), then
$$\begin{aligned} \partial _{\sigma } T_\varSigma \longrightarrow \partial _{\sigma '} (\alpha _*T)_{\varSigma '} \end{aligned}$$
satisfies the valuative criterion of properness. However this is the map of toric varieties
$$\begin{aligned} T(\sigma )_{\overline{\varSigma }_0(\sigma )} \longrightarrow (\alpha _*T)(\sigma ')_{\overline{\varSigma }_0'(\sigma ')}. \end{aligned}$$
As \(\overline{\varSigma }_0(\sigma )\) is finite, it suffices to check that
$$\begin{aligned} \bigcup _{\begin{array}{c} \tau ' \supset \sigma ' \\ \tau ' \in \varSigma _0' \end{array}} X_*(\alpha )^{-1}\left( (\tau ')^0+ \langle \sigma ' \rangle _{\mathbb {R}}\right) =\bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \left( \tau ^0+\langle \sigma \rangle _{\mathbb {R}}\right) . \end{aligned}$$
Choose a point \(P \in \sigma ^0\) such that
$$\begin{aligned} X_*(\alpha )P \in \left( X_*(\alpha )\sigma \right) ^0 \subset (\sigma ')^0. \end{aligned}$$
Then
$$\begin{aligned} \langle \sigma ' \rangle _{{\mathbb {R}}}=\sigma ' + {\mathbb {R}}X_*(\alpha )(P). \end{aligned}$$
[To see this choose nonzero vectors \(v_i\) in each one-dimensional face of \(\sigma '\). Then we can write \(X_*(\alpha )(P)=\sum _i \mu _i v_i\) with each \(\mu _i > 0\). If \(\lambda _i \in {\mathbb {R}}\), then for \(\lambda \) sufficiently large \(\lambda _i+\lambda \mu _i \in {\mathbb {R}}_{>0}\) for all i, and so
$$\begin{aligned} \left. \sum _i \lambda _i v_i=\sum _i (\lambda _i + \lambda \mu _i)v_i -\lambda X_*(\alpha )(P) \in \sigma ' + {\mathbb {R}}X_*(\alpha )(P).\right] \end{aligned}$$
Thus
$$\begin{aligned} \langle \sigma '\rangle _{\mathbb {R}}=\sigma ' + X_*(\alpha )\langle \sigma \rangle _{\mathbb {R}}. \end{aligned}$$
Hence for all \(\tau ' \in \varSigma _0'\) with \(\tau ' \supset \sigma '\), we have
$$\begin{aligned} (\tau ')^0+ \langle \sigma '\rangle _{\mathbb {R}}=(\tau ')^0 + X_*(\alpha )\langle \sigma \rangle _{\mathbb {R}}\end{aligned}$$
and so
$$\begin{aligned} X_*(\alpha )^{-1}((\tau ')^0+\langle \sigma ' \rangle _{\mathbb {R}})= \langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1} (\tau ')^0. \end{aligned}$$
We deduce that it suffices to check that
$$\begin{aligned} \langle \sigma \rangle _{\mathbb {R}}+ \bigcup _{\begin{array}{c} \tau ' \supset \sigma ' \\ \tau ' \in \varSigma _0' \end{array}} X_*(\alpha )^{-1}(\tau ')^0=\langle \sigma \rangle _{\mathbb {R}}+\bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0. \end{aligned}$$
The left hand side certainly contains the right hand side, so it suffices to prove that for all \(\tau ' \in \varSigma _0'\) with \(\tau ' \supset X_*(\alpha )\sigma \) we have
$$\langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1} \tau ' \subset \langle \sigma \rangle _{\mathbb {R}}+ \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0.$$
Let \(\pi \) denote the map
$$\begin{aligned} \pi : X_*(S)_{\mathbb {R}}\twoheadrightarrow X_*(S)_{\mathbb {R}}/\langle \sigma \rangle _{\mathbb {R}}. \end{aligned}$$
Because \(X_*(\alpha )^{-1} \tau '\) and \(\bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0\) are invariant under the action of \({\mathbb {R}}^\times _{>0}\) it suffices to find an open set \(U \subset X_*(S)_{\mathbb {R}}\) containing P such that
$$\begin{aligned} (\pi U) \cap \pi X_*(\alpha )^{-1} \tau ' \subset \pi \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0, \end{aligned}$$
or equivalently such that
$$\begin{aligned} U \cap (\langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1} \tau ') \subset \langle \sigma \rangle _{\mathbb {R}}+ \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0. \end{aligned}$$
Thus it suffices to find an open set \(U \subset X_*(S)_{\mathbb {R}}\) containing P such that
  1. (1)

    \(U \cap X_*(\alpha )^{-1}|\varSigma _0'|^0 \subset \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0\);

     
  2. (2)

    \(U \cap X_*(\alpha )^{-1} \tau ' \subset X_*(\alpha )^{-1} |\varSigma _0'|^0\);

     
  3. (3)

    and for all open \(U' \subset U\) containing P we have \(U' \cap (\langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1} \tau ')=U' \cap X_*(\alpha )^{-1} \tau '\).

     
Moreover in order to find such a \(U \ni P\) it suffices to find one satisfying each property independently and take their intersection.
One can find an open set \(U \ni P\) satisfying the first property because
$$\begin{aligned} \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0 \subset |\varSigma _0|^0 \subset X^*(\alpha )^{-1}|\varSigma _0'|^0 \end{aligned}$$
are both open inclusions.

To find \(U \ni P\) satisfying the second condition we just need to avoid the faces of \(X_*(\alpha )^{-1}\tau '\) which do not contain P.

It remains to check that we can find an open \(U \ni P\) satisfying the last condition. Suppose that \(X_*(\alpha )^{-1}\tau '\) is defined by inequalities \(\chi _i \ge 0\) for \(i=1,\ldots ,r\) with \(\chi _i \in X^*(S)_{\mathbb {R}}\). Suppose that \(\chi _i=0\) on \(\sigma \) for \(i=1,\ldots ,s\), but that \(\chi _i(P)>0\) for \(i=s+1,\ldots ,r\). It suffices to choose U so that \(\chi _i>0\) on U for \(i=s+1,\ldots ,r\). For then if \(x \in X_*(\alpha )^{-1} \tau '\) and \(y \in \langle \sigma \rangle _{\mathbb {R}}\) with \(x+y \in U\) we see that
$$\begin{aligned} \chi _i(x+y)=\chi _i(x) \ge 0 \end{aligned}$$
for \(i=1,\ldots ,s\), while \(\chi _i(x+y)>0\) for \(i=s+1,\ldots ,r\). Thus for \(U' \subset U\) we have
$$\begin{aligned} U' \cap \left( \langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1}\tau ' \right) =U' \cap X_*(\alpha )^{-1} \tau ', \end{aligned}$$
as desired. \(\square \)
By a partial fan we will mean a collection \(\varSigma _0\) of rational polyhedral cones satisfying
  • \((0) \not \in \varSigma _0\);

  • if \(\sigma _1, \sigma _2 \in \varSigma _0\), then \(\sigma _1 \cap \sigma _2\) is a face of \(\sigma _1\) and of \(\sigma _2\);

  • if \(\sigma _1, \sigma _2 \in \varSigma _0\), and if \(\sigma \supset \sigma _2\) is a face of \(\sigma _1\), then \(\sigma \in \varSigma _0\).

(Again note that \(\varSigma _0\) may not be closed under taking faces.) In this case we will let \(\widetilde{\varSigma }_0\) denote the set of faces of elements of \(\varSigma _0\) together with \(\{ 0\}\). Then \(\widetilde{\varSigma }_0\) and \(\widetilde{\varSigma }_0-\varSigma _0\) are fans, and \(\varSigma _0\) is boundary data for \(\widetilde{\varSigma }_0\). [To see this suppose that \(\tau _i\) is a face of \(\sigma _i \in \varSigma _0\) for \(i=1,2\). Then \(\sigma _1 \cap \sigma _2\) is a face of \(\sigma _1\) and so \(\tau _1 \cap \sigma _2 =\tau _1 \cap (\sigma _1 \cap \sigma _2)\) is a face of \(\sigma _1 \cap \sigma _2\) and hence of \(\sigma _2\). Thus \(\tau _1 \cap \tau _2=\tau _2 \cap (\tau _1 \cap \sigma _2)\) is a face of \(\tau _2\).] If \(\varSigma \) is a fan and \(\varSigma _0\) is boundary data for \(\varSigma \), then \(\varSigma _0\) is a partial fan, and \(\varSigma \supset \widetilde{\varSigma }_0\). Thus
$$\begin{aligned} \partial _{\varSigma _0} T_\varSigma \cong \partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}. \end{aligned}$$
If \(\varSigma _0\) and \(\varSigma _0'\) are partial fans we will say that \(\varSigma _0\) refines \(\varSigma _0'\) if every element of \(\varSigma _0\) is contained in an element of \(\varSigma _0'\) and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\). In this case \(\widetilde{\varSigma }_0\) also refines \(\widetilde{\varSigma }_0'\).
If \(\varSigma _0\) is a partial fan we will set
$$\begin{aligned} |\varSigma _0|=\{ 0\} \cup \bigcup _{\sigma \in \varSigma _0} \sigma =|\widetilde{\varSigma }_0|. \end{aligned}$$
and
$$\begin{aligned} |\varSigma _0|^0=|\varSigma _0|- \bigcup _{ \sigma \in \widetilde{\varSigma }_0-\varSigma _0} \sigma . \end{aligned}$$
We will call \(\varSigma _0\)
  • smooth if each \(\sigma \in \varSigma _0\) is smooth;

  • full if every element of \(\varSigma _0\) which is not a face of any other element of \(\varSigma _0\), has the same dimension as S;

  • open if \(|\varSigma _0|^0\) is open in \(X_*(S)_{\mathbb {R}}\);

  • finite if it has finite cardinality;

  • locally finite if for every rational polyhedral cone \(\tau \subset |\varSigma _0|\) (not necessarily in \(\varSigma _0\)) the intersection \(\tau \cap |\varSigma _0|^0\) meets only finitely many elements of \(\varSigma _0\).

If \(\varSigma _0\) is smooth, so is \(\widetilde{\varSigma }_0\).
Suppose that \(\varSigma _0\) is a partial fan. If \(\varSigma \supset \widetilde{\varSigma }_0\) is a fan then the natural maps
$$\begin{aligned} \partial _{\varSigma _0} T_{\widetilde{\varSigma }_0} \longrightarrow \partial _{\varSigma _0} T_\varSigma \end{aligned}$$
and
$$\left( T_{\widetilde{\varSigma }_0}\right) ^\wedge _{\partial _{\varSigma _0}T} \longrightarrow \left( T_{\varSigma }\right) ^\wedge _{\partial _{\varSigma _0}T}$$
are isomorphisms, and we will denote these schemes/formal schemes \(\partial _{\varSigma _0}T\) and \(T_{\varSigma _0}^\wedge \), respectively. Moreover the log structures induced on \(T_{\varSigma _0}^\wedge \) by \({\mathcal {M}}_{\widetilde{\varSigma }_0}\) and by \({\mathcal {M}}_\varSigma \) are the same and we will denote them \({\mathcal {M}}^\wedge _{\varSigma _0}\). If \(\varSigma _0' \subset \varSigma _0\) is also a partial fan, then \(T_{\varSigma _0'}^\wedge \) can be identified with the completion of \(T_{\varSigma _0}^\wedge \) along \(\partial _{\varSigma _0'}T\), and \({\mathcal {M}}_{\varSigma _0}^\wedge \) induces \({\mathcal {M}}_{\varSigma _0'}^\wedge \). If \(\sigma \in \widetilde{\varSigma }_0\) then we will let
$$\begin{aligned} \left( T_{\varSigma _0}^\wedge \right) _\sigma \end{aligned}$$
denote the restriction of \(T_{\varSigma _0}^\wedge \) to the topological space \((\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0})_\sigma \). Thus the \((T_{\varSigma _0}^\wedge )_\sigma \) for \(\sigma \in \varSigma _0\) form an affine open cover of \(T_{\varSigma _0}^\wedge \). We have
$$\begin{aligned} \left( T_{\varSigma _0}^\wedge \right) _{\{ 0\}}= \emptyset \end{aligned}$$
and
$$\left( T_{\varSigma _0}^\wedge \right) _{\sigma _1} \cap \left( T_{\varSigma _0}^\wedge \right) _{\sigma _2}= \left( T_{\varSigma _0}^\wedge \right) _{\sigma _1 \cap \sigma _2}.$$
If \(\varSigma _0'\) refines \(\varSigma _0\) then there is an induced map
$$\begin{aligned} T^\wedge _{\varSigma _0'} \longrightarrow T^\wedge _{\varSigma _0}. \end{aligned}$$
Continue to suppose that \(\varSigma _0\) is a partial fan. We will call \(\varSigma _1 \subset \varSigma _0\) boundary data if, whenever \(\sigma \in \varSigma _0\) contains \(\sigma ' \in \varSigma _1\), then \(\sigma \in \varSigma _1\). In this case \(\varSigma _1\) is a partial fan and \(T_{\varSigma _1}^\wedge \) is canonically identified with the completion of \(T_{\varSigma _0}^\wedge \) along \(\partial _{\varSigma _1} T_{\widetilde{\varSigma }_0}\).
We will also use the following notation.
  • \({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) will denote the structure sheaf of \(T_{\varSigma _0}^\wedge \).

  • \({\mathcal {I}}_{T_{\varSigma _0}^\wedge }\) will denote the completion of \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}\), an ideal of definition for \(T_{\varSigma _0}^\wedge \).

  • \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) will denote the completion of \({\mathcal {I}}_{\partial T_{\widetilde{\varSigma }_0}}\). Thus \( {\mathcal {I}}_{T_{\varSigma _0}^\wedge } \supset {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \).

  • \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\) will denote \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log {\mathcal {M}}_\varSigma ^\wedge )\), which is isomorphic to the completion of \(\varOmega ^1_{T_{\widetilde{\varSigma }_0}/{{\text {Spec}}\,}R_0}(\log \infty )\).

For \(\sigma \in \widetilde{\varSigma }_0\) recall that \({\mathcal {I}}_{\partial _{\varSigma _0}T_{\widetilde{\varSigma }_0}}^m|_{T_\sigma }\) corresponds to the ideal
$$\begin{aligned} \bigoplus _{\chi \in {\mathfrak {X}}_{\varSigma _0,\sigma ,m}} {\mathcal {L}}_T(\chi ) \end{aligned}$$
of
$$\begin{aligned} \bigoplus _{\chi \in \sigma ^\vee \cap X^*(S)} {\mathcal {L}}_T(\chi ). \end{aligned}$$
Also recall that if \(\sigma \not \in \varSigma _0\) then
$$\begin{aligned} {\mathfrak {X}}_{\varSigma _0,\sigma ,m}=\sigma ^\vee \cap X^*(S) \end{aligned}$$
for all m, while if \(\sigma \in \varSigma _0\) then
$$\begin{aligned} \bigcap _m {\mathfrak {X}}_{\varSigma _0,\sigma ,m}=\emptyset . \end{aligned}$$
By line bundle data for \(\varSigma _0\) we mean a continuous function \(\psi : |\varSigma _0| \rightarrow {\mathbb {R}}\), such that for each cone \(\sigma \in \widetilde{\varSigma }_0\), the restriction \(\psi |_{\sigma }\) equals some \(\psi _{\sigma } \in X^*(S)\). This is the same as line bundle data for the fan \(\widetilde{\varSigma }_0\), and we will write \({\mathcal {L}}_\psi ^\wedge \) for the line bundle on \(T_{\varSigma _0}^\wedge \), which is the completion of \({\mathcal {L}}_\psi /T_{\widetilde{\varSigma }_0}\). Note that
$$\begin{aligned} {\mathcal {L}}_\psi ^\wedge \otimes {\mathcal {L}}_{\psi '}^\wedge = {\mathcal {L}}_{\psi +\psi '}^\wedge , \end{aligned}$$
and that
$$\begin{aligned} ({\mathcal {L}}_\psi ^\wedge )^{\otimes -1}={\mathcal {L}}_{-\psi }^\wedge . \end{aligned}$$
We have the following examples of line bundle data.
  1. (1)

    \({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) is the line bundle associated to \(\psi \equiv 0\).

     
  2. (2)
    If \(\varSigma _0\) is smooth then \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) is the line bundle associated to the unique such function \(\psi _{\widetilde{\varSigma }_0}\) which for every one-dimensional cone \(\sigma \in \widetilde{\varSigma }_0\) satisfies
    $$\begin{aligned} \psi _{\widetilde{\varSigma }_0}(X_*(S) \cap \sigma )={\mathbb {Z}}_{\ge 0}. \end{aligned}$$
     
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of tori, and that \(\varSigma _0\) (resp. \(\varSigma _0'\)) is a partial fan for S (resp. \(S'\)). We call \(\varSigma _0\) and \(\varSigma _0'\) compatible if for every \(\sigma \in \varSigma _0\) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma _0'\) but in no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case \((\widetilde{\varSigma }_0,\varSigma _0)\) and \((\widetilde{\varSigma }_0',\varSigma _0')\) are compatible, and there is a natural morphism
$$\alpha : \left( T^\wedge _{\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( (\alpha _*T)_{\varSigma _0'}^\wedge ,{\mathcal {M}}_{\varSigma _0'}^\wedge \right) .$$
We will write
$$\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)_{\varSigma _0'}^\wedge }^1(\log \infty )=\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)^\wedge _{\varSigma _0'}}^1\left( \log {\mathcal {M}}_{\varSigma _0}^\wedge /{\mathcal {M}}^\wedge _{\varSigma _0'}\right) .$$
The following lemma follows immediately from Lemma 2.5.

Lemma 2.8

If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then there is a natural isomorphism
$$\begin{aligned} (X^*(S)/X^*(\alpha )X^*(S')) \otimes _{\mathbb {Z}}{\mathcal {O}}_{T_{\varSigma _0}^\wedge } \mathop {\longrightarrow }\limits ^{\sim }\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)_{\varSigma _0'}^\wedge }^1(\log \infty ) . \end{aligned}$$
We will call \(\varSigma _0\) and \(\varSigma _0'\) strictly compatible if they are compatible and if an element of \(\widetilde{\varSigma }_0\) lies in \(\varSigma _0\) if and only if it maps to no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case \((\widetilde{\varSigma }_0,\varSigma _0)\) and \((\widetilde{\varSigma }_0',\varSigma _0')\) are strictly compatible. We will say that
  • \(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0\);

  • and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).

If \(\alpha \) is an isogeny, if \(\varSigma _0\) and \(\varSigma _0'\) are strictly compatible and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\), then we call \(\varSigma _0\) a quasi-refinement of \(\varSigma _0'\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\). The next lemma follows immediately from Lemmas 2.6 and 2.7.

Lemma 2.9

Suppose that \(\varSigma _0'\) and \(\varSigma _0\) are strictly compatible.
  1. (1)

    \(T_{\varSigma _0}^\wedge \) is the formal completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0'}(\alpha _* T)\), and \(T_{\varSigma _0}^\wedge \) is locally (on the source) topologically of finite type over \((\alpha _*T)^\wedge _{\varSigma _0'}\).

     
  2. (2)

    If \(\varSigma _0\) is locally finite and if it is open and finite over \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).

     

Corollary 2.10

If \(\alpha \) is an isogeny, if \(\varSigma _0\) is locally finite and if \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).

If \(\varSigma _0\) and \(\varSigma _0'\) are compatible partial fans and if \(\varSigma _1' \subset \varSigma _0'\) is boundary data then \(\varSigma _0(\varSigma _1')\) will denote the set of elements \(\sigma \in \varSigma _0\) such that \(X_*(\alpha )\sigma \) is contained in no element of \(\varSigma _0'-\varSigma _1'\). It is boundary data for \(\varSigma _0\). Moreover the formal completion of \(T_{\varSigma _0}^\wedge \) along the reduced subscheme of \((\alpha _*T)_{\varSigma _1'}^\wedge \) is canonically identified with \(T^\wedge _{\varSigma _0(\varSigma _1')}\). If \(\varSigma _1'=\{ \sigma '\}\) is a singleton we will write \(\varSigma _0(\sigma ')\) for \(\varSigma _0(\{ \sigma '\})\).

3.4 Cohomology of line bundles

In this section we will compute the cohomology of line bundles on formal completions of torus embeddings. We will work throughout over a base scheme Y which is connected, separated and flat and locally of finite type over \({{\text {Spec}}\,}R_0\).

We start with some definitions. We continue to assume that S / Y is a split torus, that T / Y is an S-torsor, that \(\varSigma _0\) is a partial fan and that \(\psi \) is line bundle data for \(\varSigma _0\). If \(\sigma \in \widetilde{\varSigma }_0\) then we set
$$\begin{aligned} {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,0}=\left\{ \chi \in X^*(S) \cap \sigma ^\vee : \chi \ge \psi \,\, \mathrm{on} \,\, \sigma \right\} . \end{aligned}$$
For \(m>0\) we define \({\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,m}\) to be the set of sums of an element of \({\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,0}\) and an element of \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\). If \(\sigma \not \in \varSigma _0\) then
$$\begin{aligned} {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,m}={\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,0} \end{aligned}$$
for all m, while if \(\sigma \in \varSigma _0\)
$$\begin{aligned} \bigcap _m {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,m}=\emptyset . \end{aligned}$$
Further suppose that \(\chi \in X^*(S)\).
  • Set \(Y_\psi (\chi )=\{ x \in X^*(S)_{\mathbb {R}}: (\psi -\chi )(x)>0\} \).

  • If \(U \subset Y\) is open let \(H^j_{\varSigma _0,\psi ,m}(\chi )(U)\) denote the jth cohomology of the Cech complex with ith term
    $$\prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1} \\ \chi \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0 \cap \cdots \cap \sigma _i,0} \\ \chi \not \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0\cap \cdots \cap \sigma _i,m} \end{array}} {\mathcal {L}}_T(\chi )(U).$$
Note the examples:
  1. (1)

    \(Y_0(\chi )\cap |\varSigma _0|^0=\emptyset \) if and only if \(\chi \in |\varSigma _0|^\vee \).

     
  2. (2)

    \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0=\emptyset \) if and only if \(\chi \in |\varSigma _0|^{\vee ,0}\).

     
Also note that if \(\varSigma _0\) is finite then, for m large enough, \(H^j_{\varSigma _0,\psi ,m}(\chi )(U)\) does not depend on m. We will denote it simply \(H^j_{\varSigma _0,\psi }(\chi )(U)\). It equals the cohomology of the Cech complex
$$\prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1} \\ \sigma _0 \cap \cdots \cap \sigma _i \in \varSigma _0 \end{array}} {\mathcal {L}}_T(\chi )(U).$$

Lemma 2.11

If U is connected then
$$H_{\varSigma _0,\psi }^i(\chi )(U)=H^i_{|\varSigma _0|^0-Y_\psi (\chi )} \left( |\varSigma _0|^0,{\mathcal {L}}_T(\chi )(U)\right) .$$

Proof

Write M for \({\mathcal {L}}_T(\chi )(U)\). We follow the argument of section 3.5 of [26]. As \(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\) and \(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0 \cap Y_\psi (\chi )\) are convex, we see that
$$\begin{array}{rl} &{}H^j_{(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0)- Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, M\right) \\ &{}\quad = \left\{ \begin{array}{ll} M &{}\quad \mathrm{if}\,\, j=0 \,\, \mathrm{and} \,\, \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) \cap Y_\psi (\chi )= \emptyset \\ (0) &{}\quad \mathrm{otherwise}.\end{array} \right. \end{array}$$
(See the first paragraph of section 3.5 of [26].) Thus the ith term of our Cech complex becomes
$$\prod _{(\sigma _0,\ldots ,\sigma _i)\in \varSigma _0^{i+1}} H^0_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, M\right) .$$
Thus it suffices to show that the Cech complex with ith term
$$\prod _{(\sigma _0,\ldots ,\sigma _i )\in \varSigma _0^{i+1}} H^0_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, M\right) $$
computes
$$\begin{aligned} H^i_{|\varSigma _0|^0- Y_\psi (\chi )} (|\varSigma _0|^0, M). \end{aligned}$$
To this end choose an injective resolution
$$\begin{aligned} M \longrightarrow {\mathcal {I}}^0 \longrightarrow {\mathcal {I}}^1 \longrightarrow \cdots \end{aligned}$$
as sheaves of abelian groups on \(|\varSigma _0|^0\), and consider the double complex
$$\prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1}} H^0_{(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0)- Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, {\mathcal {I}}^j\right) .$$
We compute the cohomology of the corresponding total complex in two ways. Firstly the jth cohomology of the complex
$$\begin{array}{c} H^0_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, {\mathcal {I}}^0\right) \\ \downarrow \\ H^0_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, {\mathcal {I}}^1\right) \\ \downarrow \\ \vdots \end{array}$$
equals
$$H^j_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0,M\right) .$$
(See theorem 4.1, proposition 5.3 and theorem 5.5 of chapter II of [14].) This vanishes for \(j>0\), and so the cohomology of our total complex is the same as the cohomology of the Cech complex with ith term
$$\prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1}} H^0_{\left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\right) - Y_\psi (\chi )} \left( \sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0, M\right) .$$
Thus it suffices to identify the cohomology of our double complex with
$$\begin{aligned} H^i_{|\varSigma _0|^0- Y_\psi (\chi )} (|\varSigma _0|^0, M). \end{aligned}$$
For this it suffices to show that
$$\begin{array}{l} (0) \longrightarrow H^0_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,{\mathcal {I}}^j\right) \longrightarrow \prod _{\sigma _0 \in \varSigma _0} H^0_{\sigma _0 \cap |\varSigma _0|^0- Y_\psi (\chi )} \left( \sigma _0 \cap |\varSigma _0|^0, {\mathcal {I}}^j\right) \\ \quad \longrightarrow \prod _{(\sigma _0,\sigma _1) \in \varSigma _0^2} H^0_{(\sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0)- Y_\psi (\chi )} \left( \sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0, {\mathcal {I}}^j\right) \longrightarrow \cdots \end{array}$$
is exact for all j. Let \({\widetilde{{\mathcal {I}}}}^j\) denote the sheaf of discontinuous sections of \({\mathcal {I}}^j\), i.e. \({\widetilde{{\mathcal {I}}}}^j(V)\) denotes the set of functions which assign to each point of \(x \in V\) an element of the stalk \({\mathcal {I}}^j_x\) of \({\mathcal {I}}^j\) at x. Then \({\mathcal {I}}^j\) is a direct summand of \({\widetilde{{\mathcal {I}}}}^j\) so it suffices to show that
$$\begin{array}{l} (0) \longrightarrow H^0_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,{\widetilde{{\mathcal {I}}}}^j\right) \longrightarrow \prod _{\sigma _0 \in \varSigma _0} H^0_{\sigma _0 \cap |\varSigma _0|^0- Y_\psi (\chi )} \left( \sigma _0 \cap |\varSigma _0|^0, {\widetilde{{\mathcal {I}}}}^j\right) \\ \quad \longrightarrow \prod _{(\sigma _0,\sigma _1) \in \varSigma _0^2} H^0_{(\sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0)- Y_\psi (\chi )} \left( \sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0, {\widetilde{{\mathcal {I}}}}^j\right) \longrightarrow \cdots \end{array}$$
is exact for all j. However \({\widetilde{{\mathcal {I}}}}^j\) is the direct product over x in \(|\varSigma _0|^0\) of the sky-scraper \(\overline{{{\mathcal {I}}}}^j_x\) sheaf at x with stalk \({\mathcal {I}}^j_x\). Thus it suffices to show that
$$\begin{array}{l} (0) \longrightarrow H^0_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,\overline{{{\mathcal {I}}}}_x^j\right) \longrightarrow \prod _{\sigma _0 \in \varSigma _0} H^0_{\sigma _0 \cap |\varSigma _0|^0- Y_\psi (\chi )} \left( \sigma _0 \cap |\varSigma _0|^0, \overline{{{\mathcal {I}}}}_x^j\right) \\ \quad \longrightarrow \prod _{(\sigma _0,\sigma _1) \in \varSigma _0^2} H^0_{(\sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0)- Y_\psi (\chi )} \left( \sigma _0 \cap \sigma _1 \cap |\varSigma _0|^0, \overline{{{\mathcal {I}}}}_x^j\right) \longrightarrow \cdots \end{array}$$
is exact for all \(x \in |\varSigma _0|^0\) and for all j. If \(x \in Y_\psi (\chi )\cap |\varSigma _0|^0\) all the terms in this sequence are 0, so the sequence is certainly exact. If \(x \in |\varSigma _0|^0-Y_\psi (\chi )\), this sequence equals
$$\begin{aligned} (0) \longrightarrow {\mathcal {I}}_x^j \longrightarrow \prod _{\begin{array}{c} \sigma _0 \in \varSigma _0 \\ x \in \sigma \end{array}} {\mathcal {I}}_x^j \longrightarrow \prod _{\begin{array}{c} (\sigma _0,\sigma _1) \in \varSigma _0^2 \\ x \in (\sigma _0 \cap \sigma _1) \end{array}} {\mathcal {I}}_x^j \longrightarrow \cdots \end{aligned}$$
A standard argument shows that this is indeed exact: Choose \(\sigma \in \varSigma _0\) with \(x \in \sigma \). Suppose
$$\begin{aligned} (a(\sigma _0,\ldots ,\sigma _i)) \in \ker \left( \prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1} \\ x \in \sigma _0 \cap \cdots \cap \sigma _i \end{array}} {\mathcal {I}}_x^j \longrightarrow \prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _{i+1}) \in \varSigma _0^{i+2} \\ x \in \sigma _0 \cap \cdots \cap \sigma _{i+1} \end{array}} {\mathcal {I}}_x^j \right) . \end{aligned}$$
Define
$$\begin{aligned} \left( a'(\sigma _0,\ldots ,\sigma _{i-1})\right) \in \prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _{i-1}) \in \varSigma _0^{i} \\ x \in \sigma _0 \cap \cdots \cap \sigma _{i-1} \end{array}} {\mathcal {I}}_x^j \end{aligned}$$
by
$$\begin{aligned} a'(\sigma _0,\ldots ,\sigma _{i-1})=a(\sigma _0,\ldots ,\sigma _{i-1},\sigma ). \end{aligned}$$
If \(\partial a'\) denotes the image of \(a'\) in
$$\begin{aligned} \prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1} \\ x \in \sigma _0 \cap \cdots \cap \sigma _i \end{array}} {\mathcal {I}}_x^j \end{aligned}$$
then
$$\begin{aligned} (\partial a')(\sigma _0,\ldots ,\sigma _i)=\sum _{k=0}^i (-1)^k a(\sigma _0,\ldots ,\widehat{\sigma _k},\ldots ,\sigma _i,\sigma )=(-1)^i a(\sigma _0,\ldots ,\sigma _i), \end{aligned}$$
i.e. \(a=(-1)^i \partial a'\). \(\square \)
In general we will let \(H^i_{|\varSigma _0|^0-Y_\psi (\chi )}(|\varSigma _0|^0,{\mathcal {L}}_T(\chi ))\) denote the sheaf of \({\mathcal {O}}_Y\)-modules on Y associated to the pre-sheaf
$$U \longmapsto H^i_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,{\mathcal {L}}_T(\chi )(U)\right) .$$

Lemma 2.12

Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, let \(\psi \) be line bundle data for \(\varSigma _0\), and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is finite, non-empty and open. Then
$$\begin{aligned} R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {L}}_\psi ^\wedge =\prod _{\chi \in X^*(S)} H^i_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,{\mathcal {L}}_T(\chi )\right) . \end{aligned}$$
(Note that \(R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {L}}_\psi ^\wedge \) may not be quasi-coherent on Y. Infinite products of quasi-coherent sheaves may not be quasi-coherent.)

Proof

The left hand side is the sheaf associated to the pre-sheaf
$$\begin{aligned} U \longmapsto H^i\left( T_{\varSigma _0}^\wedge \big |_U,{\mathcal {L}}_\psi ^\wedge \right) \end{aligned}$$
and the right hand side is the sheaf associated to the pre-sheaf
$$U \longmapsto \prod _{\chi \in X^*(S)(U)} H^i_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0, {\mathcal {L}}_T(\chi )(U)\right) .$$
Thus it suffices to establish isomorphisms
$$H^i\left( T_{\varSigma _0}^\wedge \big |_U,{\mathcal {L}}_\psi ^\wedge \right) \cong \prod _{\chi \in X^*(S)(U)} H^i_{|\varSigma _0|^0-Y_\psi (\chi )}\left( |\varSigma _0|^0,{\mathcal {L}}_T(\chi )(U)\right) ,$$
compatibly with restriction, for \(U={{\text {Spec}}\,}A\), with A noetherian and \({{\text {Spec}}\,}A\) connected.
Write \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) for the closed subscheme of \(T_{\widetilde{\varSigma }_0}\) defined by \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m\). It has the same underlying topological space as \(\partial _{\varSigma _0}T_{\widetilde{\varSigma }_0}\). We will first compute
$$H^i\left( \partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\big |_U,{\mathcal {L}}_\psi \Big /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \right) ,$$
using the affine cover of \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) by the open sets \(T_\sigma \) for \(\sigma \in \varSigma _0\). This gives rise to a Cech complex with terms
$$\prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1}} \bigoplus _{\begin{array}{c} \chi \in X^*(S) \\ \chi \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0 \cap \cdots \cap \sigma _i,0} \\ \chi \not \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0 \cap \cdots \cap \sigma _i,m} \end{array}} {\mathcal {L}}_T(\chi )(U).$$
As \(\varSigma _0\) is finite, we see that
$$H^i\left( \partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\big |_U,{\mathcal {L}}_\psi \Big /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \right) =\bigoplus _{\chi \in X^*(S)} H_{\varSigma _0,\psi ,m}^i(\chi )(U).$$
Because A is noetherian, because \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) is proper over \({{\text {Spec}}\,}A\) and because \({\mathcal {L}}_\psi /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \) is a coherent sheaf on \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\), we see that the cohomology group \(H^i(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}|_U,{\mathcal {L}}_\psi /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi )\) is a finitely generated A-module, and hence, for fixed m and i, we see that the groups \(H_{\varSigma _0,\psi ,m}^i(\chi )(U)=(0)\) for all but finitely many \(\chi \). In particular
$$H^i\left( \partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\big |_U,{\mathcal {L}}_\psi \Big /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \right) =\prod _{\chi \in X^*(S)} H_{\varSigma _0,\psi ,m}^i(\chi )(U).$$
Moreover, combining this observation with the fact that \(\{ H^i_{\varSigma _0,\psi ,m}(\chi )(U)\}\) satisfies the Mittag-Leffler condition, we see that the system
$$\left\{ H^i\left( \partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\big |_U,{\mathcal {L}}_\psi \Big /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \right) \right\} $$
satisfies the Mittag-Leffler condition. Hence from proposition 0.13.3.1 of [23] we see that
$$\begin{array}{rcl} H^i(T_{\varSigma _0}^\wedge \big |_U,{\mathcal {L}}_\psi ^\wedge ) &{}\cong &{}\lim _{\leftarrow m} H^i\left( \partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}|_U,{\mathcal {L}}_\psi \Big /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \right) \\ &{} \cong &{} \prod _{\chi \in X^*(S)} \lim _{\leftarrow m} H^i_{\varSigma _0,\psi ,m}(\chi )(U), \end{array}$$
and the present lemma follows from Lemma 2.11. \(\square \)

Lemma 2.13

Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _\infty \) be a partial fan for S, let
$$\begin{aligned} \varSigma _1 \subset \varSigma _2 \subset \cdots \end{aligned}$$
be a nested sequence of partial fans with \(\varSigma _\infty =\bigcup _i \varSigma _i\) and let \(\psi \) be line bundle data for \(\varSigma _\infty \). For \(i=1,2,3,\ldots ,\infty \) let \(\pi _{\varSigma _i}^\wedge \) denote the map \(T_{\varSigma _i}^\wedge \rightarrow Y\).
Suppose that for \(i\in {\mathbb {Z}}_{>0}\) the partial fan \(\varSigma _i\) is finite, non-empty and open. Suppose also that for all \(i \in {\mathbb {Z}}_{\ge 0}\) and all connected, noetherian, affine open sets \(U \subset Y\), the inverse system
$$\begin{aligned} \left\{ H^{i}_{|\varSigma _j|^0-Y_\psi (\chi )}\left( |\varSigma _j|^0,{\mathcal {O}}_Y(U)\right) \right\} _j \end{aligned}$$
satisfies the Mittag-Leffler condition. Then
$$R^i\pi _{\varSigma _\infty ,*}^\wedge {\mathcal {L}}_\psi ^\wedge \cong \prod _{\chi \in X^*(S)} \lim _{\leftarrow j} H^i_{|\varSigma _j|^0-Y_\psi (\chi )}\left( |\varSigma _j|^0,{\mathcal {L}}_T(\chi )\right) .$$

Proof

The left hand side is the sheaf associated to the pre-sheaf
$$\begin{aligned} U \longmapsto H^i\left( T_{\varSigma _\infty }^\wedge \big |_U, {\mathcal {L}}_\psi ^\wedge \right) \end{aligned}$$
and the right hand side is the sheaf associated to the pre-sheaf
$$\begin{aligned} U \longmapsto \prod _{\chi \in X^*(S)} \lim _{\leftarrow j} H^i_{|\varSigma _j|^0-Y_\psi (\chi )}\left( |\varSigma _j|^0,{\mathcal {O}}_Y(U)\right) \otimes {\mathcal {L}}_T(\chi )(U). \end{aligned}$$
Thus it suffices to establish isomorphisms
$$\begin{aligned} H^i\left( T_{\varSigma _\infty }^\wedge \big |_U, {\mathcal {L}}_\psi ^\wedge \right) \cong \prod _{\chi \in X^*(S)(Y)} \lim _{\leftarrow j} H^i_{|\varSigma _j|^0-Y_\psi (\chi )}\left( |\varSigma _j|^0,{\mathcal {O}}_Y(U)\right) \otimes {\mathcal {L}}_T(\chi )(U), \end{aligned}$$
compatibly with restriction, for \(U={{\text {Spec}}\,}A\), with A noetherian and \({{\text {Spec}}\,}A\) connected.
We can compute \(H^i(T_{\varSigma _\infty }^\wedge |_U, {\mathcal {L}}_\psi ^\wedge )\) as the cohomology of the Cech complex with ith term
$$\prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _\infty ^{i+1}} {\mathcal {L}}_\psi ^\wedge \left( \left( T_{\varSigma _\infty }^\wedge \right) _{(\sigma _0 \cap \cdots \cap \sigma _i)}|_U\right) ,$$
and we can compute \(H^i(T_{\varSigma _j}^\wedge |_U, {\mathcal {L}}_\psi ^\wedge )\) as the cohomology of the Cech complex with ith term
$$\prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _j^{i+1}} {\mathcal {L}}_\psi ^\wedge \left( \left( T_{\varSigma _j}^\wedge \right) _{(\sigma _0 \cap \cdots \cap \sigma _i)}\big |_U\right) .$$
Note that as soon as the faces of \(\sigma \) in \(\varSigma _j\) equals the faces of \(\sigma \) in \(\varSigma _\infty \) then \((T_{\varSigma _\infty }^\wedge )_\sigma =(T_{\varSigma _j}^\wedge )_\sigma \). Thus
$$\lim _{\leftarrow j} \prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _j^{i+1}} {\mathcal {L}}_\psi ^\wedge \left( \left( T_{\varSigma _j}^\wedge \right) _{(\sigma _0 \cap \cdots \cap \sigma _i)}\big |_U\right) \cong \prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _\infty ^{i+1}} {\mathcal {L}}_\psi ^\wedge \left( \left( T_{\varSigma _\infty }^\wedge \right) _{(\sigma _0 \cap \cdots \cap \sigma _i)}\big |_U\right) ,$$
and
$$\left\{ \prod _{(\sigma _0,\ldots ,\sigma _i) \in \varSigma _j^{i+1}} {\mathcal {L}}_\psi ^\wedge \left( \left( T_{\varSigma _j}^\wedge \right) _{(\sigma _0 \cap \cdots \cap \sigma _i)(U)}\right) \right\} $$
satisfies the Mittag-Leffler condition (with j varying but i fixed).
From theorem 3.5.8 of [57] we see that there is a short exact sequence
$$\begin{aligned}\begin{array}{l} (0) \longrightarrow {\lim _{\leftarrow j}}^1 H^{i-1}\left( T_{\varSigma _j}^\wedge \big |_U, {\mathcal {L}}_\psi ^\wedge \right) \longrightarrow H^i\left( T_{\varSigma _\infty }^\wedge \big |_U, {\mathcal {L}}_\psi ^\wedge \right) \longrightarrow \lim _{\leftarrow j} H^{i}\left( T_{\varSigma _j}^\wedge \big |_U, {\mathcal {L}}_\psi ^\wedge \right) \longrightarrow (0).\end{array}\end{aligned}$$
Applying Lemma 2.12 and the fact that \(\lim _{\leftarrow }\) and \({\lim _{\leftarrow }}^1\) in the category of abelian groups commute with arbitrary products, the present lemma follows. (It follows easily from definition 3.5.1 of [57] and the exactness of infinite products in the category of abelian groups that \(\lim _{\leftarrow }\) and \({\lim _{\leftarrow }}^1\) commute with arbitrary products in the category of abelian groups.) \(\square \)

We now turn to two specific line bundles: \({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) and, in the case that \(\varSigma _0\) is smooth, \({\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \).

Lemma 2.14

Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is non-empty, finite and open and that \(|\varSigma _0|^0\) is convex.
  1. (1)
    Then
    $$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
     
  2. (2)
    If in addition \(\varSigma _0\) is smooth then
    $$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial , \varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
     

Proof

The first part follows from Lemma 2.12 because \(Y_0(\chi )\cap |\varSigma _0|^0\) is empty if \(\chi \in |\varSigma _0|^\vee \) and otherwise, being the intersection of two convex sets, it is convex.

For the second part we have that \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0 =\emptyset \) if and only if \(\chi \in |\varSigma _0|^{\vee ,0}\). Thus it suffices to show that each \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) is empty or contractible.

To this end, consider the sets
$$\begin{aligned} Y'(\chi )= \bigcup _{\begin{array}{c} \sigma \in \varSigma _0 \\ \chi \le 0 \,\,\mathrm{on}\,\,\sigma \end{array}} \sigma \end{aligned}$$
and
$$\begin{aligned} Y''(\chi )= \bigcup _{\begin{array}{c} \sigma \in \varSigma _0 \\ \chi \not > 0 \,\,\mathrm{on}\,\,\sigma -\{0\} \end{array}} \sigma . \end{aligned}$$
If \(\chi >0\) on \(\sigma -\{0\}\) then \(\chi \ge \psi _{\widetilde{\varSigma }_0}\) on \(\sigma \) so that \(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )=\emptyset \). Thus
$$Y''(\chi ) \supset Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0 \supset Y'(\chi ) \cap |\varSigma _0|^0$$
and
$$Y''(\chi ) \supset \left\{ x \in |\varSigma _0|^0: \chi (x)\le 0\right\} \supset Y'(\chi ) \cap |\varSigma _0|^0.$$
We will describe a deformation retraction
$$\begin{aligned} H: Y''(\chi ) \times [0,1] \longrightarrow Y''(\chi ) \end{aligned}$$
from \(Y''(\chi )\) to \(Y'(\chi )\), which restricts to deformation retractions
$$\left( Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0\right) \times [0,1] \longrightarrow Y_{\psi _{\widetilde{\varSigma }_0}} \cap |\varSigma _0|^0(\chi )$$
from \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) to \(Y'(\chi )\cap |\varSigma _0|^0\), and
$$\begin{aligned} \left\{ x \in |\varSigma _0|^0: \chi (x)\le 0\} \times [0,1] \longrightarrow \{ x \in |\varSigma _0|^0: \chi (x)\le 0\right\} \end{aligned}$$
from \(\{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) to \(Y'(\chi )\cap |\varSigma _0|^0\). (Recall that in particular \(H|_{Y'(\chi ) \times [0,1]}\) is just projection to the first factor.) As \(\{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) is empty or convex, it would follow that \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) is empty or contractible and the second part of the corollary would follow.
To define H it suffices to define, for each \(\sigma \in \widetilde{\varSigma }_0\) with \(\sigma \subset Y''(\chi )\), a deformation retraction
$$\begin{aligned} H_\sigma : \sigma \times [0,1] \longrightarrow \sigma \end{aligned}$$
from \(\sigma \) to \(\sigma \cap Y'(\chi )\) with the following properties:
  • If \(\sigma ' \subset \sigma \) then \(H_\sigma |_{\sigma ' \times [0,1]}=H_{\sigma '}\).

  • \(H_\sigma |_{(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0) \times [0,1]}\) is a deformation retraction from \(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) to \(\sigma \cap Y'(\chi )\cap |\varSigma _0|^0\).

  • \(H_\sigma |_{(\sigma \cap \{ x \in |\varSigma _0|^0: \chi (x)\le 0\}) \times [0,1]}\) is a deformation retraction from \(\sigma \cap \{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) to \(\sigma \cap Y'(\chi )\cap |\varSigma _0|^0\).

To define \(H_\sigma \), let \(v_1,\ldots ,v_r,w_1,\ldots ,w_s\) denote the shortest nonzero vectors in \(X_*(S)\) on each of the one-dimensional faces of \(\sigma \) (so that \(r+s=\dim \sigma \)), with the notation chosen such that \(\chi (v_i)\le 0\) for all i and \(\chi (w_j)>0\) for all j. Note that \(1-\chi (v_i) >0\) for all i and \(1-\chi (w_j) \le 0\) for all j. We set
$$\begin{aligned} H_\sigma \left( \sum _i a_i v_i + \sum _j b_i w_j,t\right) =\sum _i a_i v_i +(1-t) \sum _j b_i w_j. \end{aligned}$$
Because
$$\begin{array}{rcl} \sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0&{}=&{} \left\{ \sum _i a_i v_i+\sum _j b_jw_j: a_i,b_j \in {\mathbb {R}}_{\ge 0} \, \mathrm{and}\right. \\ &{}&{}\left. \sum _ia_i(1-\chi (v_i))+\sum _jb_j(1-\chi (w_j)) > 0\right\} \cap |\varSigma _0|^0 \end{array}$$
and
$$\begin{array}{l} \sigma \cap \{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\\ \quad =\left\{ \sum _i a_i v_i+\sum _j b_jw_j: a_i,b_j \in {\mathbb {R}}_{\ge 0} \,\, \mathrm{and} \sum _ia_i\chi (v_i)+\sum _jb_j\chi (w_j) \le 0\right\} \cap |\varSigma _0|^0 \end{array}$$
are convex sets, and because
$$\begin{aligned} \sigma \cap Y'(\chi )=\left\{ \sum _i a_i v_i: a_i \in {\mathbb {R}}_{\ge 0}\right\} , \end{aligned}$$
it is easy to check that it has the desired properties and the proof of the lemma is complete. \(\square \)

Lemma 2.15

Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let \(\alpha : S \rightarrow S'\) be an isogeny of split tori over Y, and let \(\varSigma _0'\) (resp. \(\varSigma _0\)) be a locally finite partial fan for \(S'\) (resp. S). Suppose that \(\varSigma _0'\) is full. Also suppose that \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\), and let \(\pi ^\wedge \) denote the map \(T_{\varSigma _0}^\wedge \rightarrow (\alpha _*T)_{\varSigma _0'}^\wedge \).

Then for \(i>0\) we have
$$\begin{aligned} R^i\pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=(0), \end{aligned}$$
while
$${\mathcal {O}}_{(\alpha _*T)_{\varSigma _0'}^\wedge } \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }\right) ^{\ker \alpha }.$$
If moreover \(\varSigma _0\) and \(\varSigma _0'\) are smooth then, for \(i>0\) we have
$$\begin{aligned} R^i\pi ^\wedge _* {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge =(0). \end{aligned}$$
while
$${\mathcal {I}}_{\partial ,\varSigma _0'}^\wedge \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _* {\mathcal {I}}_{\partial , \varSigma _0}^\wedge \right) ^{\ker \alpha }.$$

Proof

We may reduce to the case that \(Y={{\text {Spec}}\,}A\) is affine. The map \(\pi ^\wedge \) is proper and hence
$$\begin{aligned} R^i\pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge } \end{aligned}$$
and
$$\begin{aligned} R^i\pi ^\wedge _* {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \end{aligned}$$
and
$${{\text {coker}}\,}\left( {\mathcal {O}}_{(\alpha _*T)_{\varSigma _0'}^\wedge } \longrightarrow \left( \pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }\right) ^{\ker \alpha }\right) $$
and
$${{\text {coker}}\,}\left( {\mathcal {I}}_{\partial ,\varSigma _0'}^\wedge \longrightarrow \left( \pi ^\wedge _* {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \right) ^{\ker \alpha }\right) $$
are coherent sheaves. Thus they have closed support. Their support is also S-invariant. Thus it suffices to show that for each maximal element \(\sigma ' \in \varSigma _0'\) the space \(\partial _{\sigma '}(\alpha _*T)_{\widetilde{\varSigma }_0'}\) does not lie in the support of these sheaves. Let \(\varSigma _0(\sigma ')\) denote the subset of elements \(\sigma \in \varSigma _0\) which lie in \(\sigma '\), but in no face of \(\sigma '\). Then \(\varSigma _0(\sigma ')\) is a partial fan and \(T^\wedge _{\varSigma _0(\sigma ')}\) equals the formal completion of \(T^\wedge _{\varSigma _0}\) along \(\partial _{\sigma '}T_{\widetilde{\varSigma }_0'}\). Thus the formal completion of the above four sheaves along \(\partial _{\sigma '}T_{\widetilde{\varSigma }_0'}\) equal the corresponding sheaf for the pair \(\varSigma _0(\sigma ')\) and \(\{ \sigma '\}\), so that we are reduced to the case that \(\varSigma _0'=\{ \sigma '\}\) is a singleton.
In the case that \(\varSigma _0'=\{\sigma '\}\) then \((\alpha _*T)_{\varSigma _0'}\) and Y have the same underlying topological space. Let \(\pi _1^\wedge \) denote the map of ringed spaces \(T^\wedge _{\varSigma _0} \rightarrow Y\). Then it suffices to show that for \(i>0\) we have
$$\begin{aligned} R^i\pi ^\wedge _{1,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=(0) \end{aligned}$$
and
$$\begin{aligned} R^i\pi ^\wedge _{1,*} {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge =(0); \end{aligned}$$
and that we have
$${\mathcal {O}}_{(\alpha _*T)_{\varSigma _0'}^\wedge } \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _{1,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }\right) ^{\ker \alpha }$$
and
$${\mathcal {I}}_{\partial ,\varSigma _0'}^\wedge \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _{1,*} {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \right) ^{\ker \alpha }.$$
This follows from Lemma 2.14. (Note that
$$\begin{aligned} \prod _{\chi \in |\varSigma _0|^\vee \cap X^*(S)} {\mathcal {L}}(\chi )=\bigoplus _{\xi \in (\ker \alpha )^\vee } \prod _{\begin{array}{c} \chi \in |\varSigma _0|^\vee \cap X^*(S) \\ \chi |_{\ker \alpha =\xi } \end{array}} {\mathcal {L}}(\chi ), \end{aligned}$$
where \(\ker \alpha \) acts on the \(\xi \) term via \(\xi \); and that
$$\left\{ \chi \in |\varSigma _0|^\vee \cap X^*(S): \chi |_{\ker \alpha }=1\right\} = |\varSigma _0|^\vee \cap X^*(S')=|\{ \sigma '\}|^{\vee } \cap X^*(S').$$
These assertions remain true with \(|\varSigma _0|^{\vee ,0}\) replacing \(|\varSigma _0|^\vee \) and \(|\{ \sigma '\}|^{\vee ,0}\) replacing \(|\{\sigma '\}|^{\vee }\).)

\(\square \)

Lemma 2.16

Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map of ringed spaces \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is non-empty, full, locally finite and open, and that \(|\varSigma _0|^0\) is convex.
  1. (1)
    Then
    $$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{}\quad \mathrm{if}\,\, i=0 \\ (0) &{}\quad \mathrm{otherwise.}\end{array} \right. $$
     
  2. (2)
    If in addition \(\varSigma _0\) is smooth then
    $$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{}\quad \mathrm{otherwise.}\end{array} \right. $$
     

Proof

Let \(\sigma _1,\sigma _2,\ldots \) be an enumeration of the 1 cones in \(\widetilde{\varSigma }_0\). Let \(\varDelta ^{(i)} \subset |\varSigma |\) denote the convex hull of \(\sigma _1,\ldots ,\sigma _i\). It is a rational polyhedral cone contained in \(|\varSigma _0|\), and there exists \(i_0\) such that for \(i \ge i_0\) the cone \(\varDelta ^{(i)}\) will have the same dimension as \(X_*(S)_{\mathbb {R}}\). Let \(\partial \varDelta ^{(i)}\) denote the union of the proper faces of \(\varDelta ^{(i)}\); and let \(\varDelta ^{(i),c}\) denote the closure of \(|\varSigma _0|-\varDelta ^{(i)}\) in \(|\varSigma _0|\).

Define recursively fans \(\varSigma ^{(i)}\) and boundary data \(\varSigma ^{(i)}_0\) as follows. We set \(\varSigma ^{(i_0-1)}=\widetilde{\varSigma }_0\) and \(\varSigma ^{(i_0-1)}_0=\varSigma _0\). For \(i\ge i_0\) set
$$\varSigma ^{(i)}=\left\{ \sigma \cap \varDelta ^{(i)}, \sigma \cap \partial \varDelta ^{(i)}, \sigma \cap \varDelta ^{(i),c}: \sigma \in \varSigma ^{(i-1)}\right\} .$$
Then \(\varSigma ^{(i)}\) refines \(\varSigma ^{(i-1)}\) and we choose \(\varSigma ^{(i)}_0\) to be the unique subset of \(\varSigma ^{(i)}\) such that \((\varSigma ^{(i)},\varSigma _0^{(i)})\) refines \((\varSigma ^{(i-1)},\varSigma _0^{(i-1)})\). Then \(\widetilde{\varSigma _0^{(i)}}=\varSigma ^{(i)}\). We also check by induction on i that
  • \(\varSigma ^{(i)} \cup \varSigma ^{(i-1)}-(\varSigma ^{(i)}\cap \varSigma ^{(i-1)})\) is finite;

  • and \(\varSigma ^{(i)}_0\) is locally finite.

(The point being that the local finiteness of \(\varSigma ^{(i-1)}_0\) implies that only finitely many elements of \(\varSigma ^{(i-1)}_0\), and hence of \(\varSigma ^{(i-1)}\), meet both \(\varDelta ^{(i)}-\partial \varDelta ^{(i)}\) and \(X_*(S)_{\mathbb {R}}-\varDelta ^{(i)}\).)
Now define \(\varSigma ^{(\infty )}\) (resp. \(\varSigma ^{(\infty )}_0\)) to be the set of cones that occur in \(\varSigma ^{(i)}\) (resp. \(\varSigma _0^{(i)}\)) for infinitely many i. Alternatively
$$\begin{aligned} \varSigma ^{(\infty )}=\bigcup _i \{ \sigma \in \varSigma ^{(i)}: \sigma \subset \varDelta ^{(i)} \}. \end{aligned}$$
Also let \(\varSigma ^{(\infty ),{{\text {sm}}}}\) denote a smooth refinement of \(\varSigma ^{(\infty )}\) and let \(\varSigma _0^{(\infty ),{{\text {sm}}}}\) denote the elements of \(\sigma \in \varSigma ^{(\infty ),{{\text {sm}}}}\) for which there exists \(\tau \in \varSigma _0\) such that \(\sigma \subset \tau \) and \(\sigma \cap \tau ^0 \ne \emptyset \). (See Lemma 2.4.) Then \(\varSigma ^{(\infty ),{{\text {sm}}}}\) is a fan, \(\varSigma _0^{(\infty ),{{\text {sm}}}}\) provides locally finite boundary data for \(\varSigma ^{(\infty ),{{\text {sm}}}}\), we have \(\widetilde{\varSigma _0^{(\infty ),{{\text {sm}}}}}=\varSigma ^{(\infty ),{{\text {sm}}}}\), and \((\varSigma ^{(\infty ),{{\text {sm}}}},\varSigma ^{(\infty ),{{\text {sm}}}}_0)\) refines \((\widetilde{\varSigma }_0,\varSigma _0)\). Moreover \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\) is open. We also define \(\varSigma ^{(\infty ),{{\text {sm}}}}_i\) to be the set of \(\sigma \in \varSigma ^{(\infty ),{{\text {sm}}}}_0\) such that \(\sigma \subset \varDelta ^{(i)}\) but \(\sigma \not \subset \partial \varDelta ^{(i)}\). Note that:
  • \(\varSigma ^{(\infty ),{{\text {sm}}}}_i\) is finite and open;

  • \(\varSigma ^{(\infty ),{{\text {sm}}}}_i \supset \varSigma ^{(\infty ),{{\text {sm}}}}_{i-1}\);

  • \(|\varSigma ^{(\infty ),{{\text {sm}}}}_i|^0=\varDelta ^{(i)} - \partial \varDelta ^{(i)}\) is convex;

  • and \(\varSigma ^{(\infty ),{{\text {sm}}}}_0=\bigcup _{i>0} \varSigma ^{(\infty ),{{\text {sm}}}}_i\).

(For the last of these properties use the fact that \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\) is open.)

By Lemma 2.15 it suffices to prove this lemma after replacing the pair \(\varSigma _0\) by \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\). This lemma then follows from Lemmas 2.13 and 2.14. \(\square \)

3.5 The case of a disconnected base

Throughout this section we will continue to assume that Y is a separated scheme, flat and locally of finite type over \({{\text {Spec}}\,}R_0\). We prove nothing new in this section, we simply explain how to re-express the last two sections in a way that makes sense over a disconnected base, so that the results we have already established immediately extend.

Let S be a split torus over Y and let T / Y be an S-torsor. By a rational polyhedral cone \(\sigma \) in \(X_*(S)_{\mathbb {R}}\) we shall mean a locally constant sheaf of subsets \(\sigma \subset X_*(S)_{\mathbb {R}}\), such that
  • for each connected open \(U \subset Y\) the set \(\sigma (U) \subset X_*(S)_{\mathbb {R}}(U)\) is either empty or a rational polyhedral cone,

  • and the locus where \(\sigma \ne \emptyset \) is non-empty and connected. We call this locus the support of \(\sigma \).

We call \(\sigma '\) a face of \(\sigma \) if for each open connected U either \(\sigma (U)=\sigma '(U)=\emptyset \) or the cone \(\sigma '(U)\) is a face of \(\sigma (U)\). We call \(\sigma \) smooth if each \(\sigma (U)\) is smooth. By a fan \(\varSigma \) in \(X_*(S)_{\mathbb {R}}\) we mean a set of rational polyhedral cones in \(X_*(S)_{\mathbb {R}}\), such that
  • if \(\sigma \in \varSigma \) then so is any face \(\sigma '\) of \(\sigma \);

  • if \(\sigma ,\sigma ' \in \varSigma \) then \(\sigma \cap \sigma '\) is either empty or a face of \(\sigma \) and \(\sigma '\),

  • any connected component of Y arises as the support of some element of \(\varSigma \).

Thus to give a fan in \(X_*(S)_{\mathbb {R}}\) is the same as giving a fan in \(X_*(S)_{\mathbb {R}}(Z)\) for each connected component Z of Y. If U is a non-empty connected open in Y then we set
$$\begin{aligned} \varSigma (U)=\{ \sigma (U): \sigma \in \varSigma \} -\{ \emptyset \} . \end{aligned}$$
It is a fan for \(X_*(S)_{\mathbb {R}}(U)\).
We call \(\varSigma \) smooth (resp. full, resp. finite) if each \(\varSigma (U)\) is. We define a locally constant sheaf \(|\varSigma |\) of subsets of \(X_*(S)_{\mathbb {R}}\) by setting
$$\begin{aligned} |\varSigma |(U)= \bigcup _{\sigma \in \varSigma } \sigma (U) \end{aligned}$$
(resp.
$$\begin{aligned} |\varSigma |^*(U)= \bigcup _{\sigma \in \varSigma } (\sigma (U) -\{0\})) \end{aligned}$$
for U any connected open subset of Y. We will call \(|\varSigma |\) (resp. \(\vert \varSigma \vert ^*\)) convex if \(|\varSigma |(U)\) (resp. \(|\varSigma |^*(U)\)) is for each connected open \(U\subset Y\). We also define locally constant sheaves of subsets \(|\varSigma |^{\vee }\) and \(|\varSigma |^{\vee ,0}\) of \(X^*(S)_{\mathbb {R}}\) by setting
$$|\varSigma |^{\vee }(U)=\left\{ \chi \in X^*(S)_{\mathbb {R}}(U): \chi (|\varSigma |(U)) \subset {\mathbb {R}}_{\ge 0}\right\} =\bigcap _{\sigma \in \varSigma } \sigma (U)^{\vee }$$
and
$$|\varSigma |^{\vee ,0}(U)=\left\{ \chi \in X^*(S)_{\mathbb {R}}(U): \chi (|\varSigma |^*(U)) \subset {\mathbb {R}}_{>0}\right\} =\bigcap _{\sigma \in \varSigma } \sigma (U)^{\vee ,0}.$$
We call \(\varSigma '\) a refinement of \(\varSigma \) if each \(\varSigma '(U)\) is a refinement of \(\varSigma (U)\) for each open, connected U. Any fan \(\varSigma \) has a smooth refinement \(\varSigma '\) such that every smooth cone \(\sigma \in \varSigma \) also lies in \(\varSigma '\).
To a fan \(\varSigma \) one can attach a scheme \(T_\varSigma \) flat and separated over Y and locally (on \(T_\varSigma \)) of finite type over Y, together with an action of S and an S-equivariant embedding \(T \hookrightarrow T_\varSigma \). It has an open cover \(\{ T_\sigma \}_{\sigma \in \varSigma }\), with each \(T_\sigma \) relatively affine over Y. Over a connected open \(U \subset Y\) this restricts to the corresponding picture for \({\varSigma (U)}\). We write \({\mathcal {O}}_{T_\varSigma }\) for the structure sheaf of \(T_\varSigma \). If \(\varSigma \) is smooth then \(T_\varSigma /Y\) is smooth. If \(\varSigma \) is finite and \(|\varSigma |=X_*(S)_{\mathbb {R}}\), then \(T_\varSigma \) is proper over Y. If \(\varSigma '\) refines \(\varSigma \) then there is an S-equivariant proper map
$$\begin{aligned} T_{\varSigma '} \rightarrow T_\varSigma \end{aligned}$$
which restricts to the identity on T.
By boundary data we shall mean a proper subset \(\varSigma _0 \subset \varSigma \) such that \(\varSigma - \varSigma _0\) is a fan. If \(U\subset Y\) is a connected open we set
$$\begin{aligned} \varSigma _0(U)=\{ \sigma (U): \sigma \in \varSigma _0\} -\{ \emptyset \}. \end{aligned}$$
If \(\varSigma _0\) is boundary data, then we can associate to it a closed subscheme \(\partial _{\varSigma _0} T_\varSigma \subset T_\varSigma \), which over a connected open \(U\subset Y\) restricts to \(\partial _{\varSigma _0(U)} (T|_U)_{\varSigma (U)} \subset (T|_U)_{\varSigma (U)}\).

In the case that \(\varSigma _0\) is the set of elements of \(\varSigma \) of dimension bigger than 0 we shall simply write \(\partial T_\varSigma \) for \(\partial _{\varSigma _0}T_\varSigma \). Thus \(T=T_\varSigma -\partial T_\varSigma \). We will write \({\mathcal {I}}_{\partial T_\varSigma }\) for the ideal of definition of \(\partial T_\varSigma \) in \({\mathcal {O}}_{T_\varSigma }\). We will also write \({\mathcal {M}}_\varSigma \rightarrow {\mathcal {O}}_{T_\varSigma }\) for the associated log structure and \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log \infty )\) for the log differentials \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log {\mathcal {M}}_\varSigma )\).

If \(\varSigma \) is smooth then \(\partial T_\varSigma \) is a simple normal crossings divisor on \(T_\varSigma \) relative to Y.

If \(\sigma \in \varSigma \) has positive dimension and if \(\varSigma _0\) denotes the set of elements of \(\varSigma \) which have \(\sigma \) for a face, then we will write \(\partial _\sigma T_\varSigma \) for \(\partial _{\varSigma _0} T_\varSigma \). It is connected and flat over Y, and, locally on Y, it has codimension in \(T_\varSigma \) equal to the dimension of \(\sigma \). If \(\varSigma \) is smooth then each \(\partial _\sigma T_\varSigma \) is smooth over Y. The schemes \(\partial _{\sigma _1} T_\varSigma ,\ldots ,\partial _{\sigma _s} T_\varSigma \) intersect if and only if \(\sigma _1, \ldots , \sigma _s\) are all contained in some \(\sigma \in \varSigma \). In this case the intersection equals \(\partial _\sigma T_\varSigma \) for the smallest such \(\sigma \). We set
$$\begin{aligned} \partial _i T_\varSigma =\coprod _{\dim \sigma =i} \partial _\sigma T_\varSigma . \end{aligned}$$
If the connected components of Y are irreducible then each \(\partial _\sigma T_\varSigma \) is irreducible. Moreover the irreducible components of \(\partial T_\varSigma \) are the \(\partial _\sigma T_\varSigma \) as \(\sigma \) runs over one-dimensional elements of \(\varSigma \). If \(\varSigma \) is smooth then we see that \({\mathcal {S}}(\partial T_\varSigma )\) is the delta set with cells in bijection with the elements of \(\varSigma \) with dimension bigger than 0 and with the same ‘face relations’. In particular it is in fact a simplicial complex and
$$|{\mathcal {S}}(\partial T_\varSigma )|= \coprod _{Z \in \pi _0(Y)} |\varSigma |^*(Z)/{\mathbb {R}}^{\times }_{>0}.$$
We will call \(\varSigma _0\) open (resp. finite, resp. locally finite) if \(\varSigma _0(U)\) is for each open connected \(U \subset Y\). If \(\varSigma _0\) is finite and open, then \(\partial _{\varSigma _0} T_\varSigma \) is proper over Y.
By a partial fan in \(X_*(S)\) we mean a collection \(\varSigma _0\) of rational polyhedral cones in \(X_*(S)\) such that
  • \(\varSigma _0\) does not contain \((0) \subset X_*(S)(U)_{\mathbb {R}}\) for any open connected U;

  • if \(\sigma _1, \sigma _2 \in \varSigma _0\) then \(\sigma _1 \cap \sigma _2\) is either empty or a face of \(\sigma _1\) and of \(\sigma _2\);

  • if \(\sigma _1, \sigma _2 \in \varSigma _0\) and if \(\sigma \supset \sigma _2\) is a face of \(\sigma _1\), then \(\sigma \in \varSigma _0\).

In this case we will let \(\widetilde{\varSigma }_0\) denote the set of faces of elements of \(\varSigma _0\) together with \(\{0\}\) supported on any connected component of Y. It is a fan, and \(\varSigma _0\) is boundary data for \(\widetilde{\varSigma }_0\). By boundary data \(\varSigma _1\) for \(\varSigma _0\) we shall mean a subset \(\varSigma _1 \subset \varSigma _0\) such that if \(\sigma \in \varSigma _0\) contains \(\sigma _1 \in \varSigma _1\), then \(\sigma \in \varSigma _1\). In this case \(\varSigma _1\) is again a partial fan and boundary data for \(\widetilde{\varSigma }_0\). We say that a partial fan \(\varSigma _0\) for \(X_*(S)\) refines a partial fan \(\varSigma _0'\) for \(X_*(S)\) if every element of \(\varSigma _0\) lies in an element of \(\varSigma _0'\) and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\).
If \(\varSigma _0\) is a partial fan we define locally constant sheaves of subsets \(|\varSigma _0|\), \(|\varSigma _0|^*\), \(|\varSigma _0|^\vee \) and \(|\varSigma _0|^{\vee ,0}\) of \(X_*(S)_{\mathbb {R}}\) or \(X^*(S)_{\mathbb {R}}\) to be \(|\widetilde{\varSigma }_0|\), \(|\widetilde{\varSigma }_0|^*\), \(|\widetilde{\varSigma }_0|^\vee \) and \(|\widetilde{\varSigma }_0|^{\vee ,0}\), respectively. We also define a sheaf of subsets \(|\varSigma _0|^0\) by
$$\begin{aligned} |\varSigma _0|^0(U)=|\widetilde{\varSigma }_0|(U)\,\,\, -\bigcup _{ \sigma \in \widetilde{\varSigma }_0(U)-\varSigma _0(U) } \sigma \end{aligned}$$
for any connected open set \(U \subset Y\). We will call \(|\varSigma _0|\) (resp. \(|\varSigma _0|^0\)) convex if \(|\varSigma _0|(U)\) (resp. \(|\varSigma _0|^0(U)\)) is convex for all open connected subsets \(U \subset Y\).

We will call \(\varSigma _0\) smooth (resp. full, resp. open, resp. finite, resp. locally finite) if \(\varSigma _0(U)\) is for each \(U \subset Y\) open and connected. We will call \(\varSigma _0\) non-degenerate if for each non-empty connected open subset \(U \subset Y\) the set \(\varSigma _0(U)\) is non-empty.

If \(\varSigma _0\) is a partial fan we will write
$$\begin{aligned} \partial _{\varSigma _0} T \end{aligned}$$
for \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\);
$$\begin{aligned} T^\wedge _{\varSigma _0} \end{aligned}$$
for the completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\); and
$$\begin{aligned} {\mathcal {M}}^\wedge _{\varSigma _0} \longrightarrow {\mathcal {O}}_{T_{\varSigma _0}^\wedge } \end{aligned}$$
for the log structure induced by \({\mathcal {M}}_{\widetilde{\varSigma }_0}\). We make the following definitions.
  • \({\mathcal {I}}_{T_{\varSigma _0}^\wedge }\) will denote the completion of \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}\), the sheaf of ideals defining \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\). It is an ideal of definition for \(T_{\varSigma _0}^\wedge \).

  • \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) will denote the completion of \({\mathcal {I}}_{\partial T_{\widetilde{\varSigma }_0}}\), the sheaf of ideals defining \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\). Thus \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \subset {\mathcal {I}}_{T_{\varSigma _0}^\wedge }\).

  • \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\) will denote \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log {\mathcal {M}}_\varSigma ^\wedge )\).

We will write
$$\begin{aligned} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}_T(\chi ) \end{aligned}$$
(resp.
$$\begin{aligned} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}_T(\chi )) \end{aligned}$$
for the sheaf (of abelian groups) on Y such that for any connected open subset \(U \subset Y\) we have
$$\left. \left( \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}_T(\chi )\right) \right| _U=\prod _{\chi \in |\varSigma _0|^\vee (U) \cap X^*(S)(U)} {\mathcal {L}}_T(\chi )$$
(resp.
$$\left. \left. \left( \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}_T(\chi )\right) \right| _U=\prod _{\chi \in |\varSigma _0|^{\vee ,0}(U) \cap X^*(S)(U)} {\mathcal {L}}_T(\chi )\right) .$$
Suppose that \(\alpha : S \rightarrow S'\) is a surjective map of split tori over Y. Then \(X^*(\alpha ): X^*(S') \hookrightarrow X^*(S)\) and \(X_*(\alpha ): X_*(S)_{\mathbb {R}}\twoheadrightarrow X_*(S')_{\mathbb {R}}\). We call fans \(\varSigma \) for \(X_*(S)\) and \(\varSigma '\) for \(X_*(S')\) compatible if for all \(\sigma \in \varSigma \) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma '\). In this case the map \(\alpha : T \rightarrow \alpha _* T\) extends to an S-equivariant map
$$\begin{aligned} \alpha : T_\varSigma \longrightarrow (\alpha _*T)_{\varSigma '}. \end{aligned}$$
We will write
$$\begin{aligned} \varOmega _{T_\varSigma /(\alpha _* T)_{\varSigma '}}^1(\log \infty )=\varOmega _{T_\varSigma /(\alpha _* T)_{\varSigma '}}^1(\log {\mathcal {M}}_\varSigma /{\mathcal {M}}_{\varSigma '}). \end{aligned}$$
The following lemma is an immediate consequence of Lemma 2.5.

Lemma 2.17

If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then \(\alpha : (T_\varSigma ,{\mathcal {M}}_\varSigma ) \rightarrow ((\alpha _*T)_{\varSigma '},{\mathcal {M}}_{\varSigma '})\) is log smooth, and there is a natural isomorphism
$$\begin{aligned} (X^*(S)/X^*(\alpha )X^*(S')) \otimes _{\mathbb {Z}}{\mathcal {O}}_{T_\varSigma } \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{T_\varSigma /(\alpha _*T)_{\varSigma '}}(\log \infty ). \end{aligned}$$

If \(\alpha \) is an isogeny, if \(\varSigma \) and \(\varSigma '\) are compatible, and if every element of \(\varSigma '\) is a finite union of elements of \(\varSigma \), then we call \(\varSigma \) a quasi-refinement of \(\varSigma '\). In that case the map \(\alpha : T_{\varSigma } \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.

Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of tori, and that \(\varSigma _0\) (resp. \(\varSigma _0'\)) is a partial fan for S (resp. \(S'\)). We call \(\varSigma _0\) and \(\varSigma _0'\) compatible if for every \(\sigma \in \varSigma _0\) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma _0'\) but in no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case there is a natural morphism
$$\alpha : (T^\wedge _{\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \longrightarrow \left( (\alpha _*T)_{\varSigma _0'}^\wedge ,{\mathcal {M}}_{\varSigma _0'}^\wedge \right) .$$
We will write
$$\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)_{\varSigma _0'}^\wedge }^1(\log \infty )=\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)^\wedge _{\varSigma _0'}}^1\left( \log {\mathcal {M}}_{\varSigma _0}^\wedge \Big /{\mathcal {M}}^\wedge _{\varSigma _0'}\right) .$$
The following lemma follows immediately from Lemma 2.8.

Lemma 2.18

If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then there is a natural isomorphism
$$\begin{aligned} (X^*(S)/X^*(\alpha )X^*(S')) \otimes _{\mathbb {Z}}{\mathcal {O}}_{T_{\varSigma _0}^\wedge } \mathop {\longrightarrow }\limits ^{\sim }\varOmega _{T_{\varSigma _0}^\wedge /(\alpha _* T)_{\varSigma _0'}^\wedge }^1(\log \infty ) . \end{aligned}$$
We will call \(\varSigma _0\) and \(\varSigma _0'\) strictly compatible if they are compatible and if an element of \(\widetilde{\varSigma }_0\) lies in \(\varSigma _0\) if and only if it maps to no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). We will say that
  • \(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0(U)\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0(U)\) for all connected opens \(U \subset Y\);

  • and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).

If \(\alpha \) is an isogeny, if \(\varSigma _0\) and \(\varSigma _0'\) are strictly compatible and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\), then we call \(\varSigma _0\) a quasi-refinement of \(\varSigma _0'\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\). The next lemma follows immediately from Lemma 2.9.

Lemma 2.19

Suppose that \(\varSigma _0'\) and \(\varSigma _0\) are strictly compatible.
  1. (1)

    \(T_{\varSigma _0}^\wedge \) is the formal completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0'}(\alpha _* T)\), and \(T_{\varSigma _0}^\wedge \) is locally (on the source) topologically of finite type over \((\alpha _*T)^\wedge _{\varSigma _0'}\).

     
  2. (2)

    If \(\varSigma _0\) is locally finite and if it is open and finite over \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).

     

Corollary 2.20

If \(\alpha \) is an isogeny, if \(\varSigma _0\) is locally finite and if \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).

If \(\varSigma _0\) and \(\varSigma _0'\) are compatible partial fans and if \(\varSigma _1' \subset \varSigma _0'\) is boundary data then \(\varSigma _0(\varSigma _1')\) will denote the set of elements \(\sigma \in \varSigma _0\) such that \(X_*(\alpha )\sigma \) is contained in no element of \(\varSigma _0'-\varSigma _1'\). It is boundary data for \(\varSigma _0\). Moreover the formal completion of \(T_{\varSigma _0}^\wedge \) along the reduced subscheme of \((\alpha _*T)_{\varSigma _1'}^\wedge \) is canonically identified with \(T^\wedge _{\varSigma _0(\varSigma _1')}\). If \(\varSigma _1'=\{ \sigma '\}\) is a singleton we will write \(\varSigma _0(\sigma ')\) for \(\varSigma _0(\{ \sigma '\})\).

The next two lemmas follow immediately from Lemmas 2.15 and 2.16, respectively.

Lemma 2.21

Let Y be a separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let \(\alpha : S \rightarrow S'\) be an isogeny of split tori over Y, and let \(\varSigma _0'\) (resp. \(\varSigma _0\)) be a locally finite partial fan for \(S'\) (resp. S). Suppose that Y is separated and locally noetherian, that \(\varSigma _0'\) is full and that \(\varSigma _0\) is locally finite. Also suppose that \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\). Let \(\pi ^\wedge \) denote the map \(T_{\sigma _0}^\wedge \rightarrow (\alpha _*T)^\wedge _{\sigma _0'}\).

Then for \(i>0\) we have
$$\begin{aligned} R^i\pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=(0), \end{aligned}$$
while
$${\mathcal {O}}_{(\alpha _*T)_{\varSigma _0'}^\wedge } \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _{*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }\right) ^{\ker \alpha }.$$
If moreover \(\varSigma _0\) and \(\varSigma _0'\) are smooth then, for \(i>0\) we have
$$\begin{aligned} R^i\pi ^\wedge _* {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge =(0). \end{aligned}$$
while
$${\mathcal {I}}_{\partial ,\varSigma _0'}^\wedge \mathop {\longrightarrow }\limits ^{\sim }\left( \pi ^\wedge _* {\mathcal {I}}_{\partial , \varSigma _0}^\wedge \right) ^{\ker \alpha }.$$

Lemma 2.22

Let Y be a separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y is an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that Y is separated and locally noetherian, that \(\varSigma _0\) is non-degenerate, full, locally finite and open and that \(|\varSigma _0|^0\) is convex.
  1. (1)
    Then
    $$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
     
  2. (2)
    If in addition \(\varSigma _0\) is smooth then
    $$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial , \varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
     

4 Shimura varieties

In this section we will describe the Shimura varieties associated to \(G_n\) and the mixed Shimura varieties associated to \(G_n^{(m)}\) and \(\widetilde{{G}}_n^{(m)}\). We assume that all schemes discussed in this section are locally noetherian.

4.1 Some Shimura varieties

4.1.1 Moduli problems

By a \(G_n\)-abelian scheme over a scheme \(Y/{\mathbb {Q}}\) we shall mean an abelian scheme A / Y of relative dimension \(n[F: {\mathbb {Q}}]\) together with an embedding
$$\begin{aligned} i: F \hookrightarrow {{\text {End}}}(A/Y)_{\mathbb {Q}}\end{aligned}$$
such that \({{\text {Lie}}\,}A\) is a locally free right \({\mathcal {O}}_Y \otimes _{\mathbb {Q}}F\)-module of rank n. By a morphism (resp. quasi-isogeny) of \(G_n\)-abelian schemes we mean a morphism (resp. quasi-isogeny) of abelian schemes which commutes with the F-action. If (Ai) is a \(G_n\)-abelian scheme then we give \(A^\vee \) the structure \((A^\vee ,i^\vee )\) of a \(G_n\)-abelian scheme by setting \(i^\vee (a)=i({}^ca)^\vee \). By a quasi-polarization of a \(G_n\)-abelian scheme (Ai) / Y we shall mean a quasi-isogeny \(\lambda : A \rightarrow A^\vee \) of \(G_n\)-abelian schemes, some \({\mathbb {Q}}^\times \)-multiple of which is a polarization. (Note that according to this convention, if \(\lambda \) is a polarization, then \(-\lambda \) is a quasi-polarization.) If \(Y={{\text {Spec}}\,}k\) with k a field, we will let \(\langle \,\,\,,\,\,\,\rangle _\lambda \) denote the Weil pairing induced on the adelic Tate module VA (see section 23 of [47]).

Lemma 3.1

If k is a field of characteristic 0 and if \((A,i,\lambda )/k\) is a quasi-polarized \(G_n\)-abelian scheme, then \(V_p(A \times \overline{{k}})\) is a free \(F_p\)-module of rank 2n.

Proof

We may suppose that k is a finitely generated field extension of \({\mathbb {Q}}\), which we may embed into \({\mathbb {C}}\). Then
$$\begin{aligned} \left( V_p(A \times \overline{{k}}) \otimes _{{\mathbb {Q}}_p,\imath } {\mathbb {C}}\right) \cong ({{\text {Lie}}\,}A_{{\overline{{y}}}} \otimes _{k} {\mathbb {C}}) \oplus ({{\text {Lie}}\,}A_{{\overline{{y}}}} \otimes _{k,c} {\mathbb {C}}), \end{aligned}$$
so that \(V_p(A \times \overline{{k}}) \otimes _{{\mathbb {Q}}_p,\imath } {\mathbb {C}}\) is a free \(F \otimes _{\mathbb {Q}}{\mathbb {C}}\)-module. As \(F \otimes _{\mathbb {Q}}{\mathbb {C}}=F_p \otimes _{{\mathbb {Q}}_p, \imath } {\mathbb {C}}\) we deduce that \(V_p (A \times \overline{{k}})\) is a free \(F_p\)-module, as desired. \(\square \)
By an ordinary \(G_n\)-abelian scheme over a \({\mathbb {Z}}_{(p)}\)- scheme Y we shall mean an abelian scheme A / Y of relative dimension \(n[F: {\mathbb {Q}}]\), such that for each geometric point \({\overline{{y}}}\) of Y we have \(\# A[p](k({\overline{{y}}})) \ge p^{n[F: {\mathbb {Q}}]}\), together with an embedding
$$\begin{aligned} i: {\mathcal {O}}_{F,(p)} \hookrightarrow {{\text {End}}}(A/Y)_{{\mathbb {Z}}_{(p)}} \end{aligned}$$
such that \({{\text {Lie}}\,}A\) is a locally free right \({\mathcal {O}}_Y \otimes _{{\mathbb {Z}}_{(p)}} {\mathcal {O}}_{F,(p)}\)-module of rank n. By a morphism of ordinary \(G_n\)-abelian schemes we mean a morphism of abelian schemes which commutes with the \({\mathcal {O}}_{F,(p)}\)-action. If (Ai) is an ordinary \(G_n\)-abelian scheme then we give \(A^\vee \) the structure, \((A^\vee ,i^\vee )\), of a \(G_n\)-abelian scheme by setting \(i^\vee (a)=i({}^ca)^\vee \). By a prime-to- p quasi-polarization of an ordinary \(G_n\)-abelian scheme (Ai) / Y we shall mean a prime-to-p quasi-isogeny \(\lambda : A \rightarrow A^\vee \) of ordinary \(G_n\)-abelian schemes, some \({\mathbb {Z}}^\times _{(p)}\)-multiple of which is a prime-to-p polarization.
If U is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty })\) then by a U-level structure on a quasi-polarized \(G_n\)-abelian scheme \((A,i,\lambda )\) over a connected scheme \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\) with a geometric point \({\overline{{y}}}\), we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant U-orbit \([\eta ]\) of pairs \((\eta _0,\eta _1)\) of \({\mathbb {A}}^\infty \)-linear isomorphisms
$$\begin{aligned} \eta _0: {\mathbb {A}}^{\infty }_{\overline{{y}}}\mathop {\longrightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty }(1)_{\overline{{y}}}=V{\mathbb {G}}_{m,{\overline{{y}}}} \end{aligned}$$
and
$$\begin{aligned} \eta _1: V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty } \mathop {\longrightarrow }\limits ^{\sim }VA_{\overline{{y}}}\end{aligned}$$
such that
$$\begin{aligned} \eta _1(ax)=i(a) \eta _1(x) \end{aligned}$$
for all \(a \in F\) and \(x \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty }\), and such that
$$\begin{aligned} \left\langle \eta _1 x , \eta _1 y\right\rangle _\lambda =\eta _0 \langle x,y \rangle _n \end{aligned}$$
for all \(x,y \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty }\). This definition is independent of the choice of geometric point \({\overline{{y}}}\) of Y. By a U-level structure on a quasi-polarized \(G_n\)-abelian scheme \((A,i,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\), we mean the collection of a U-level structure over each connected component of Y. If \([(\eta _0,\eta _1)]\) is a level structure we define \(||\eta _0|| \in {\mathbb {Q}}^\times _{>0}\) by
$$\begin{aligned} ||\eta _0|| \eta _0 {\widehat{{\mathbb {Z}}}}={\widehat{{\mathbb {Z}}}}(1). \end{aligned}$$
Now suppose that \(U^p\) is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) and that \(N_1 \le N_2\) are non-negative integers. By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme \((A,i,\lambda )\) over a connected scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) with a geometric point \({\overline{{y}}}\), we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p\)-orbit \([\eta ]\) of four-tuples \((\eta _0^p,\eta _1^p,C,\eta _p)\) consisting of
  • an \({\mathbb {A}}^{\infty ,p}\)-linear isomorphism \(\eta _0^p: {\mathbb {A}}^{\infty ,p}_{\overline{{y}}}\mathop {\longrightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1)_{\overline{{y}}}=V^p{\mathbb {G}}_{m,{\overline{{y}}}}\);

  • an \({\mathbb {A}}_F^{\infty ,p}\)-linear isomorphism
    $$\begin{aligned} \eta _1^p: V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }V^pA_{\overline{{y}}}\end{aligned}$$
    such that
    $$\begin{aligned} \langle \eta _1^p x , \eta _1^p y \rangle _\lambda =\eta _0 \langle x,y \rangle _n \end{aligned}$$
    for all \(x,y \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty ,p}\);
  • a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(C \subset A[p^{N_2}]\), such that for every geometric point \(\widetilde{{y}}\) of Y there is an \({\mathcal {O}}_{F,(p)}\)-invariant sub-Barsotti–Tate group \(\widetilde{{C}}_{\widetilde{{y}}} \subset A_{\widetilde{{y}}}[p^\infty ]\) with the following properties
    • \(C_{\widetilde{{y}}}=\widetilde{{C}}_{\widetilde{{y}}}[p^{N_2}]\),

    • for all N the subgroup scheme \(\widetilde{{C}}_{\widetilde{{y}}}[p^N]\) is isotropic in \(A[p^N]_{\widetilde{{y}}}\) for the \(\lambda \)-Weil pairing,

    • \(A_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}}\) is ind-etale,

    • the Tate module \(T(A_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}})\) is free over \({\mathcal {O}}_{F,p}\) of rank n;

  • and an isomorphism
    $$\begin{aligned} \eta _p: p^{-N_1}\varLambda _n\big /\big (p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\big ) \mathop {\longrightarrow }\limits ^{\sim }A[p^{N_1}]/(A[p^{N_1}] \cap C) \end{aligned}$$
    such that
    $$\begin{aligned} \eta _p(ax)=i(a) \eta _p(x) \end{aligned}$$
    for all \(a \in {\mathcal {O}}_{F,(p)}\) and \(x \in p^{-N_1}\varLambda _n/(p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n)\).
This definition is independent of the choice of the geometric point \({\overline{{y}}}\in Y\). By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme \((A,i,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\), we mean the collection of a \(U^p(N_1,N_2)\)-level structure over each connected component of Y. If \([(\eta _0^p,\eta _1^p,C,\eta _p)]\) is a level structure we define \(||\eta _0^p|| \in {\mathbb {Z}}^\times _{(p),>0}\) by
$$\begin{aligned} ||\eta _0^p|| \eta _0^p {\widehat{{\mathbb {Z}}}}^p={\widehat{{\mathbb {Z}}}}^p(1). \end{aligned}$$
By a quasi-isogeny (resp. prime-to-p quasi-isogeny) between quasi-polarized, \(G_n\)-abelian schemes with U-level structures (resp. ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structures)
$$\begin{aligned} (\beta ,\delta ): (A, i, \lambda , [\eta ]) \longrightarrow (A',i',\lambda ',[\eta ']) \end{aligned}$$
we mean a quasi-isogeny (resp. prime-to-p quasi-isogeny) of abelian schemes \(\beta \in {{\text {Hom}}}(A,A')_{\mathbb {Q}}\) (resp. \(\beta \in {{\text {Hom}}}(A,A')_{{\mathbb {Z}}_{(p)}}\)) and \(\delta \in {\mathbb {Q}}^\times \) (resp. \(\delta \in {\mathbb {Z}}_{(p)}^\times \)) such that
  • \(\beta \circ i(a)=i'(a) \circ \beta \) for all \(a \in F\) (resp. \({\mathcal {O}}_{F,(p)}\));

  • \(\delta \lambda =\beta ^\vee \circ \lambda ' \circ \beta \);

  • \([(\delta \eta _0,(V \beta ) \circ \eta _1)]=[\eta ']\) (resp. \([(\delta \eta _0^p,(V^p \beta ) \circ \eta _1^p, \beta C, \beta \circ \eta _p)]=[\eta ']\)).

Lemma 3.2

Suppose that T is an \({\mathcal {O}}_{F,p}\)-module, which is free over \({\mathcal {O}}_{F,p}\) of rank 2n, with a perfect alternating pairing
$$\begin{aligned} \langle \,\,\,,\,\,\, \rangle : T \times T \longrightarrow {\mathbb {Z}}_p \end{aligned}$$
such that
$$\begin{aligned} \langle ax,y \rangle =\langle x,{}^cay \rangle \end{aligned}$$
for all \(x,y \in T\) and \(a \in {\mathcal {O}}_{F,p}\). Also suppose that \(\widetilde{{T}}\subset T\) is a sub-\({\mathcal {O}}_{F,p}\)-module which is isotropic for \(\langle \,\,\,,\,\,\,\rangle \) and such that \(T/\widetilde{{T}}\) is free of rank n over \({\mathcal {O}}_{F,p}\). Finally suppose that
$$\eta _p: p^{-N_1}\varLambda _n\Big /\left( p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\right) \mathop {\longrightarrow }\limits ^{\sim }p^{-N_1}T\Big / \left( p^{-N_1}\widetilde{{T}}+T\right) $$
is an \({\mathcal {O}}_{F,p}\)-module isomorphism.
Consider the set \([\eta ]\) of isomorphisms
$$\begin{aligned} \eta : \varLambda _n \otimes {\mathbb {Z}}_p \mathop {\longrightarrow }\limits ^{\sim }T \end{aligned}$$
such that
  • \(\eta (a x)=a \eta (x)\) for all \(a \in {\mathcal {O}}_{F,(p)}\);

  • there exists \(\delta \in {\mathbb {Z}}_p^\times \) such that
    $$\begin{aligned} \langle \eta x, \eta y \rangle =\delta \langle x,y\rangle _n \end{aligned}$$
    for all \(x,y \in \varLambda _n \otimes {\mathbb {Z}}_p\);
  • \(\eta ((p^{-N_2} \varLambda _{n,(n)})\otimes {\mathbb {Z}}_p + \varLambda _{n}\otimes {\mathbb {Z}}_p)=p^{-N_2} \widetilde{{T}}+T\);

  • the map
    $$\begin{aligned} p^{-N_1}\varLambda _n\big /\big (p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\big ) \mathop {\longrightarrow }\limits ^{\sim }p^{-N_1}T\big /\big (p^{-N_1}\widetilde{{T}}+T\big ) \end{aligned}$$
    induced by \(\eta \) equals \(\eta _p\).
Then \([\eta ]\) is non-empty and a single \(U_p(N_1,N_2)\)-orbit.

Proof

Let \(e_1,\ldots ,e_n\) denote a \({\mathcal {O}}_{F,p}\)-basis of \(T/\widetilde{{T}}\). Note that \(\langle \,\,\,,\,\,\,\rangle \) induces a perfect pairing between \(\widetilde{{T}}\) and \(T/\widetilde{{T}}\). We recursively lift the \(e_i\) to elements \(\widetilde{{e}}_i \in T\) with \(\widetilde{{e}}_i\) orthogonal to the \({\mathcal {O}}_{F,p}\) span of the \(\widetilde{{e}}_j\) for \(j< i\). Suppose that \(\widetilde{{e}}_1,\ldots ,\widetilde{{e}}_{i-1}\) have already been chosen. Choose some lift \(e_i'\) of \(e_i\). Then choose \(t \in \widetilde{{T}}\) such that
  • \(\langle t , x \rangle =\langle e_i',x \rangle \) for all \(x \in \bigoplus _{j=1}^{i-1} {\mathcal {O}}_{F,p} \widetilde{{e}}_j\),

  • and \(\langle t, \alpha e_i' \rangle =\langle e_i'/2,\alpha e_i'\rangle \) for all \(\alpha \in {\mathcal {O}}_{F,p}^{c=-1}\).

(If \(p=2\) some explanation is required as to why we can do this. The map
$$\begin{array}{rcl} {\mathcal {O}}_{F,p} &{} \longrightarrow &{} {\mathbb {Z}}_p \\ \alpha &{} \longmapsto &{} \left\langle e_i',\alpha e_i'\right\rangle \end{array}$$
is of the form
$$\begin{aligned} \alpha \longmapsto {{\text {tr}}}_{F/{\mathbb {Q}}}( \beta \alpha ) \end{aligned}$$
for some \(\beta \in ({\mathcal {D}}_{F,p}^{-1})^{c=-1}\). Because \(p=2\) is unramified in \(F/F^+\), we can write \(\beta =\gamma -{}^c\gamma \) for some \(\gamma \in {\mathcal {D}}_{F,p}^{-1}\). Thus the second condition can be replaced by the condition
$$\begin{aligned} \left\langle t, \alpha e_i'\right\rangle ={{\text {tr}}}_{F/{\mathbb {Q}}}( \gamma \alpha ) \end{aligned}$$
for all \(\alpha \in {\mathcal {O}}_{F,p}^{c=-1}\). Now it is clear that the required element t exists.) Then take \(\widetilde{{e}}_i=e_i'-t\). Then \(\widetilde{{e}}_i\) is orthogonal to \(\bigoplus _{j=1}^{i-1} {\mathcal {O}}_{F,p} \widetilde{{e}}_j\). Moreover for \(\alpha \in {\mathcal {O}}_{F,p}\) we have
$$\begin{array}{rcl} \left\langle \widetilde{{e}}_i,\alpha \widetilde{{e}}_i \right\rangle &{}=&{} \left\langle e_i',\alpha e_i'\right\rangle -\left\langle t, (\alpha -{}^c\alpha )e_i' \right\rangle \\ &{}=&{} \left\langle e_i',\alpha e_i'\right\rangle -\left\langle e_i'/2, (\alpha -{}^c\alpha )e_i' \right\rangle \\ &{}=&{} \left( \left\langle e_i', \alpha e_i' \right\rangle + \left\langle e_i', {}^c\alpha e_i' \right\rangle \right) \big /2 \\ &{}=&{} 0. \end{array}$$
Thus we can write
$$\begin{aligned} T=\widetilde{{T}}\oplus \widetilde{{T}}' \end{aligned}$$
with \(\widetilde{{T}}'\) an isotropic \({\mathcal {O}}_{F,p}\)-subspace of T, which is free over \({\mathcal {O}}_{F,p}\) of rank n. We see that
$$\begin{aligned} \widetilde{{T}}' \cong {{\text {Hom}}}_{{\mathbb {Z}}_p}(\widetilde{{T}}, {\mathbb {Z}}_p). \end{aligned}$$
The lemma now follows without difficulty. \(\square \)

Corollary 3.3

If Y is a \({\mathbb {Q}}\)-scheme with geometric point \({\overline{{y}}}\), if \((A,i,\lambda )/Y\) is an ordinary \(G_n\)-abelian scheme and if \([(\eta _0^p,\eta _1^p,C, \eta _p)]\) is a \(U^p(N_1,N_2)\)-level structure on \((A,i,\lambda )\), then there is a unique \(U_p(N_1,N_2)\)-orbit of pairs of isomorphisms
$$\begin{aligned} \eta _{0,p}: {\mathbb {Z}}_{p,{\overline{{y}}}} \mathop {\longrightarrow }\limits ^{\sim }{\mathbb {Z}}_p(1)_{\overline{{y}}}\end{aligned}$$
and
$$\begin{aligned} \eta _{1,p}: \varLambda _n \otimes {\mathbb {Z}}_p \mathop {\longrightarrow }\limits ^{\sim }T_pA_{\overline{{y}}}\end{aligned}$$
such that
  • \(\eta _{1,p}(a x)=a \eta _{1,p}(x)\) for all \(a \in {\mathcal {O}}_{F,(p)}\),

  • \(\langle \eta _{1,p}x,\eta _{1,p}y\rangle _\lambda =\eta _{0,p}\langle x,y \rangle _n\) for all \(x,y \in \varLambda _n \otimes {\mathbb {Z}}_p\),

  • \(\eta _{1,p} p^{-N_2} \varLambda _{n,(n)}/\varLambda _{n,(n)}=C\),

  • \(\eta _{1,p}\) induces \(\eta _p\).

Proof

This follows on combining Lemmas 3.1 and 3.2. \(\square \)

Corollary 3.4

Suppose that Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Q}}\). There is a natural bijection between prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure and quasi-isogeny classes of quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure.

Proof

We may assume that Y is connected with geometric point \({\overline{{y}}}\). We will show both sets are in natural bijection with the set of prime-to-p quasi-isogeny classes of four-tuples \((A,i,\lambda ,[\eta ])\), where (Ai) is a \(G_n\)-abelian scheme, \(\lambda \) is a prime-to-p quasi-polarization of (Ai), and \([\eta ]\) is a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p(N_1,N_2)\)-orbit of pairs \((\eta _0,\eta _{1})\), where
  • \(\eta _0: {\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1) \times {\mathbb {Z}}_p(1)\),

  • and \(\eta _1: \varLambda _n \otimes ({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p) \mathop {\rightarrow }\limits ^{\sim }V^pA_{\overline{{y}}}\times T_pA_{\overline{{y}}}\) satisfies
    $$\begin{aligned} \eta _0 \langle x,y \rangle _n=\langle \eta _1 x, \eta _1 y\rangle _\lambda . \end{aligned}$$
There is a natural map from this set to the set of quasi-isogeny classes of quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure, which is easily checked to be a bijection. The bijection between this set and the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized \(G_n\)-abelian schemes with a \(U^p(N_1,N_2)\)-level structure, follows by the usual arguments (see, for instance, section III.1 of [29]) from Corollary 3.3. \(\square \)

4.1.2 Hecke actions

If \((A,i,\lambda ,[\eta ])/Y\) is a quasi-polarized, \(G_n\)-abelian scheme with U-level structure and if \(g \in G_n({\mathbb {A}}^{\infty })\) with \(U' \supset g^{-1} U g\), then we can define a quasi-polarized, \(G_n\)-abelian scheme with \(U'\)-level structure \((A,i,\lambda ,[\eta ])g /Y\) by
$$\begin{aligned} (A,i,\lambda ,[(\eta _0,\eta _1)])g=(A,i,\lambda ,[(\nu (g) \eta _0, \eta _1 \circ g]). \end{aligned}$$
This action takes one quasi-isogeny class to another.
If \((A,i,\lambda ,[\eta ])/Y\) is an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \(U^p(N_1,N_2)\)-level structure and if \(g\in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) with
$$\begin{aligned} (U')^p(N_1',N_2') \supset g^{-1} U^p(N_1,N_2) g \end{aligned}$$
(so that in particular \(N_i \ge N_i'\) for \(i=1,2\)), then we can define an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \((U')^p(N_1',N_2')\)-level structure \((A,i,\lambda ,[\eta ])g /Y\) by
$$\left( A,i,\lambda ,\left[ \left( \eta _0^p,\eta _1^p,C,\eta _p\right) \right] \right) g= \left( A,i,\lambda ,\left[ \left( \nu (g^p) \eta _0^p, \eta _1^p \circ g^p,C[p^{N_2'}],\eta _p \circ g_p\right) \right] \right) .$$
Recall the definition of \(\varsigma _p\) towards the end of Sect. 1.2. If \((U')^p(N_1',N_2') \supset \varsigma _p^{-1} U^p(N_1,N_2) \varsigma _p\) (so that in particular \(N_1 \ge N_1'\) and \(N_2> N_2'\)), then we can define an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \((U')^p(N_1',N_2')\)-level structure \((A,i,\lambda ,[\eta ])\varsigma _p/Y\) by
$$\begin{array}{l} \left( A,i,\lambda ,\left[ \left( \eta _0^p,\eta _1^p,C,\eta _p\right) \right] \right) \varsigma _p =\left( A/C[p],i,F(\lambda ) ,\left[ \left( p\eta _0^p, F(\eta _1^p), C[p^{1+N_2'}]\big /C[p], F(\eta _p)\right) \right] \right) ; \end{array}$$
where
$$F(\lambda ): A/C[p] \mathop {\longrightarrow }\limits ^{\lambda } A^\vee /\lambda C[p]=A^\vee /C[p]^\perp \mathop {\longrightarrow }\limits ^{\sim }(A/C[p])^\vee $$
with the last isomorphism being induced by the dual of the map \(A/C[p] \rightarrow A\) induced by multiplication by p on A; where \(F(\eta _1^p)\) is the composition of \(\eta _1^p\) with the natural map \(V^pA \mathop {\rightarrow }\limits ^{\sim }V^p(A/C[p])\); and where \(F(\eta _p)\) is the composition of \(\eta _p\) with the natural identification
$$A[p^{N_1'}]/ (C \cap A[p^{N_1'}])=(A/C[p])[p^{N_1'}]/ (C[p^{1+N_2'}]/C[p] \cap (A/C[p])[p^{N_1'}]).$$
If Y is an \({\mathbb {F}}_p\)-scheme then \(\varsigma _p\) is the composite of pull-back by absolute Frobenius followed by forgetting some of the structure.

Together these two definitions give an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). This action takes one prime-to-p quasi-isogeny class to another.

With these definitions the correspondence of Corollary 3.4 is \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant.

4.1.3 Representability

If U is a neat open compact subgroup of \(G_n({\mathbb {A}}^\infty )\) then the functor that sends a (locally noetherian) scheme \(S/{\mathbb {Q}}\) to the set of quasi-isogeny classes of polarized \(G_n\)-abelian schemes with U-level structures is represented by a quasi-projective scheme \(X_{n,U}\) which is smooth of relative dimension \(n^2[F^+: {\mathbb {Q}}]\) over \({\mathbb {Q}}\). Let
$$\begin{aligned} \left[ \left( A^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \right] \big / X_{n,U} \end{aligned}$$
denote the universal equivalence class of polarized \(G_n\)-abelian schemes with U-level structure. If \(U' \supset g^{-1}Ug\) then there is a map \(g: X_{n,U} \rightarrow X_{n,U'}\) arising from \((A^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/X_{n,U}\) and the universal property of \(X_{n,U'}\). This makes \(\{ X_{n,U}\}\) an inverse system of schemes with right \(G_n({\mathbb {A}}^{\infty })\)-action. The maps g are finite etale. If \(U_1 \subset U_2\) is a normal subgroup then \(X_{n,U_1}/X_{n,U_2}\) is Galois with group \(U_2/U_1\).
There are identifications of topological spaces:
$$X_{n,U}({\mathbb {C}}) \cong G_n({\mathbb {Q}})^+\big \backslash \left( G_n({\mathbb {A}}^\infty )/U \times {\mathfrak {H}}_n^+\right) \cong G_n({\mathbb {Q}})\big \backslash \left( G_n({\mathbb {A}}^\infty )/U \times {\mathfrak {H}}_n^\pm \right) $$
compatible with the right action of \(G_n({\mathbb {A}}^{\infty })\). (See sections 7 and 8 of [39]. Note that \(\ker ^1({\mathbb {Q}},G_n)=(0)\), as is explained in section 7 of [39].) More precisely we associate to \((g,I) \in G_n({\mathbb {A}}^{\infty })/U \times {\mathfrak {H}}_n^+\) the torus \((\varLambda _n \otimes _{\mathbb {Z}}{\mathbb {R}})/\varLambda _n\) with complex structure coming from I; with polarization corresponding to the Riemann form given by \(\langle \,\,\, ,\,\,\,\rangle \); and with level structure coming from
$$\begin{aligned} \eta _1: \varLambda _n \otimes {\mathbb {A}}^{\infty } \mathop {\longrightarrow }\limits ^{g} \varLambda _n \otimes {\mathbb {A}}^{\infty }=V((\varLambda _n \otimes _{\mathbb {Z}}{\mathbb {R}})/\varLambda _n) \end{aligned}$$
and
$$\begin{array}{rcl} \eta _0: {\mathbb {A}}^{\infty } &{}\mathop {\longrightarrow }\limits ^{\sim }&{}{\mathbb {A}}^{\infty }(1) \\ x &{} \longmapsto &{} -\nu (g) x \zeta , \end{array}$$
where \(\zeta =\lim _{\leftarrow N} e^{2 \pi i/N} \in \widehat{{\mathbb {Z}}}(1)\). We deduce that
$$\begin{array}{rcl} \pi _0(X_{n,U} \times {{\text {Spec}}\,}\overline{{{\mathbb {Q}}}}) &{} \cong &{} G_n({\mathbb {Q}}) \backslash G_n({\mathbb {A}})\big /\big (U G_n({\mathbb {R}})^+\big ) \\ &{} \cong &{} G_n({\mathbb {Q}}) \big \backslash \left( G_n({\mathbb {A}}^\infty )\big /U \times \pi _0(G_n({\mathbb {R}}))\right) \\ &{} \cong &{} C_n({\mathbb {Q}}) \backslash C_n({\mathbb {A}}) \big / U C_n({\mathbb {R}})^0 . \end{array}$$
If \(U^p\) is neat then the functor that sends a scheme \(Y/{\mathbb {Z}}_{(p)}\) to the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure is represented by a scheme \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) quasi-projective over \({\mathbb {Z}}_{(p)}\). (See theorems 3.4.1.9 and 3.4.2.5 in [44]. Note that, by theorem 3.4.1.9 in [44], the naive moduli problem there is already smooth, and hence the submoduli problem with the right Lie algebra condition agrees with the normalization in theorem 3.4.2.5 in [44].) Let
$$\begin{aligned} \left[ \left( {\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \right] / {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\end{aligned}$$
denote the universal equivalence class. If \(g \in G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \((U^p)'(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\), then there is a quasi-finite map
$$\begin{aligned} g: {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\longrightarrow {\mathcal {X}}_{(U^p)'(N_1',N_2')}^{{\text {ord}}}\end{aligned}$$
arising from \(({\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) and the universal property of \({\mathcal {X}}_{n,(U^p)'(N_1',N_2')}^{{\text {ord}}}\). If \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map g is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(U^p(N_1,N_2)\) is a normal subgroup of \((U^p)'(N_1',N_2)\) then \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}/{\mathcal {X}}_{n,(U^p)'(N_1',N_2)}^{{\text {ord}}}\) is Galois with group \((U^p)'(N_1')/U^p(N_1)\). There are \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant identifications
$$\begin{aligned} {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\times {{\text {Spec}}\,}{\mathbb {Q}}\cong X_{n,U^p(N_1,N_2)}. \end{aligned}$$
The scheme \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) is smooth over \({\mathbb {Z}}_{(p)}\) of relative dimension \(n^2[F^+: {\mathbb {Q}}]\). (By the Serre–Tate theorem (see [36]) the formal completion of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) at a point x in the special fibre is isomorphic to
$$\begin{aligned} {{\text {Hom}}}_{{\mathbb {Z}}_p}\left( S\left( T_p{\mathcal {A}}^{{\text {univ}}}_x\right) ,\widehat{{\mathbb {G}}}_m\right) . \end{aligned}$$
This is formally smooth as long as \(S(T_p{\mathcal {A}}^{{\text {univ}}}_x) \cong S({\mathcal {O}}_{F,p}^n)\) is torsion free. This module is torsion free because in the case \(p=2\) we are assuming that \(p=2\) is unramified in \(F/F^+\).) Suppose that \(g \in G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \((U^p)'(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\), then the quasi-finite map
$$\begin{aligned} g: {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\longrightarrow {\mathcal {X}}_{n,(U^p)'(N_1',N_2')}^{{\text {ord}}}\end{aligned}$$
is in fact flat, because it is a quasi-finite map between locally noetherian regular schemes which are equidimensional of the same dimension. (See pages 507 and 508 of [37].)
On \({\mathbb {F}}_p\)-fibres the map
$$\begin{aligned} \varsigma _p: {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2+1)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$
is the absolute Frobenius map composed with the forgetful map \(1: {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2+1)} \rightarrow {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) (for any \(N_2 \ge N_1 \ge 0\)). Thus if \(N_2>0\), then the quasi-finite, flat map
$$\begin{aligned} \varsigma _p: {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2+1)} \longrightarrow {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \end{aligned}$$
has all its fibres of degree \(p^{n^2[F^+: {\mathbb {Q}}]}\) and hence is finite flat of this degree. (A flat, quasi-finite morphism \(f: X \rightarrow Y\) between noetherian schemes with constant fibre degree is proper and hence, by theorem 8.11.1 of [24], finite. We give the argument for properness. By the valuative criterion we may reduce to the case \(Y={{\text {Spec}}\,}B\) for a DVR B with fraction field L. By theorem 8.12.6 of [24] X is a dense open subscheme of \({{\text {Spec}}\,}A\), for A a finite B algebra. Let I denote the ideal of A consisting of all \({\mathfrak {m}}_B\)-torsion elements. If \(f \in A\) and \({{\text {Spec}}\,}A_f \subset X\), then by flatness the map \(A \rightarrow A_f\) factors through A / I. Thus \(X \subset {{\text {Spec}}\,}A/I\) and in fact \(I=(0)\), so that A / B is finite flat. Because an open subscheme is determined by its points, we conclude that we must have \(X={{\text {Spec}}\,}A'\) for some \(A \subset A' \subset A \otimes _B L\). By the constancy of the fibre degree we conclude that \(A'\) is finite over B.) We deduce that for any \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}}}\), if \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \), then the map
$$\begin{aligned} g: {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\longrightarrow {\mathcal {X}}_{(U^p)'(N_1',N_2')}^{{\text {ord}}}\end{aligned}$$
is finite.

Lemma 3.5

Write \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) for the completion of \({\mathcal {X}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) along its \({\mathbb {F}}_p\)-fibre. If \(N_2'>N_2 \ge N_1\) then the map
$$\begin{aligned} 1: {\mathcal {X}}_{n,U^p(N_1,N_2')}^{{{\text {ord}}},\wedge } \longrightarrow {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge } \end{aligned}$$
is an isomorphism.

Proof

The map has an inverse which sends a tuple \([({\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}}, [\eta ^{{\text {univ}}}])]\) over \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) to
$$\left[ \left( {\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}}, \left[ \left( \eta _0^{{{\text {univ}}}, p}, \eta _1^{{{\text {univ}}},p},{\mathcal {A}}^{{\text {univ}}}\left[ p^{N_2'}\right] ^0 , \eta _p^{{\text {univ}}}\right) \right] \right) \right] $$
over \({\mathcal {X}}_{n,U^p(N_1,N_2')}^{{{\text {ord}}},\wedge }\). \(\square \)
Thus we will denote \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) simply
$$\begin{aligned} {\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N_1)}. \end{aligned}$$
Then \(\{ {\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N)} \}\) is a system of p-adic formal schemes with right \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action. We will write \(\overline{{X}}^{{\text {ord}}}_{n,U^p(N)}\) for the reduced subscheme of \({\mathfrak {X}}_{n,U^p(N)}^{{\text {ord}}}\).

Throughout the paper we will use usual Roman letters, such as X, for ‘Shimura-like’ varieties of finite type over \({\mathbb {Q}}\), cursive letters, such as \({\mathcal {X}}\), for models of them of finite type over \({\mathbb {Z}}_{(p)}\), over-lined usual Roman letters, such as \(\overline{{X}}\), for their \({\mathbb {F}}_p\)-fibre, and Gothic letters, such as \({\mathfrak {X}}\), for their formal completion along this special fibre.

4.2 Some Kuga–Sato varieties

Recall that a semi-abelian scheme is a smooth separated commutative group scheme such that each geometric fibre is the extension of an abelian scheme by a torus. To a semi-abelian scheme G / Y one can associate an etale constructible sheaf of abelian groups \(X^*(G)\), the ‘character group of the toric part of G’. See theorem I.2.10 of [17]. If \(X^*(G)\) is locally constant then G is an extension
$$\begin{aligned} (0) \longrightarrow S_G \longrightarrow G \longrightarrow A_G \longrightarrow (0) \end{aligned}$$
of a uniquely determined abelian scheme \(A_G\) by a uniquely determined split torus \(S_G\) with character group \(X^*(G)\). By an isogeny (resp. prime-to-p isogeny) of semi-abelian schemes we mean a morphism which is quasi-finite and surjective (resp. quasi-finite and surjective and whose geometric fibres have orders relatively prime to p). If Y is locally noetherian, then by a quasi-isogeny (resp. prime-to- p quasi-isogeny) \(\alpha : G \rightarrow G'\) we mean an element of \({{\text {Hom}}}(G,G')_{{\mathbb {Q}}}\) (resp. \({{\text {Hom}}}(G,G')_{{\mathbb {Z}}_{(p)}}\)) with an inverse in \({{\text {Hom}}}(G',G)_{{\mathbb {Q}}}\) (resp. \({{\text {Hom}}}(G',G)_{{\mathbb {Z}}_{(p)}}\)).
Suppose that \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\) is a locally noetherian scheme. By a \(G_n^{(m)}\)-semi-abelian scheme G over Y we mean a triple (Gij) where
  • G / Y is a semi-abelian scheme,

  • \(i: F \hookrightarrow {{\text {End}}}(G)_{\mathbb {Q}}\),

  • and \(j: F^m \mathop {\rightarrow }\limits ^{\sim }X^*(G)_{\mathbb {Q}}\) is an F-linear isomorphism;

  • such that \({{\text {Lie}}\,}A_G\) is a free \({\mathcal {O}}_Y \otimes _{\mathbb {Q}}F\) module of rank \(n[F: {\mathbb {Q}}]\).

Then \(A_G\) is a \(G_n\)-abelian scheme. By a quasi-isogeny of \(G_n^{(m)}\)-semi-abelian schemes we mean a quasi-isogeny of semi-abelian schemes
$$\begin{aligned} \beta : G \rightarrow G' \end{aligned}$$
such that
$$\begin{aligned} i'(a) \circ \beta =\beta \circ i(a) \end{aligned}$$
for all \(a \in F\), and
$$\begin{aligned} j=X^*(\beta ) \circ j'. \end{aligned}$$
Note that, if \({\overline{{y}}}\) is a geometric point of Y, then j induces a map
$$\begin{aligned} j^*: VS_{G,{\overline{{y}}}} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}\left( F^m,V{\mathbb {G}}_{m,{\overline{{y}}}}\right) . \end{aligned}$$
By a quasi-polarization of (Gij) we mean a quasi-polarization of \(A_G\).
If Y is connected and \({\overline{{y}}}\) is a geometric point of Y and if \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) is a neat open compact subgroup then by a U level structure on a quasi-polarized \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant U-orbit of pairs \((\eta _0,\eta _1)\) where
$$\begin{aligned} \eta _0: {\mathbb {A}}^\infty \mathop {\longrightarrow }\limits ^{\sim }V{\mathbb {G}}_{m,{\overline{{y}}}} \end{aligned}$$
is an \({\mathbb {A}}^\infty \)-linear map, and where
$$\begin{aligned} \eta _1: \varLambda _n^{(m)} \otimes _{\mathbb {Z}}{\mathbb {A}}^\infty \longrightarrow VG_{\overline{{y}}}\end{aligned}$$
is an \({\mathbb {A}}_F^\infty \)-linear map such that
$$\begin{aligned} \eta _1|_{{{\text {Hom}}}_{\mathbb {Z}}({\mathcal {O}}_F^m,{\mathbb {A}}^\infty )}=(j^*)^{-1}\circ {{\text {Hom}}}(1_{F^m},\eta _0) \end{aligned}$$
and
$$\begin{aligned}{}[(\eta _0, \eta _1 \bmod VS_{G,{\overline{{y}}}})] \end{aligned}$$
is a U-level structure on \(A_G\). This is canonically independent of \({\overline{{y}}}\). We define a U level structure on a \(G_n^{(m)}\)-semi-abelian scheme over a general locally noetherian scheme Y to be such a level structure over each connected component of Y. By a quasi-isogeny between two quasi-polarized, \(G_n^{(m)}\)-semi-abelian schemes with U-level structure
$$\begin{aligned} (\beta ,\delta ): (G,i,j,\lambda ,[(\eta _0,\eta _1)]) \longrightarrow (G',i',j',\lambda ',[(\eta _0',\eta _1')]) \end{aligned}$$
we mean a quasi-isogeny
$$\begin{aligned} \beta : (G,i,j) \longrightarrow (G',i',j') \end{aligned}$$
and an element \(\delta \in {\mathbb {Q}}^\times \) such that
$$\begin{aligned} \delta \lambda =\beta ^\vee \circ \lambda ' \circ \beta \end{aligned}$$
and
$$\begin{aligned} \left[ \left( \eta _0',\eta _1'\right) \right] =[( \delta \eta _0, V(\beta ) \circ \eta _1)]. \end{aligned}$$
If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(U'\supset g^{-1}Ug\) then we define a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with \(U'\)-level structure
$$\begin{aligned} (G,i,j,\lambda ,[(\eta _0,\eta _1)])g=(G,i,j,\lambda ,[(\nu (g)\eta _0,\eta _1\circ g)]). \end{aligned}$$
The quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])g\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(\gamma \in GL_m(F)\) and \(U' \supset \gamma U\) then we define a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with \(U'\)-level structure
$$\begin{aligned} \gamma (G,i,j,\lambda ,[(\eta _0,\eta _1)])=(G,i,j \circ \gamma ^{-1},\lambda ,[(\eta _0,\eta _1 \circ \gamma ^{-1})]). \end{aligned}$$
The quasi-isogeny class of \(\gamma (G,i,j,\lambda ,[(\eta _0,\eta _1)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \). If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(m'\le m\) and if \(U'\supset i_{m',m}^*U\), then we define a quasi-polarized, \(G_n^{(m')}\)-semi-abelian scheme with \(U'\)-level structure
$$\pi _{m,m'} (G,i,j,\lambda ,[(\eta _0,\eta _1)])= \left( G/S,i,j \circ i_{m',m},\lambda ,\left[ \left( \eta _0,\eta _1'\right) \right] \right) ,$$
where \(S \subset S_G\) is the subtorus with
$$\begin{aligned} X^*(S)=X^*(S_G) / \left( X^*(S_G) \cap j \circ i_{m',m} F^{m'}\right) \end{aligned}$$
and where
$$\begin{aligned} \eta _1' \circ i_{m',m}^*=\eta _1 \bmod VS. \end{aligned}$$
The quasi-isogeny class of \(\pi _{m,m'}(G,i,j,\lambda ,[(\eta _0,\eta _1)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). If \(\gamma \in Q_{m,m'}(F)\) then \(\pi _{m,m'} \circ \gamma =\overline{\gamma }\circ \pi _{m,m'}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}(F)\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) then \(\pi _{m,m'} \circ g=i_{m',m}^*(g)\circ \pi _{m,m'}\).
If U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) then the functor which sends a locally noetherian scheme \(Y/{\mathbb {Q}}\) to the set of quasi-isogeny classes of quasi-polarized \(G_n^{(m)}\)-semi-abelian schemes with U-level structure is represented by a quasi-projective scheme \(A_{n,U}^{(m)}\), which is smooth of dimension \(n(n+2m)[F^+: {\mathbb {Q}}]\). (See proposition 1.3.2.14 of [44].) We remark that according to our notational conventions we have
$$\begin{aligned} A_{n,U}^{(0)}=X_{n,U}. \end{aligned}$$
Let
$$\begin{aligned} \left[ \left( G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \right] /A_{n,U}^{(m)} \end{aligned}$$
denote the universal quasi-isogeny class of quasi-polarized \(G_n^{(m)}\)-semi-abelian schemes with U-level structure. If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(U_1,U_2\) are neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with \(U_2 \supset g^{-1}U_1g\) then there is a map
$$\begin{aligned} g: A^{(m)}_{n,U_1} \longrightarrow A^{(m)}_{n,U_2} \end{aligned}$$
arising from \((G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/A^{(m)}_{n,U_1}\) and from the universal property of \(A_{n,U_2}^{(m)}\). Similarly if \(\gamma \in GL_m(F)\) and \(U_1,U_2\) are neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with \(U_2 \supset \gamma U_1\) then there is a map
$$\begin{aligned} \gamma : A^{(m)}_{n,U_1} \longrightarrow A^{(m)}_{n,U_2} \end{aligned}$$
arising from \(\gamma (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])/A^{(m)}_{n,U_1}\) and from the universal property of \(A_{n,U_2}^{(m)}\). Moreover if \(m' \le m\), if \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(U'\) denotes the image of U in \(G_n^{(m')}({\mathbb {A}}^\infty )\), then there is a smooth projective map
$$\begin{aligned} \pi _{A^{(m)}_n/A^{(m')}_n}: A^{(m)}_{n,U} \longrightarrow A_{n,U'}^{(m')} \end{aligned}$$
arising from \(\pi _{m,m'} (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])/A^{(m)}_{n,U}\) and the universal property of \(A_{n,U'}^{(m')}\). (We will sometimes write \(\pi _{A^{(m)}_n/X_n}\) for \(\pi _{A^{(m)}_n/A^{(0)}_n}\).) We see that these actions have the following properties.
  • \(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).

  • If \(\gamma \in Q_{m,m'}(F)\) then \(\pi _{A^{(m)}_n/A^{(m')}_n} \circ \gamma =\overline{\gamma }\circ \pi _{A^{(m)}_n/A^{(m')}_n}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}(F)\).

  • \(\pi _{A^{(m)}_n/A^{(m')}_n} \circ g=g'\circ \pi _{A^{(m)}_n/A^{(m')}_n}\), where \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^\infty )\).

Moreover we have the following properties.
  • The maps g and \(\gamma \) are finite etale. The maps \(\pi _{m,m'}\) are smooth and projective.

  • If \(U_1 \subset U_2\) is an open normal subgroup of a neat open compact subgroup then \(A_{n,U_1}^{(m)}/A_{n,U_2}^{(m)}\) is Galois with group \(U_2/U_1\).

  • If \(U=U' \ltimes M\) with \(U' \subset G_n({\mathbb {A}}^\infty )\) and \(M \subset {{\text {Hom}}}_n^{(m)}({\mathbb {A}}^\infty )\) then \(A^{(m)}_{n,U}/X_{n,U'}\) is an abelian scheme of relative dimension \(mn[F: {\mathbb {Q}}]\).

  • In general \(A^{(m)}_{n,U}\) is a principal homogenous space for \(A^{(m)}_{n,U' \ltimes (U \cap {{\text {Hom}}}^{(m)}_n({\mathbb {A}}^\infty ))}\) over \(X_{n,U'}\), where \(U'\) denotes the image of U in \(G_n({\mathbb {A}}^\infty )\).

  • There are \(G^{(m)}_{n}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant homeomorphisms
    $$A^{(m)}_{n,U}({\mathbb {C}}) \cong G^{(m)}_{n}({\mathbb {Q}}) \big \backslash G^{(m)}_{n}({\mathbb {A}}) \big / \left( U \times U_{n,\infty }^0A_n({\mathbb {R}})^0\right) .$$
Moreover in the case \(U=U' \ltimes M\), if \(G^{{\text {univ}}}/A_{n,U}^{(m)}\) and \(A^{{\text {univ}}}/X_{n,U'}\) are chosen so that \(\pi _{A_n^{(m)}/X_{n}}^* A^{{\text {univ}}}\cong A_{G^{{\text {univ}}}}\), then there is a \({\mathbb {Q}}\)-linear map
$$\begin{aligned} i^{(m)}_{A^{{\text {univ}}}}: F^m \longrightarrow {{\text {Hom}}}_{/X_{n,U'}}\left( A_{n,U}^{(m)}, (A^{{\text {univ}}})^\vee \right) _{\mathbb {Q}}\end{aligned}$$
with the following properties.
  • If \(a \in F\) then
    $$\begin{aligned} i^{(m)}_{A^{{\text {univ}}}}(ax)=i^{{{\text {univ}}},\vee }({}^ca) \circ i^{(m)}_{A^{{\text {univ}}}}(x). \end{aligned}$$
  • If \((\beta ,\delta )\) is a quasi-isogeny
    $$\left( G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \longrightarrow \left( G^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]\right) ,$$
    then
    $$\begin{aligned} \beta ^\vee \circ i^{(m)}_{(A^{{\text {univ}}})'}(x)=i^{(m)}_{A^{{\text {univ}}}}(x). \end{aligned}$$
    In particular \(i^{(m)}_{A^{{\text {univ}}}}\) depends only on \(A^{{\text {univ}}}\) and not on \(G^{{\text {univ}}}\).
  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\gamma \in GL_m(F)\) then
    $$\begin{aligned} i^{(m)}_{A^{{\text {univ}}}}(x) \circ g=i^{(m)}_{g^* A^{{\text {univ}}}}(x) \end{aligned}$$
    and
    $$i^{(m)}_{A^{{\text {univ}}}}(x) \circ \gamma =i^{(m)}_{\gamma ^* A^{{\text {univ}}}}(\gamma ^{-1} x).$$
  • If \(e_1,\ldots ,e_m\) denotes the standard basis of \(F^m\) then
    $$\begin{aligned}&i_{A^{{\text {univ}}}}=||\eta _0^{{\text {univ}}}||^{-1}\left( (\lambda ^{{\text {univ}}})^{-1}\circ i^{(m)}_{A^{{\text {univ}}}}(e_1),\ldots , (\lambda ^{{\text {univ}}})^{-1} \circ i^{(m)}_{A^{{\text {univ}}}}(e_m)\right) : A_{n,U}^{(m)} \longrightarrow (A^{{\text {univ}}})^m \end{aligned}$$
    is a quasi-isogeny. If \((\beta ,\delta )\) is a quasi-isogeny
    $$\begin{aligned} (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]) \longrightarrow (G^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]), \end{aligned}$$
    then
    $$\begin{aligned} \beta ^{\oplus m} \circ i_{A^{{\text {univ}}}}=i_{(A^{{\text {univ}}})'}. \end{aligned}$$
  • The map
    $$\begin{array}{rcccc} \eta _{n,U}^{(m)}: {{\text {Hom}}}_F(F^m,V_n) \otimes _{\mathbb {Q}}{\mathbb {A}}^\infty &{}\mathop {\rightarrow }\limits ^{\sim }&{} V(A^{{\text {univ}}})^m&{}\mathop {\rightarrow }\limits ^{\sim }&{} VA_{n,U}^{(m)} \\ f &{} \mapsto &{} (\eta _1^{{\text {univ}}}(f(e_1)),\ldots ,\eta _1^{{\text {univ}}}(f(e_m))) &{}&{}\\ &{}&{} x &{} \mapsto &{} V(i_{A^{{\text {univ}}}})^{-1} x \end{array} $$
    is an isomorphism, which does not depend on the choice of \(G^{{\text {univ}}}\). It satisfies
    $$\begin{aligned} \eta _{n,U}^{(m)} M=TA_{n,U}^{(m)}. \end{aligned}$$
(See lemmas 1.3.2.7 and 1.3.2.50, propositions 1.3.2.14, 1.3.2.24 and 1.3.2.55, theorem 1.3.3.15, and remark 1.3.3.33 of [44]; and section 3.5 of [43].)
Note that
$$\begin{aligned} i_{A^{{\text {univ}}}} \circ g=i_{g^* A^{{\text {univ}}}} \end{aligned}$$
and
$$\begin{aligned} i_{A^{{\text {univ}}}} \circ \gamma ={}^t\gamma ^{-1} \circ i_{\gamma ^* A^{{\text {univ}}}}. \end{aligned}$$
Define
$$i_\lambda ^{(m)} : F^m \otimes _{F,c}F^m \longrightarrow {{\text {Hom}}}_{/X_{n,U'}}\left( A_{n,U}^{(m)}, A_{n,U}^{(m),\vee }\right) _{\mathbb {Q}}$$
by
$$\begin{aligned} i_\lambda ^{(m)}(x \otimes y)=||\eta _0^{{\text {univ}}}||^{-1} i^{(m)}_{A^{{\text {univ}}}}(x)^\vee \circ \lambda ^{{{\text {univ}}},-1} \circ i^{(m)}_{A^{{\text {univ}}}}(y). \end{aligned}$$
This does not depend on the choice of \(A^{{\text {univ}}}\). We have
$$\begin{aligned} i_\lambda ^{(m)}(x \otimes y)^\vee =i_\lambda ^{(m)}(y \otimes x). \end{aligned}$$
Moreover
$$\begin{aligned} \left( i_{A^{{\text {univ}}}}^{-1}\right) ^\vee \circ i_\lambda ^{(m)}(x \otimes y) \circ i_{A^{{\text {univ}}}}^{-1}=(\lambda ^{{\text {univ}}})^{\oplus m} \circ i^{{\text {univ}}}({}^{c,t}xy). \end{aligned}$$
If \(a \in (F^m \otimes _{F,c} F^m)^{{{\text {sw}}}=1}\) has image in \(S(F^m)\) lying in \(S(F^m)^{>0}\) then
$$\begin{aligned} \left( i_{A^{{\text {univ}}}}^{-1}\right) ^\vee \circ i_\lambda ^{(m)}(a) \circ i_{A^{{\text {univ}}}}^{-1}=(\lambda ^{{\text {univ}}})^{\oplus m} \circ i^{{\text {univ}}}(a') \end{aligned}$$
for some matrix \(a' \in M_{m \times m}(F)^{t=c}\) all whose eigenvalues are positive real numbers. (See Sect. 1.1 for the definition of \({{\text {sw}}}\).) Thus \(i_\lambda ^{(m)}(a)\) is a quasi-polarization. (See the end of section 21 of [47].)
Now suppose that \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) is a locally noetherian scheme. By an ordinary \(G_n^{(m)}\)-emi-abelian scheme G over Y we mean a triple (Gij) where
  • G / Y is a semi-abelian scheme such that \(X^*(G)\) is locally constant over Y, and such that \(\# G[p](k({\overline{{y}}})) \ge p^{(n+m)[F: {\mathbb {Q}}]}\) for each geometric point \({\overline{{y}}}\) of Y,

  • \(i: {\mathcal {O}}_{F,(p)} \hookrightarrow {{\text {End}}}(G)_{{\mathbb {Z}}_{(p)}}\) such that \({{\text {Lie}}\,}A_G\) is a free \({\mathcal {O}}_Y \otimes _{{\mathbb {Z}}_{(p)}} {\mathcal {O}}_{F,(p)}\) module of rank \(n[F: {\mathbb {Q}}]\),

  • and \(j: {\mathcal {O}}_{F,(p)}^m \mathop {\rightarrow }\limits ^{\sim }X^*(G)_{{\mathbb {Z}}_{(p)}}\) is a \({\mathcal {O}}_{F,(p)}\)-linear isomorphism.

Then \(A_G\) is an ordinary \(G_n\)-abelian scheme. By a prime-to- p quasi-isogeny of ordinary \(G_n^{(m)}\)-semi-abelian schemes we mean a prime-to-p quasi-isogeny of semi-abelian schemes
$$\begin{aligned} \beta : G \rightarrow G' \end{aligned}$$
such that
$$\begin{aligned} i'(a) \circ \beta =\beta \circ i(a) \end{aligned}$$
for all \(a \in {\mathcal {O}}_{F,(p)}\), and
$$\begin{aligned} j=X^*(\beta ) \circ j'. \end{aligned}$$
Note that, if \({\overline{{y}}}\) is a geometric point of Y, then j induces a map
$$j^*: V^pS_{G,{\overline{{y}}}} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{{\mathbb {Z}}_{(p)}} \left( {\mathcal {O}}_{F,(p)}^m,V^p{\mathbb {G}}_{m,{\overline{{y}}}}\right) .$$
By a prime-to- p quasi-polarization of (Gij) we shall mean a prime-to-p quasi-polarization of \(A_G\).
If Y is connected and \({\overline{{y}}}\) is a geometric point of Y, if \(U^p \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) is a neat open compact subgroup and if \(N_2 \ge N_1 \ge 0\) then by a \(U^p(N_1,N_2)\) level structure on a prime-to-p quasi-polarized ordinary \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p\)-orbit \([\eta ]\) of five-tuples \((\eta _0^p,\eta _1^p,C,D,\eta _p)\) consisting of
  • an \({\mathbb {A}}^{\infty ,p}\)-linear isomorphism \(\eta _0^p: {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1)_{\overline{{y}}}=V^p{\mathbb {G}}_{m,{\overline{{y}}}}\);

  • an \({\mathbb {A}}_F^{\infty ,p}\)-linear isomorphism
    $$\begin{aligned} \eta _1^p: \varLambda _n^{(m)} \otimes _{{\mathbb {Z}}} {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }V^pG_{\overline{{y}}}\end{aligned}$$
    such that \(\eta _1^p|_{{{\text {Hom}}}_{\mathbb {Z}}({\mathcal {O}}_F^m,{\mathbb {A}}^{\infty ,p})}=(j^*)^{-1} \circ {{\text {Hom}}}(1_{{\mathcal {O}}_F^m},\eta _0^p)\);
  • a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(C \subset G[p^{N_2}]\), such that for every geometric point \(\widetilde{{y}}\) of Y there is an \({\mathcal {O}}_{F,(p)}\)-invariant sub-Barsotti–Tate group \(\widetilde{{C}}_{\widetilde{{y}}} \subset G_{\widetilde{{y}}}[p^\infty ]\) with the following properties
    • \(C_{\widetilde{{y}}}=\widetilde{{C}}_{\widetilde{{y}}}[p^{N_2}]\),

    • \(\widetilde{{C}}_{\widetilde{{y}}} \supset S_{G,\widetilde{{y}}}[p^\infty ]\),

    • for all N the subgroup scheme \(\widetilde{{C}}_{\widetilde{{y}}}[p^N]/S_{G,\widetilde{{y}}}[p^N]\) is isotropic in \(A_G[p^N]_{\widetilde{{y}}}\) for the \(\lambda \)-Weil pairing,

    • \(G_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}}\) is ind-etale,

    • the Tate module \(T(G_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}})\) is free over \({\mathcal {O}}_{F,p}\) of rank n;

  • a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(D \subset C[p^{N_1}]\) such that
    $$\begin{aligned} D \mathop {\rightarrow }\limits ^{\sim }C[p^{N_1}]/S_G[p^{N_1}]; \end{aligned}$$
  • and an isomorphism
    $$\begin{aligned} \eta _p: p^{-N_1}\varLambda _n\Big /\left( p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\right) \mathop {\longrightarrow }\limits ^{\sim }G[p^{N_1}]/C[p^{N_1}] \end{aligned}$$
    such that
    $$\begin{aligned} \eta _p(ax)=i(a) \eta _p(x) \end{aligned}$$
    for all \(a \in {\mathcal {O}}_{F,(p)}\) and \(x \in p^{-N_1}\varLambda _n/(p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n)\);
such that
$$\begin{aligned} \left[ \left( \eta ^p_0, \eta ^p_1 \bmod V^pS_G, C/S_G[p^{N_2}], \eta _p\right) \right] \end{aligned}$$
is a \(U^p(N_1,N_2)\)-level structure for \((A_G,i,\lambda )\). This definition is independent of the choice of geometric point \({\overline{{y}}}\) of Y. By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\), we mean the collection of a \(U^p(N_1,N_2)\)-level structure over each connected component of Y.
By a prime-to-p quasi-isogeny between two quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian schemes with \(U^p(N_1,N_2)\)-level structure
$$\begin{aligned} (\beta ,\delta ): (G,i,j,\lambda ,[(\eta _0,\eta _1)]) \longrightarrow \left( G',i',j',\lambda ',\left[ \left( \eta _0',\eta _1'\right) \right] \right) \end{aligned}$$
we mean a prime-to-p quasi-isogeny
$$\begin{aligned} \beta : (G,i,j) \longrightarrow (G',i',j') \end{aligned}$$
and an element \(\delta \in {\mathbb {Z}}_{(p)}^\times \) such that
$$\begin{aligned} \delta \lambda =\beta ^\vee \circ \lambda ' \circ \beta \end{aligned}$$
and
$$\left[ \left( \left( \eta _0^p\right) ',\left( \eta _1^p\right) ',C',D',\eta _p'\right) \right] = \left[ \left( \delta \eta ^p_0, V^p(\beta ) \circ \eta ^p_1, \beta C, \beta D, \beta \circ \eta _p\right) \right] .$$
If \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and if
$$\begin{aligned} (U^p)'\left( N_1',N_2'\right) \supset g^{-1}U^p(N_1,N_2)g \end{aligned}$$
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
$$\left( G,i,j,\lambda ,\left[ \left( \eta ^p_0,\eta ^p_1,C,D,\eta _p\right) \right] \right) g= \left( G,i,j,\lambda ,\left[ \left( \nu (g)\eta _0^p,\eta ^p_1\circ g^p,C,D,\eta _p \circ g_p\right) \right] \right) .$$
The prime-to-p quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])g\) only depends on the prime-to-p quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\). Similarly, if
$$\begin{aligned} \left( G,i,j,\lambda ,\left[ \left( \eta _0^p,\eta _1^p,C,D,\eta _p\right) \right] \right) \end{aligned}$$
is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure and if
$$\begin{aligned} (U^p)'\left( N_1',N_2'\right) \supset \varsigma _p^{-1} U^p(N_1,N_2) \varsigma _p, \end{aligned}$$
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
$$\begin{array}{l} \left( G,i,j,\lambda ,\left[ \left( \eta _0^p,\eta _1^p,C,D,\eta _p\right) \right] \right) \varsigma _p \\ \quad =\left( G/C[p],i,pj ,F(\lambda ), \left[ \left( p\eta _0^p,F\left( \eta ^p_1\right) ,C [p^{1+N_2'}]/C[p],(D'/C[p])[p^{N_1'}],F(\eta _p)\right) \right] \right) ; \end{array}$$
where
$$\begin{aligned} F(\lambda ): A_G/C[p] \mathop {\longrightarrow }\limits ^{\lambda } A_G^\vee /\lambda C[p]=A_G^\vee \big /C[p]^\perp \mathop {\longrightarrow }\limits ^{\sim }(A_G/C[p])^\vee \end{aligned}$$
with the latter isomorphism being induced by the dual of the map \(A_G/C[p] \rightarrow A_G\) induced by multiplication by p on \(A_G\); where \(F(\eta _1^p)\) is the composition of \(\eta _1^p\) with the natural map \(V^pG \mathop {\rightarrow }\limits ^{\sim }V^p(G/C[p])\); where \(D'\) denotes the pre-image of D under the multiplication by p map \(C \rightarrow C\); and where \(F(\eta _p)\) is the composition of \(\eta _p\) with the natural identification
$$G[p^{N_1'}]/(C \cap G[p^{N_1'}])=(G/C[p])[p^{N_1'}]/ (C[p^{1+N_2'}]/C[p] \cap (G/C[p])[p^{N_1'}]).$$
Together these two definitions give an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\).
If \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and if
$$\begin{aligned} (U^p)'\left( N_1',N_2'\right) \supset \gamma U^p(N_1,N_2) \end{aligned}$$
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
$$\gamma \left( G,i,j,\lambda ,\left[ \left( \eta _0^p,\eta _1^p,C,D,\eta _p\right) \right] \right) = \left( G,i,j \circ \gamma ^{-1},\lambda ,\left[ \left( \eta _0^p,\eta ^p_1 \circ \gamma ^{-1},C,D,\eta _p\right) \right] \right) .$$
The prime-to-p quasi-isogeny class of \(\gamma (G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \). If \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(m'\le m\) and if \((U^p)'(N_1',N_2')\supset i_{m',m}^*U^p(N_1,N_2)\), then we define a quasi-polarized, ordinary \(G_n^{(m')}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
$$\pi _{m,m'} \left( G,i,j,\lambda , \left[ \left( \eta ^p_0,\eta ^p_1,C,D,\eta _p\right) \right] \right) = \left( G/S,i,j \circ i_{m',m},\lambda ,\left[ \left( \eta ^p_0,(\eta ^p_1)',C',D',\eta _p\right) \right] \right) ,$$
where \(S \subset S_G\) is the subtorus with
$$\begin{aligned} X^*(S)=X^*(S_G) \Big / \left( X^*(S_G) \cap j \circ i_{m',m} {\mathcal {O}}_{F,(p)}^{m'}\right) \end{aligned}$$
and where
$$\begin{aligned} (\eta ^p_1)' \circ i_{m',m}^*=\eta ^p_1 \bmod V^pS \end{aligned}$$
and \(C'\) (resp. \(D'\)) denotes the image of C (resp. D). The prime-to-p quasi-isogeny class of \(\pi _{m,m'}(G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\). If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\pi _{m,m'} \circ \gamma =\overline{\gamma }\circ \pi _{m,m'}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}({\mathcal {O}}_{F,(p)})\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) then \(\pi _{m,m'} \circ g=i_{m',m}^*(g)\circ \pi _{m,m'}\).
For each \(m \ge 0\) there is a system of \({\mathbb {Z}}_{(p)}\)-schemes \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \}\) as \(U^p\) runs over neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and \(N_1,N_2\) run over integers with \(N_2 \ge N_1 \ge 0\), together with the following extra structures:
  • If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(U_2^p(N_{21},N_{22}) \supset g^{-1}U_1^p(N_{11},N_{12})g\) then there is a quasi-finite, flat map
    $$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})}. \end{aligned}$$
  • If \(m'\le m\) and if \((U^p)'\) denotes the image of \(U^p\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), then there is a smooth projective map with geometrically connected fibres
    $$\begin{aligned} \pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2)}. \end{aligned}$$
  • If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(U^p_2 \supset \gamma U_1^p\) then there is a finite etale map
    $$\begin{aligned} \gamma : {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_1,N_2)}. \end{aligned}$$
Moreover there is a canonical prime-to-p quasi-isogeny class of ordinary \(G_n^{(m)}\)-semi-abelian schemes with \(U^p(N_1,N_2)\) level structure
$$\begin{aligned} \left( {\mathcal {G}}^{{\text {univ}}}, i^{{\text {univ}}}, j^{{\text {univ}}}, \lambda ^{{\text {univ}}}, [\eta ^{{\text {univ}}}]\right) \big / {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \end{aligned}$$
These enjoy the following properties:
  • \({\mathcal {A}}^{(0),{{\text {ord}}}}_{n,U^p(N_1,N_2)}={\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\). (We will sometimes write \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {X}}^{{\text {ord}}}_n}\) instead of \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/A^{(0),{{\text {ord}}}}_n}\).) This identification is \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) equivariant.

  • \(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).

  • If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n} \circ \gamma =\overline{\gamma }\circ \pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}({\mathcal {O}}_{F,(p)})\).

  • \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}} \circ g=g'\circ \pi _{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}}\), where \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).

  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), then the induced map
    $$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow g^* {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})} \end{aligned}$$
    over \({\mathcal {X}}^{{\text {ord}}}_{n,U_1^p(N_{11},N_{12})}\) is finite flat of degree \(p^{nm[F: {\mathbb {Q}}]}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), then this map is also etale.
  • If \(U_1^p \subset U_2^p\) is an open normal subgroup of a neat open compact of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and if \(N_{11}\ge N_{21}\), then \({\mathcal {A}}_{n,U_1^p(N_{11},N_2)}^{(m),{{\text {ord}}}}/{\mathcal {A}}_{n,U_2^p(N_{21},N_2)}^{(m),{{\text {ord}}}}\) is Galois with Galois group \(U_2^p(N_{21})/U^p_1(N_{11})\).

  • On \({\mathbb {F}}_p\)-fibres the map
    $$\begin{aligned} \varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2+1)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$
    equals the composition of the absolute Frobenius map with the forgetful map (for any \(N_2 \ge N_1 \ge 0\)).
  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(U_2^p(N_{21},N_{22}) \supset g^{-1}U^p_1(N_{11},N_{12})g\) then the pull-back
    $$g^* \left( {\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, [\eta _2^{{\text {univ}}}]\right) $$
    is prime-to-p quasi-isogenous to the tuple \(({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])g\).
  • If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(U_2^p(N_{21},N_{22}) \supset \gamma U^p_1(N_{11},N_{12})\) then the pull-back
    $$\gamma ^* \left( {\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, \left[ \eta _2^{{\text {univ}}}\right] \right) $$
    is prime-to-p quasi-isogenous to the tuple \(\gamma ({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])\).
  • If \(m' \le m\) and if \(U_2^p(N_{21},N_{22}) \supset i_{m',m}^* U^p_1(N_{11},N_{12})\) then the pull-back \(\pi _{{\mathcal {A}}_n^{(m)}/{\mathcal {A}}_n^{(m')}}^* ({\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, [\eta _2^{{\text {univ}}}])\) is prime-to-p quasi-isogenous to the tuple \(\pi _{m,m'} ({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])\).

  • If \(U^p=(U^p)' \ltimes M^p\) with \((U^p)' \subset G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p \subset {{\text {Hom}}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\) then
    $$\begin{aligned} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\big /{\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)} \end{aligned}$$
    is an abelian scheme of relative dimension \(mn[F: {\mathbb {Q}}]\).
  • In general \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) is a principal homogenous space for the abelian scheme \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,((U^p)' \ltimes M^p)(N_1,N_2)}\) over \({\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}\), where \((U^p)'\) denotes the image of \(U^p\) in \(G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p=U^p \cap {{\text {Hom}}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\).

  • There are natural identifications
    $${\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong A^{(m)}_{n,U^p(N_1,N_2)}.$$
    These identifications are compatible with the identifications
    $${\mathcal {X}}_{n,(U^p)'(N_1,N_2)}^{{{\text {ord}}}} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong X_{n,(U^p)'(N_1,N_2)}$$
    and the maps \(\pi _{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}}\) and \(\pi _{A^{(m)}_n/A_n^{(m')}}\). They are also equivariant for the actions of the semi-group \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and the group \(GL_m({\mathcal {O}}_{F,(p)})\).
Moreover in the case \(U^p=(U^p)' \ltimes M^p\), if \({\mathcal {G}}^{{\text {univ}}}/{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) and \({\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) are chosen so that \(\pi ^*_{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {X}}_n^{{\text {ord}}}} {\mathcal {A}}^{{\text {univ}}}\cong A_{{\mathcal {G}}^{{\text {univ}}}}\), then there is a \({\mathbb {Z}}_{(p)}\)-linear map
$$i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}: {\mathcal {O}}_{F,(p)}^m \longrightarrow {{\text {Hom}}}_{/{\mathcal {X}}_{n,(U^p)'(N_1,N_2)}^{{\text {ord}}}}\left( {\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}, ({\mathcal {A}}^{{\text {univ}}}/C^{{\text {univ}}}[p^{N_1}])^\vee \right) _{{\mathbb {Z}}_{(p)}}$$
with the following properties.
  • If \(a \in {\mathcal {O}}_{F,(p)}\) then
    $$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(ax)=i^{{{\text {univ}}},\vee }({}^ca) \circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x). \end{aligned}$$
  • If \((\beta ,\delta )\) is a prime-to-p quasi-isogeny
    $$\left( {\mathcal {G}}^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \longrightarrow \left( {\mathcal {G}}^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]\right) ,$$
    then
    $$\begin{aligned} \beta ^\vee \circ i^{(m)}_{({\mathcal {A}}^{{\text {univ}}})'}(x)=i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x). \end{aligned}$$
    In particular \(i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}\) depends only on \({\mathcal {A}}^{{\text {univ}}}\) and not on \({\mathcal {G}}^{{\text {univ}}}\).
  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) then
    $$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x) \circ g=i^{(m)}_{g^* {\mathcal {A}}^{{\text {univ}}}}(x) \end{aligned}$$
    and
    $$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x) \circ \gamma =i^{(m)}_{\gamma ^* {\mathcal {A}}^{{\text {univ}}}}(\gamma ^{-1} x). \end{aligned}$$
  • If \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_{F,(p)}^m\) then
    $$i_{{\mathcal {A}}^{{\text {univ}}}}=\left| \left| \eta ^{p,{{\text {univ}}}}_0\right| \right| ^{-1}\left( \left( \lambda (N_1)^{{\text {univ}}}\right) ^{-1}\circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(e_1),\ldots , \left( \lambda (N_1)^{{\text {univ}}}\right) ^{-1} \circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(e_m) \right) $$
    is a prime-to-p quasi-isogeny
    $${\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \longrightarrow ({\mathcal {A}}^{{\text {univ}}}/C^{{\text {univ}}}[p^{N_1}])^m.$$
    Here \(\lambda (N_1)^{{\text {univ}}}\) refers to the prime-to-p quasi-polarization \({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}] \rightarrow ({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}])^\vee \) for which the composite
    $$\begin{aligned} {\mathcal {A}}^{{\text {univ}}}\longrightarrow {\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}] \mathop {\longrightarrow }\limits ^{\lambda (N_1)^{{\text {univ}}}} ({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}])^\vee \longrightarrow {\mathcal {A}}^{{{\text {univ}}},\vee } \end{aligned}$$
    equals \(p^{N_1} \lambda ^{{\text {univ}}}\).
    We have
    $$\begin{aligned} \beta ^{\oplus m} \circ i_{{\mathcal {A}}^{{\text {univ}}}}=i_{({\mathcal {A}}^{{\text {univ}}})'}. \end{aligned}$$
    The composite map
    $$\begin{aligned} \begin{array}{rcl} \eta _{n,U^p(N_1,N_2)}^{(m)}: {{\text {Hom}}}_{{\mathcal {O}}_{F}}({\mathcal {O}}_{F}^m,\varLambda _n) \otimes _{\mathbb {Z}}{\mathbb {A}}^{\infty ,p} &{}\longrightarrow &{} V^p({\mathcal {A}}^{{\text {univ}}})^m \\ &{} \mathop {\longrightarrow }\limits ^{p^{-N_1}} &{}V^p({\mathcal {A}}^{{\text {univ}}}/C^{{\text {univ}}}[p^{N_1}])^m \\ &{} \longrightarrow &{} V^p{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}, \end{array} \end{aligned}$$
    where the first maps sends
    $$\begin{aligned} f \longmapsto \left( \eta _1^{p,{{\text {univ}}}}(f(e_1)),\ldots ,\eta _1^{p,{{\text {univ}}}}(f(e_m))\right) \end{aligned}$$
    and the third map sends
    $$\begin{aligned} x\longmapsto V^p(i_{{\mathcal {A}}^{{\text {univ}}}})^{-1} x, \end{aligned}$$
    is an isomorphism, which does not depend on the choice of \({\mathcal {G}}^{{\text {univ}}}\). It satisfies
    $$\begin{aligned} \eta _{n,U^p(N_1,N_2)}^{(m)} M^p=T^p{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}. \end{aligned}$$
(See lemmas 5.2.4.7 and 7.1.2.1, propositions 5.2.4.13, 5.2.4.25 and 7.1.2.5, remarks 7.1.2.38 and 7.1.4.27, and theorem 7.1.4.1 of [44].)
We deduce the following additional properties:
  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map \(g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \rightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})}\) is etale. If further \(N_{12}=N_{22}\), then it is finite etale.

  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), if \(N_{22}>0\) and if \(p^{N_{12}-N_{22}}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then
    $$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})} \end{aligned}$$
    is finite. If \(N_2>0\) then the finite flat map
    $$\begin{aligned} \varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2+1)} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \end{aligned}$$
    has degree \(p^{n(n+2m)[F^+: {\mathbb {Q}}]}\).
  • $$\begin{aligned} i_{{\mathcal {A}}^{{\text {univ}}}} \circ g=i_{g^* {\mathcal {A}}^{{\text {univ}}}} \end{aligned}$$
    and
    $$\begin{aligned} i_{{\mathcal {A}}^{{\text {univ}}}} \circ \gamma ={}^t\gamma ^{-1} \circ i_{\gamma ^* {\mathcal {A}}^{{\text {univ}}}}. \end{aligned}$$
Also in this case define
$$i_\lambda ^{(m)} : {\mathcal {O}}_{F,(p)}^m \otimes _{{\mathcal {O}}_{F,(p)},c}{\mathcal {O}}_{F,(p)}^m \longrightarrow {{\text {Hom}}}_{/{\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}}\left( {\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}, A_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}},\vee }\right) _{{\mathbb {Z}}_{(p)}}$$
by
$$\begin{aligned} i_\lambda ^{(m)}(x \otimes y)=\left| \left| \eta _0^{p,{{\text {univ}}}}\right| \right| ^{-1} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x)^\vee \circ \left( \lambda (N_1)^{{\text {univ}}}\right) ^{-1} \circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(y). \end{aligned}$$
This does not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). We have
$$\begin{aligned} i_\lambda ^{(m)}(x \otimes y)^\vee =i_\lambda ^{(m)}(y \otimes x). \end{aligned}$$
Moreover
$$\left( i_{{\mathcal {A}}^{{\text {univ}}}}^{-1}\right) ^\vee \circ i_\lambda ^{(m)}(x \otimes y) \circ i_{{\mathcal {A}}^{{\text {univ}}}}^{-1}=\left( \lambda (N_1)^{{{\text {univ}}}}\right) ^{ \oplus m} \circ i^{{\text {univ}}}({}^{c,t}{xy}).$$
If \(a \in ({\mathcal {O}}_{F,(p)}^m \otimes _{{\mathcal {O}}_{F,(p)},c} {\mathcal {O}}_{F,(p)}^m)^{{{\text {sw}}}=1}\) has image in \(S({\mathcal {O}}_{F,(p)}^m)\) lying in \(S({\mathcal {O}}_{F,(p)}^m)^{>0}\) then
$$\begin{aligned} \left( i_{{\mathcal {A}}^{{\text {univ}}}}^{-1}\right) ^\vee \circ i_\lambda ^{(m)}(a) \circ i_{{\mathcal {A}}^{{\text {univ}}}}^{-1}=\left( \lambda (N_1)^{{\text {univ}}}\right) ^{\oplus m} \circ i^{{\text {univ}}}(a') \end{aligned}$$
for some matrix \(a' \in M_{m \times m}({\mathcal {O}}_{F,(p)})^{t=c}\) all whose eigenvalues are positive real numbers. Thus \(i_\lambda ^{(m)}(a)\) is a quasi-polarization. (See the end of section 21 of [47].)
The completion of \({\mathcal {A}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) along its \({\mathbb {F}}_p\)-fibre does not depend on \(N_2\), so we will denote it
$$\begin{aligned} {\mathfrak {A}}^{(m),{{\text {ord}}}}_{U^p(N_1)}. \end{aligned}$$
(See theorem 7.1.4.1 of [44].) Then \(\{ {\mathfrak {A}}^{(m),{{\text {ord}}}}_{U^p(N)}\}\) is a system of p-adic formal schemes with a right \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action and a left \(GL_m({\mathcal {O}}_{F,(p)})\)-action. There is an equivariant map
$$\left\{ {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N)}\right\} \longrightarrow \left\{ {\mathfrak {X}}^{{\text {ord}}}_{n,(U')^p(N)} \right\} .$$
We will write \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N)}\) for the reduced subscheme of \({\mathfrak {A}}_{n,U^p(N)}^{(m),{{\text {ord}}}}\).

4.3 Some mixed Shimura varieties

If \(\widetilde{{U}}\) (resp. \(\widetilde{{U}}^p\)) is a neat open compact subgroup of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\)) we will denote by \(S_{n,\widetilde{{U}}}^{(m)}\) (resp. \({\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\)) the split torus over \({{\text {Spec}}\,}{\mathbb {Q}}\) (resp. \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\)) with
$$\begin{aligned} X_*\left( S_{n,\widetilde{{U}}}^{(m)}\right) =Z(N_n^{(m)})({\mathbb {Q}}) \cap \widetilde{{U}}\subset {{\text {Herm}}}^{(m)}({\mathbb {Q}}) \end{aligned}$$
(resp.
$$\begin{aligned} X_*\left( {\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\right) =Z(N_n^{(m)})({\mathbb {Z}}_{(p)}) \cap \widetilde{{U}}^p \subset {{\text {Herm}}}^{(m)}({\mathbb {Z}}_{(p)})). \end{aligned}$$
If \(g \in \widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty })\) (resp. \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and \(\widetilde{{U}}_2 \supset g^{-1}\widetilde{{U}}_1g\) (resp. \(\widetilde{{U}}_2^p \supset g^{-1}\widetilde{{U}}_1^pg\)) we get a map
$$\begin{aligned} g: S_{n,\widetilde{{U}}_1}^{(m)} \longrightarrow S_{n,\widetilde{{U}}_2}^{(m)} \end{aligned}$$
(resp.
$$\begin{aligned} \left. g: {\mathcal {S}}_{n,\widetilde{{U}}_1^p}^{(m),{{\text {ord}}}} \longrightarrow {\mathcal {S}}_{n,\widetilde{{U}}^p_2}^{(m),{{\text {ord}}}}\right) \end{aligned}$$
corresponding to
$$||\nu (g)||: X_*\left( S_{n,\widetilde{{U}}_1}^{(m)}\right) \longrightarrow X_* \left( S_{n,\widetilde{{U}}_2}^{(m)}\right) $$
(resp.
$$||\nu (g)||: X_*\left( {\mathcal {S}}_{n,\widetilde{{U}}_1^p}^{(m),{{\text {ord}}}}\right) \longrightarrow \left. X_*\left( {\mathcal {S}}_{n,\widetilde{{U}}^p_2}^{(m),{{\text {ord}}}}\right) \right) ,$$
where we think of the domain and codomain both as subspaces of \({{\text {Herm}}}^{(m)}\). If \(\gamma \in GL_m({\mathbb {Q}})\) (resp. \(GL_m({\mathbb {Z}}_{(p)})\)) and \(\widetilde{{U}}_2 \supset \gamma \widetilde{{U}}_1\) (resp. \(\widetilde{{U}}_2^p \supset \gamma \widetilde{{U}}_1^p\)) we get a map
$$\begin{aligned} \gamma : S_{n,\widetilde{{U}}_1}^{(m)} \longrightarrow S_{n,\widetilde{{U}}_2}^{(m)} \end{aligned}$$
(resp.
$$\begin{aligned} \left. \gamma : {\mathcal {S}}_{n,\widetilde{{U}}_1^p}^{(m),{{\text {ord}}}} \longrightarrow {\mathcal {S}}_{n,\widetilde{{U}}^p_2}^{(m),{{\text {ord}}}}\right) \end{aligned}$$
corresponding to
$$\begin{aligned} \gamma : X_*\left( S_{n,\widetilde{{U}}_1}^{(m)}\right) \longrightarrow X_*\left( S_{n,\widetilde{{U}}_2}^{(m)}\right) \end{aligned}$$
(resp.
$$\left. \gamma : X_*\left( {\mathcal {S}}_{n,\widetilde{{U}}_1^p}^{(m),{{\text {ord}}}}\right) \longrightarrow X_* \left( {\mathcal {S}}_{n,\widetilde{{U}}^p_2}^{(m),{{\text {ord}}}}\right) \right) ,$$
where again we think of the domain and codomain both as subspaces of \({{\text {Herm}}}^{(m)}\). If \(m_1 \ge m_2\) and if \(\widetilde{{U}}_2\) (resp. \(\widetilde{{U}}_2^p\)) is the image of \(\widetilde{{U}}_1\) (resp. \(\widetilde{{U}}_1^p\)) in \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^{\infty ,p})\)), then our chosen map \({{\text {Herm}}}^{(m_1)} \rightarrow {{\text {Herm}}}^{(m_2)}\) induces a map
$$\begin{aligned} S_{n,\widetilde{{U}}_1}^{(m_1)} \longrightarrow S_{n,\widetilde{{U}}_2}^{(m_2)} \end{aligned}$$
(resp.
$$\begin{aligned} \left. {\mathcal {S}}_{n,\widetilde{{U}}_1^p}^{(m_1),{{\text {ord}}}} \longrightarrow {\mathcal {S}}_{n,\widetilde{{U}}_2^p}^{(m_2),{{\text {ord}}}}\right) . \end{aligned}$$
As \(\widetilde{{U}}\) runs over neat open compact subgroups of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^\infty )\), there is a system of \(S_{n,\widetilde{{U}}}^{(m)}\)-torsors
$$\begin{aligned} T^{(m)}_{n,\widetilde{{U}}}=\underline{{{\text {Spec}}\,}} \bigoplus _{\chi \in X^*(S^{(m)}_{n,\widetilde{{U}}})} {\mathcal {L}}^{(m)}_{n,\widetilde{{U}}}(\chi ) \end{aligned}$$
over \(A^{(m)}_{n,\widetilde{{U}}}\) together with the following extra structures:
  • If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) and \(\widetilde{{U}}_1,\widetilde{{U}}_2\) are neat open compact subgroups of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) with \(\widetilde{{U}}_2 \supset g^{-1}\widetilde{{U}}_1g\) then there is a finite etale map
    $$\begin{aligned} g: T^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m)}_{n,\widetilde{{U}}_2} \end{aligned}$$
    compatible with the maps \(g: A^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m)}_{n,\widetilde{{U}}_2}\) and \(g: S^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m)}_{n,\widetilde{{U}}_2}\).
  • If \(\gamma \in GL_m(F)\) and \(\widetilde{{U}}_1,\widetilde{{U}}_2\) are neat open compact subgroups of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) with \(\widetilde{{U}}_2 \supset \gamma \widetilde{{U}}_1\) then there is a finite etale map
    $$\begin{aligned} \gamma : T^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m)}_{n,\widetilde{{U}}_2}, \end{aligned}$$
    compatible with the maps \(\gamma : A^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m)}_{n,\widetilde{{U}}_2}\) and \(\gamma : S^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m)}_{n,\widetilde{{U}}_2}\).
  • If \(m_1 \ge m_2\) and \(\widetilde{{U}}_2\) is the image of \(\widetilde{{U}}_1\) in \(\widetilde{{G}}^{(m_2)}({\mathbb {A}}^\infty )\), then there is a map
    $$\begin{aligned} T^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m_2)}_{n,\widetilde{{U}}_2} \end{aligned}$$
    compatible with the maps \(S^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m_2)}_{n,\widetilde{{U}}_2}\) and \(A^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m_2)}_{n,\widetilde{{U}}_2}\).
These enjoy the following properties:
  • \(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).

  • If \(\widetilde{{U}}_1 \subset \widetilde{{U}}_2\) is an open normal subgroup of a neat open compact subgroup of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\), then \(T_{n,\widetilde{{U}}_1}^{(m)}/T_{n,\widetilde{{U}}_2}^{(m)}\) is Galois with group \(\widetilde{{U}}_2/\widetilde{{U}}_1\).

  • The maps \(T^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m_2)}_{n,\widetilde{{U}}_2}\) are compatible with the actions of \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\) and \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\) and the map \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\rightarrow \widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\), and also with the action of \(Q_{m_1,m_2}(F)\).

  • Suppose that \(\widetilde{{U}}=U' \ltimes M\) with \(U' \subset G_n({\mathbb {A}}^\infty )\) and \(M \subset N_n^{(m)}({\mathbb {A}}^\infty )\). Also suppose that
    $$\begin{aligned} \chi \in X^*(S^{(m)}_{n,\widetilde{{U}}}) \subset S(F^m) \end{aligned}$$
    is sufficiently divisible. Then we can find \(a \in F^m \otimes _{F,c} F^m\) lifting \(\chi \) such that
    $$\begin{aligned} i^{(m)}_\lambda (a): A^{(m)}_{n,\widetilde{{U}}} \longrightarrow \left( A^{(m)}_{n,\widetilde{{U}}}\right) ^\vee \end{aligned}$$
    is a homomorphism. For any such a
    $$\begin{aligned} {\mathcal {L}}^{(m)}_{n,\widetilde{{U}}}(\chi )=\left( 1,i^{(m)}_\lambda (a)\right) ^* {\mathcal {P}}_{A^{(m)}_{n,\widetilde{{U}}}}. \end{aligned}$$
  • If \(\chi \in X^*(S^{(m)}_{n,\widetilde{{U}}}) \cap S(F^m)^{>0}\) then \({\mathcal {L}}_{n,\widetilde{{U}}}^{(m)}(\chi )\) is relatively ample for \(A^{(m)}_{n,\widetilde{{U}}}/X_{n,\widetilde{{U}}}\).

  • There are \(\widetilde{{G}}^{(m)}_{n}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant homeomorphisms
    $$T^{(m)}_{n,\widetilde{{U}}}({\mathbb {C}}) \cong \widetilde{{G}}^{(m)}_{n}({\mathbb {Q}}) \Big \backslash \widetilde{{G}}^{(m)}_{n}({\mathbb {A}}) {{\text {Herm}}}^{(m)}({\mathbb {C}})\Big /\left( \widetilde{{U}}\times U_{n,\infty }^0A_n({\mathbb {R}})^0\right) .$$
(See lemmas 1.3.2.25 and 1.3.2.72, and propositions 1.3.2.31, 1.3.2.45 and 1.3.2.90 of [44]; section 3.6 of [43]; and the second paragraph of Sect. 3.2 above.)
Similarly as \(\widetilde{{U}}^p\) runs over neat open compact subgroups of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\) and \(N_1,N_2\) run over integers with \(N_2 \ge N_1 \ge 0\), there is a system of \({\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\)-torsors
$${\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}=\underline{{{\text {Spec}}\,}} \bigoplus _{\chi \in X^*\left( {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\right) } {\mathcal {L}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}(\chi )$$
over \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\) together with the following extra structures:
  • If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{U}}_2^p(N_{21},N_{22}) \supset g^{-1}\widetilde{{U}}_1^p(N_{11},N_{12})g\) then there is a quasi-finite, flat map
    $$\begin{aligned} g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})} \end{aligned}$$
    compatible with the map \(g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) and the map \(g: {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}\).
  • If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\widetilde{{U}}_2^p \supset \gamma \widetilde{{U}}_1^p\) then there is a finite etale map
    $$\begin{aligned} \gamma : {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}, \end{aligned}$$
    compatible with the maps
    $$\begin{aligned} \gamma : {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m)}_{n,\widetilde{{U}}_2^p(N_1,N_2)} \end{aligned}$$
    and
    $$\begin{aligned} \gamma : {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}. \end{aligned}$$
  • If \(m_1 \ge m_2\) and \(\widetilde{{U}}_2^p\) is the image of \(\widetilde{{U}}_1^p\) in \(\widetilde{{G}}^{(m_2)}({\mathbb {A}}^{\infty ,p})\), then there is a map
    $$\begin{aligned} {\mathcal {T}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)} \end{aligned}$$
    compatible with the map \({\mathcal {S}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}\) and the map \({\mathcal {A}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}\).
These enjoy the following properties:
  • \(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).

  • If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map \(g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) is etale. If further \(N_{12}=N_{22}\), then it is finite etale.

  • The maps \({\mathcal {T}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}\) are compatible with the actions of \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and the map \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\rightarrow \widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\), and with the action of \(Q_{m_1,m_2}({\mathcal {O}}_{F,(p)})\).

  • If \(\widetilde{{U}}_1^p \subset \widetilde{{U}}_2^p\) is an open normal subgroup of a neat open compact of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\), and if \(N_{11} \ge N_{21}\) then \({\mathcal {T}}_{n,\widetilde{{U}}_1^p(N_{11},N_2)}^{(m),{{\text {ord}}}}/{\mathcal {T}}_{n,\widetilde{{U}}_2^p(N_{21},N_2)}^{(m),{{\text {ord}}}}\) is Galois with Galois group \(\widetilde{{U}}_2^p(N_{21})/\widetilde{{U}}_1^p(N_{11})\).

  • If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), if \(N_{22}>0\) and if \(p^{N_{12}-N_{22}}\nu (g_p) \in {\mathbb {Z}}_p^\times \), then the map \(g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) is finite. If \(N_{2}>0\) then the finite flat map
    $$\begin{aligned} \varsigma _p: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{1},N_{2}+1)} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{1},N_{2})} \end{aligned}$$
    has degree \(p^{(n+m)^2[F^+: {\mathbb {Q}}]}\).
  • On the \({\mathbb {F}}_p\)-fibre
    $$\begin{aligned} \varsigma _p: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2+1)}\times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$
    equals the composition of the absolute Frobenius map with the forgetful map (for any \(N_2 \ge N_1 \ge 0\)).
  • Suppose that \(\widetilde{{U}}^p=(U^p)' \ltimes M^p\) with \((U^p)' \subset G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p \subset N_n^{(m)}({\mathbb {A}}^{\infty ,p})\). Also suppose that
    $$\begin{aligned} \chi \in X^*\left( {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p}\right) \subset S({\mathcal {O}}_{F,(p)}^m) \end{aligned}$$
    is sufficiently divisible. Then we can find \(a \in {\mathcal {O}}_{F,(p)}^m \otimes _{{\mathcal {O}}_{F,(p)}} {\mathcal {O}}_{F,(p)}^m\) lifting \(\chi \) such that
    $$i^{(m)}_\lambda (a): {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow \left( {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\right) ^\vee $$
    is a homomorphism. For any such a
    $$\begin{aligned} {\mathcal {L}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}(\chi )=\left( 1,i^{(m)}_\lambda (a)\right) ^* {\mathcal {P}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}}. \end{aligned}$$
  • If \(\chi \in X^*({\mathcal {S}}^{(m)}_{n,\widetilde{{U}}^p}) \cap S({\mathcal {O}}_{F,(p)}^m)^{>0}\) then \({\mathcal {L}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}}(\chi )\) is relatively ample for
    $$\begin{aligned} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\big /{\mathcal {X}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{{\text {ord}}}. \end{aligned}$$
  • There are natural identifications
    $$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong T^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)}. \end{aligned}$$
    These identifications are compatible with the identifications
    $$\begin{aligned} {\mathcal {A}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong A^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)} \end{aligned}$$
    and the maps
    $$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow {\mathcal {A}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}} \end{aligned}$$
    and
    $$\begin{aligned} T^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow A^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)}. \end{aligned}$$
    The identifications are also equivariant for the actions of the semi-group \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and the group \(GL_m({\mathcal {O}}_{F,(p)})\).
(See lemmas 5.2.4.26 and 7.1.2.22, propositions 5.2.4.30, 5.2.4.41 and 7.1.2.36, and remark 7.1.2.38 of [44].)

4.4 Vector bundles

4.4.1 Vector bundles on Shimura varieties in characteristic zero

Suppose that U is a neat open compact subgroup of \(G_n({\mathbb {A}}^\infty )\). We will let \(\varOmega _{n,U}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\). This is a locally free sheaf of rank \(n[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \(A^{{\text {univ}}}\). (Because, by the neatness of U, there is a unique quasi-isogeny between any two universal four-tuples \((A^{{\text {univ}}}, i^{{\text {univ}}}, \lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])\).) The system of sheaves \(\{ \varOmega _{n,U}\}\) has an action of \(G_n({\mathbb {A}}^\infty )\). There is a natural isomorphism between \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\) and the pull-back of \(\varOmega _{n,U}\) from \(X_{n,U}\) to \(A^{{\text {univ}}}\). We will write
$$\begin{aligned} \omega _U=\omega _{n,U}=\wedge ^{n[F: {\mathbb {Q}}]} \varOmega _{n,U}. \end{aligned}$$
Similarly, if \(\pi : A^{{\text {univ}}}\rightarrow X_{n,U}\) is the structural map, then the sheaf
$$R^i\pi _* \varOmega ^j_{A^{{\text {univ}}}/X_{n,U}} \cong (\wedge ^j \varOmega _{n,U}) \otimes R^i\pi _* {\mathcal {O}}_{A^{{\text {univ}}}}$$
is locally free and canonically independent of the choice of \(A^{{\text {univ}}}\). These sheaves again have an action of \(G_n({\mathbb {A}}^\infty )\).

We will also write \(\varXi _{n,U}={\mathcal {O}}_{X_{n,U}}(||\nu ||)\) for the sheaf \({\mathcal {O}}_{X_{n,U}}\) but with the \(G_n({\mathbb {A}}^\infty )\)-action multiplied by \(||\nu ||\).

For any \(m \in {\mathbb {Z}}\) such that \(m \lambda ^{{\text {univ}}}\) is a true isogeny we get a class
$$\begin{array}{rcl} \left[ (1,[m]\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{A^{{{\text {univ}}}}}\right] &{}\in &{} H^1\left( A^{{{\text {univ}}}},{\mathcal {O}}^\times _{A^{{{\text {univ}}}}}\right) \\ &{}\longrightarrow &{} H^0\left( X_{n,U},R^1\pi _* {\mathcal {O}}^\times _{A^{{{\text {univ}}}}}\right) \\ &{}\mathop {\longrightarrow }\limits ^{d \log }&{} H^0\left( X_{n,U}, R^1\pi _* \varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\right) . \end{array}$$
The class
$$\left[ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{A^{{{\text {univ}}}}}\right] =\left[ (1,[m]\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{A^{{{\text {univ}}}}}\right] \Big /m \in H^0\left( X_{n,U}, R^1\pi _* \varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\right) $$
is well defined independently of m. We obtain an embedding
$$\begin{aligned} \varXi _{n,U} \hookrightarrow R^1\pi _* \varOmega ^1_{A^{{\text {univ}}}/X_{n,U}} \end{aligned}$$
sending 1 to \(||\eta ^{{\text {univ}}}|| [ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{A^{{{\text {univ}}}}}]\). (See Sect. 3.1 for the definition of \(||\eta ^{{\text {univ}}}||\).) These maps are compatible with the isomorphisms
$$R^1\pi _* \varOmega ^1_{A^{{\text {univ}}}/X_{n,U}} \mathop {\longrightarrow }\limits ^{\sim }R^1\pi _* \varOmega ^1_{A^{{{\text {univ}}},\prime }/X_{n,U}}$$
induced by the unique quasi-isogeny between two universal four-tuples. They are also \(G_n({\mathbb {A}}^\infty )\)-equivariant.
The composites of induced maps
$$\begin{array}{rcl} {{\text {Hom}}}(\varOmega _{n,U}, \varXi _{n,U}) &{}\hookrightarrow &{} {{\text {Hom}}}\left( \varOmega _{n,U}, R^1\pi _* \varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\right) \\ &{}\quad \mathop {\longleftarrow }\limits ^{\sim } &{} {{\text {Hom}}}\left( \varOmega _{n,U},\varOmega _{n,U} \otimes R^1\pi _* {\mathcal {O}}_{A^{{\text {univ}}}}\right) \\ &{}\quad \mathop {\longrightarrow }\limits ^{{{\text {tr}}}} &{} R^1\pi _* {\mathcal {O}}_{A^{{\text {univ}}}} \end{array}$$
are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms, independent of the choice of \(A^{{\text {univ}}}\). Moreover the short exact sequence
$$(0) \longrightarrow \varOmega ^1_{X_{n,U}} \otimes {\mathcal {O}}_{A^{{\text {univ}}}} \longrightarrow \varOmega ^1_{A^{{\text {univ}}}} \longrightarrow \varOmega _{n,U} \otimes {\mathcal {O}}_{A^{{\text {univ}}}} \longrightarrow (0)$$
gives rise to a map
$$\begin{array}{rcl} \varOmega _{n,U} &{}\longrightarrow &{} \varOmega ^1_{X_{n,U}} \otimes R^1\pi _* {\mathcal {O}}_{A^{{\text {univ}}}} \\ &{} \mathop {\longleftarrow }\limits ^{\sim } &{} \varOmega ^1_{X_{n,U}} \otimes {{\text {Hom}}}(\varOmega _{n,U}, \varXi _{n,U}) \end{array}$$
and hence to a map
$$\begin{aligned} \varOmega _{n,U}^{\otimes 2} \longrightarrow \varOmega ^1_{X_{n,U}}\otimes \varXi _{n,U}. \end{aligned}$$
These maps do not depend on the choice of \(A^{{\text {univ}}}\) and are \(G_n({\mathbb {A}}^\infty )\)-equivariant. They further induce \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms
$$\begin{aligned} S(\varOmega _{n,U}) \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{X_{n,U}}\otimes \varXi _{n,U}, \end{aligned}$$
which again do not depend on the choice of \(A^{{\text {univ}}}\). (See, for instance, propositions 2.1.7.3 and 2.3.5.2 of [41]. This is referred to as the ‘Kodaira–Spencer isomorphism’.)
Let \({\mathcal {E}}_{U}\) denote the principal \(L_{n,(n)}\)-bundle on \(X_{n,U}\) in the Zariski topology defined by setting, for \(W \subset X_{n,U}\) a Zariski open, \({\mathcal {E}}_{U}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi _{n,U}|_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\begin{aligned} \xi _1: \varOmega _{n,U} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}( V_n/V_{n,(n)}, {\mathcal {O}}_W). \end{aligned}$$
We define the \(L_{n,(n)}\)-action on \({\mathcal {E}}_{U}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\mathcal {E}}_{U} \}\) has an action of \(G_n({\mathbb {A}}^\infty )\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(L_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}_{U,\rho }\) over \(X_{n,U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}_{U,\rho }(W)\) to be the set of \(L_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}_{U}|_W \rightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}_{U,\rho } \}\) is a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )\)-action over the system of schemes \(\{ X_{n,U} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n({\mathbb {A}}^\infty )\), then the natural map
$$\begin{aligned} g^*{\mathcal {E}}_{U,\rho } \longrightarrow {\mathcal {E}}_{U',\rho } \end{aligned}$$
is an isomorphism.
In the case \(R_0={\mathbb {C}}\), the holomorphic vector bundle on \(X_{n,U}({\mathbb {C}})\) associated to \({\mathcal {E}}_{U,\rho }\) is
$$\begin{aligned} {\mathfrak {E}}_{U,\rho }=G_n({\mathbb {Q}}) \big \backslash \left( G_n({\mathbb {A}}^\infty )\big /U \times {\mathfrak {E}}_\rho \right) \end{aligned}$$
over
$$\begin{aligned} X_{n,U}({\mathbb {C}})=G_n({\mathbb {Q}}) \big \backslash \left( G_n({\mathbb {A}}^\infty )\big /U \times {\mathfrak {H}}_n^\pm \right) . \end{aligned}$$
(See Sect. 1.1 for the definition of the holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^{\pm }\).)
Note that
$$\begin{aligned} {\mathcal {E}}_{U,{{\text {Std}}}^\vee } \cong \varOmega _{n,U} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}_{U,\nu ^{-1}} \cong \varXi _{n,U} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}_{U,\wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee } \cong \omega _U \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}_{U,{{\text {KS}}}} \cong \varOmega ^1_{X_{n,U}}. \end{aligned}$$
(See Sect. 1.2 for the definition of the representation \({{\text {KS}}}\).)

4.4.2 Vector bundles on Kuga–Sato varieties in characteristic zero

Suppose now that U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n({\mathbb {A}}^\infty )\). We will let \(\varOmega _{n,U}^{(m)}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{G^{{\text {univ}}}/A_{n,U}^{(m)}}\). This is a locally free sheaf of rank \((n+m)[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \(G^{{\text {univ}}}\). The system of sheaves \(\{ \varOmega ^{(m)}_{n,U}\}\) has actions of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and of \(GL_m(F)\). Moreover there is an exact sequence
$$\begin{aligned} (0) \longrightarrow \pi _{A_n^{(m)}/X_n}^* \varOmega _{n,U'} \longrightarrow \varOmega _{n,U}^{(m)} \longrightarrow F^m \otimes _{\mathbb {Q}}{\mathcal {O}}_{A_{n,U}^{(m)}} \longrightarrow (0) \end{aligned}$$
which is equivariant for the actions of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\).
Let \({\mathcal {E}}_{U}^{(m)}\) denote the principal \(R^{(m)}_{n,(n)}\)-bundle on \(A^{(m)}_{n,U}\) in the Zariski topology defined by setting, for \(W \subset A^{(m)}_{n,U}\) a Zariski open, \({\mathcal {E}}_{U}^{(m)}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi _{n,U}|_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\xi _1: \varOmega _{n,U}^{(m)} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}\left( V_n/V_{n,(n)}\oplus {{\text {Hom}}}_{\mathbb {Q}}(F^m,{\mathbb {Q}}) , {\mathcal {O}}_W\right) $$
satisfies
$$\begin{aligned} \xi _1: \varOmega _{n,U} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}\left( V_n/V_{n,(n)}, {\mathcal {O}}_W\right) \end{aligned}$$
and induces the canonical isomorphism
$$\begin{aligned} F^m \otimes _{\mathbb {Q}}{\mathcal {O}}_W \longrightarrow {{\text {Hom}}}_{\mathbb {Q}}\left( {{\text {Hom}}}_{\mathbb {Q}}(F^m,{\mathbb {Q}}) , {\mathcal {O}}_W\right) . \end{aligned}$$
We define the \(R^{(m)}_{n,(n)}\)-action on \({\mathcal {E}}_{U}^{(m)}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\mathcal {E}}_{U}^{(m)} \}\) has an action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and of \(GL_m(F)\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R^{(m)}_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{(m)}_{U,\rho }\) over \(A^{(m)}_{n,U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{(m)}_{U,\rho }(W)\) to be the set of \(R^{(m)}_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}^{(m)}_{U}\big |_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{(m)}_{U,\rho } \}\) is a system of locally free sheaves with both \(G^{(m)}_n({\mathbb {A}}^\infty )\)-action and \(GL_m(F)\)-action over the system of schemes \(\{ A^{(m)}_{n,U} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\gamma \in GL_m(F)\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{U,\rho }^{(m)} \longrightarrow {\mathcal {E}}_{U',\rho }^{(m)} \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{U,\rho }^{(m)} \longrightarrow {\mathcal {E}}_{U',\rho }^{(m)} \end{aligned}$$
are isomorphisms. If \(\rho \) factors through \(R^{(m)}_{n,(n)} \twoheadrightarrow L_{n,(n)}\) then \({\mathcal {E}}^{(m)}_{U,\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}_{U,\rho }\) from \(X_{n,U}\). In general \(W_\rho \) has a filtration by \(R_{n,(n)}^{(m)}\)-invariant direct summands such that the action of \(R^{(m)}_{n,(n)}\) on each graded piece factors through \(L_{n,(n)}\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(L_{n,(n),{{\text {herm}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{(m)}_{U,\rho }\) has a \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}_{U,\rho '}\) from \(X_{n,U}\).

4.4.3 Vector bundles on Shimura varieties in mixed characteristic

Similarly suppose that \(U^p\) is a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\), and that \(N_2 \ge N_1 \ge 0\) are integers. We will let \(\varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\) denote the pull-back by the identity section of \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}}\). This is a locally free sheaf of rank \(n[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). (Because, by the neatness of \(U^p\), there is a unique prime-to-p quasi-isogeny between any two universal four-tuples \(({\mathcal {A}}^{{\text {univ}}}, i^{{\text {univ}}}, \lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])\).) The system of sheaves \(\{ \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). There is a natural isomorphism between \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\) and the pull-back of \(\varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\). We will write
$$\begin{aligned} \omega _{U^p(N_1,N_2)}=\omega _{n,U^p(N_1,N_2)}=\wedge ^{n[F: {\mathbb {Q}}]} \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}. \end{aligned}$$
We will also write \(\varXi _{n,U^p(N_1,N_2)}={\mathcal {O}}_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}(||\nu ||)\) for the sheaf \({\mathcal {O}}_{{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}}\) but with the \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action multiplied by \(||\nu ||\).
For any \(m \in {\mathbb {Z}}\) such that \(p {\not | }m\) and \(m\lambda ^{{\text {univ}}}\) is a true isogeny we get a class
$$\begin{array}{rcl} \left[ (1,[m]\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{{\mathcal {A}}^{{{\text {univ}}}}}\right] &{}\in &{} H^1\left( {\mathcal {A}}^{{{\text {univ}}}},{\mathcal {O}}^\times _{{\mathcal {A}}^{{{\text {univ}}}}}\right) \\ &{}\longrightarrow &{} H^0\left( {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}},R^1\pi _* {\mathcal {O}}^\times _{{\mathcal {A}}^{{{\text {univ}}}}}\right) \\ &{}\mathop {\longrightarrow }\limits ^{d \log }&{} H^0\left( {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}, R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\right) . \end{array}$$
The class
$$\begin{array}{rcl} \left[ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{{\mathcal {A}}^{{{\text {univ}}}}}\right] &{}=&{} \left[ (1,[m]\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{{\mathcal {A}}^{{{\text {univ}}}}}\right] \big /m \\ &{}\in &{} H^0\left( {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}, R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\right) \end{array}$$
is well defined independently of m. We obtain an embedding
$$\begin{aligned} \varXi _{n,U^p(N_1,N_2)}^{{\text {ord}}}\hookrightarrow R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \end{aligned}$$
sending 1 to \(||\eta ^{{\text {univ}}}|| [ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{{\mathcal {A}}^{{{\text {univ}}}}}]\). These maps are compatible with the isomorphisms
$$\begin{aligned} R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \mathop {\rightarrow }\limits ^{\sim }R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{{\text {univ}}},\prime }/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \end{aligned}$$
induced by the unique prime-to-p quasi-isogeny between two universal four-tuples. They are also \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant.
The composites of induced maps
$$\begin{array}{rl} &{} {{\text {Hom}}}\left( \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}, \varXi _{n,U^p(N_1,N_2)}^{{\text {ord}}}\right) \\ &{}\quad \hookrightarrow {{\text {Hom}}}\left( \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}, R^1\pi _* \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\right) \\ &{}\quad \mathop {\longleftarrow }\limits ^{\sim } {{\text {Hom}}}\left( \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)},\varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\otimes R^1\pi _* {\mathcal {O}}_{{\mathcal {A}}^{{\text {univ}}}}\right) \\ &{}\quad \mathop {\longrightarrow }\limits ^{{{\text {tr}}}} R^1\pi _* {\mathcal {O}}_{{\mathcal {A}}^{{\text {univ}}}} \end{array}$$
are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant isomorphisms, independent of the choice of \({\mathcal {A}}^{{\text {univ}}}\). Moreover the short exact sequence
$$\begin{aligned} (0) \longrightarrow \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \otimes {\mathcal {O}}_{{\mathcal {A}}^{{\text {univ}}}} \longrightarrow \varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}} \longrightarrow \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)} \otimes {\mathcal {O}}_{{\mathcal {A}}^{{\text {univ}}}} \longrightarrow (0) \end{aligned}$$
gives rise to a map
$$\begin{array}{rcl} \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}&{}\longrightarrow &{} \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \otimes R^1\pi _* {\mathcal {O}}_{{\mathcal {A}}^{{\text {univ}}}} \\ &{} \mathop {\longleftarrow }\limits ^{\sim } &{} \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}} \otimes {{\text {Hom}}}\left( \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}, \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\right) \end{array}$$
and hence to a map
$$\left( \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\right) ^{\otimes 2} \longrightarrow \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\otimes \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)}.$$
These maps do not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\) and are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant. They further induce \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) isomorphisms
$$S \left( \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\right) \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\otimes \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)},$$
which again do not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). (See, for instance, proposition 3.4.3.3 of [44].)
Let \({\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) denote the principal \(L_{n,(n)}\)-bundle on \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2)}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\big |_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\begin{aligned} \xi _1: \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{{\mathbb {Z}}} (\varLambda _n/\varLambda _{n,(n)}, {\mathcal {O}}_W). \end{aligned}$$
We define the \(L_{n,(n)}\)-action on \({\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}} \}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of the algebraic group \(L_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}\) over \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}(W)\) to be the set of \(L_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2)}|_W \rightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action over the system of schemes \(\{ {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0 \} \). The maps
$$\begin{aligned} g^*{\mathcal {E}}_{U^p(N_1,N_2),\rho } \longrightarrow {\mathcal {E}}_{(U^p)'(N_1',N_2'),\rho } \end{aligned}$$
are isomorphisms. The pull-back of \({\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) to
$$\begin{aligned} {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0[1/p] \end{aligned}$$
is canonically identified with the sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho \otimes _{R_0} R_0[1/p]}\). This identification is \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant.
Note that
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),{{\text {Std}}}^\vee } \cong \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\nu ^{-1}} \cong \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee } \cong \omega ^{{\text {ord}}}_{U^p(N_1,N_2)} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),{{\text {KS}}}} \cong \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}. \end{aligned}$$

4.4.4 Vector bundles on Kuga–Sato varieties in mixed characteristic

Suppose now that \(U^p\) is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n({\mathbb {A}}^{\infty ,p})\). We will let \(\varOmega _{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{{\mathcal {G}}^{{\text {univ}}}/{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}}\). This is a locally free sheaf of rank \((n+m)[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \({\mathcal {G}}^{{\text {univ}}}\). The system of sheaves \(\{ \varOmega ^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\}\) has actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and of \(GL_m({\mathcal {O}}_{F,(p)})\). Moreover there is an exact sequence
$$\begin{aligned} (0) \rightarrow \pi _{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {X}}^{{\text {ord}}}_n}^* \varOmega ^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)} \rightarrow \varOmega _{n,U^p(N_1,N_2)}^{(m)} \rightarrow {\mathcal {O}}_{F,(p)}^m \otimes _{\mathbb {Q}}{\mathcal {O}}_{{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m)}} \rightarrow (0) \end{aligned}$$
which is equivariant for the actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(GL_m({\mathcal {O}}_{F,(p)})\).
Let \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) denote the principal \(R^{(m)}_{n,(n)}\)-bundle on \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\big |_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\xi _1: \varOmega _{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}\left( \varLambda _n/\varLambda _{n,(n)}\oplus {{\text {Hom}}}\left( {\mathcal {O}}_F^m,{\mathbb {Z}}\right) , {\mathcal {O}}_W\right) $$
satisfies
$$\begin{aligned} \xi _1: \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}( \varLambda _n/\varLambda _{n,(n)}, {\mathcal {O}}_W) \end{aligned}$$
and induces the canonical isomorphism
$$\begin{aligned} {\mathcal {O}}_{F,(p)}^m \otimes _{{\mathbb {Z}}_{(p)}} {\mathcal {O}}_W \longrightarrow {{\text {Hom}}}\left( {{\text {Hom}}}\left( {\mathcal {O}}_F^m,{\mathbb {Z}}\right) , {\mathcal {O}}_W\right) . \end{aligned}$$
We define the \(R^{(m)}_{n,(n)}\)-action on \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \}\) has an action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and of \(GL_m({\mathcal {O}}_{F,(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of \(R^{(m)}_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) over \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{(m)}_{U,\rho }(W)\) to be the set of \(R^{(m)}_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2)}\big |_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(G^{(m)}_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(GL_m({\mathcal {O}}_{F,(p)})\)-action over the system of schemes \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{U^p(N_1,N_2),\rho }^{(m),{{\text {ord}}}} \longrightarrow {\mathcal {E}}_{(U^p)'(N_1',N_2'),\rho }^{(m),{{\text {ord}}}} \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{U^p(N_1,N_2),\rho }^{(m),{{\text {ord}}}} \longrightarrow {\mathcal {E}}_{(U^p)'(N_1',N_2'),\rho }^{(m),{{\text {ord}}}} \end{aligned}$$
are isomorphisms. If \(\rho \) factors through \(R^{(m)}_{n,(n)} \twoheadrightarrow L_{n,(n)}\) then \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^{{\text {ord}}}_{U^p(N_1,N_2),\rho }\) from \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\). In general \(W_\rho \) has a filtration by \(R_{n,(n)}^{(m)}\)-invariant local direct summands such that the action of \(R^{(m)}_{n,(n)}\) on each graded piece factors through \(L_{n,(n)}\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(L_{n,(n),{{\text {herm}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) has a \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m({\mathcal {O}}_{F,(p)})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^{{\text {ord}}}_{U^p(N_1,N_2),\rho '}\) from \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\).

4.4.5 Higher direct images from Kuga–Sato varieties to Shimura varieties, characteristic zero case

If \(m \ge m'\) and if U is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) then the sheaf
$$\begin{aligned} R^j\pi _{A^{(m)}_{n}/A^{(m')}_n,*} \varOmega ^i_{A^{(m)}_{n,U}/A^{(m')}_{n,U'}} \end{aligned}$$
depends only on \(U'\) and not on U. We will denote it
$$\begin{aligned} \left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U'}. \end{aligned}$$
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(g^{-1}U_1g \subset U_2\) then there is a natural isomorphism
$$g: (g')^*\left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U_2'} \mathop {\longrightarrow }\limits ^{\sim }\left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U_1'},$$
where \(g'\) (resp. \(U_1'\), resp. \(U_2'\)) denotes the image of g (resp. \(U_1\), resp. \(U_2\)) in \(G_n^{(m')}({\mathbb {A}}^\infty )\). This isomorphism only depends on \(g'\), \(U_1'\) and \(U_2'\) and not on g, \(U_1\) and \(U_2\). This gives the system of sheaves \(\{ (R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n})_{U'} \}\) a left action of \(G_n^{(m')}({\mathbb {A}}^\infty )\). Also if \(\gamma \in Q_{m,m'}(F)\) then \(\gamma : A^{(m)}_{n,U} \rightarrow A^{(m)}_{n,\gamma U}\) gives a natural isomorphism
$$\gamma : \left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U'} \mathop {\longrightarrow }\limits ^{\sim }\left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U'},$$
which depends only on \(U'\) and not on U. This gives the system of sheaves
$$\begin{aligned} \left\{ \left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U'} \right\} \end{aligned}$$
a right action of \(Q_{m,m'}(F)\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \(U_1' \supset U_2'\) and \(g' \in U_2'\) normalizes \(U_1'\) then on
$$\left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U_2'} \cong \left( R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n}\right) _{U_1'} \otimes _{{\mathcal {O}}_{A^{(m')}_{n,U_1'}}} {\mathcal {O}}_{A^{(m')}_{n,U_2'}}$$
the actions of g and \(1 \otimes g\) agree. Moreover if U is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) then the natural map
$$\pi _{A^{(m)}_n/A^{(m')}_n}^* \left( \pi _* \varOmega ^1_{A^{(m)}_n/A^{(m')}_n}\right) _{U'} \longrightarrow \varOmega ^1_{A^{(m)}_{n,U}/A^{(m')}_{n,U'}}$$
is an isomorphism. These isomorphisms are equivariant for the actions of the groups \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(Q_{m,m'}(F)\).
The natural maps
$$\wedge ^i \left( \pi _* \varOmega ^1_{A_n^{(m)}/A^{(m')}_n}\right) _{U'} \otimes \wedge ^j \left( R^1\pi _* {\mathcal {O}}_{A_n^{(m)}}\right) _{U'} \longrightarrow \left( R^j\pi _* \varOmega ^i_{A_n^{(m)}/A^{(m')}_n}\right) _{U'}$$
are \(G_n^{(m')}({\mathbb {A}}^\infty )\) and \(Q_{m,m'}(F)\) equivariant isomorphisms.
Suppose that U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) and \(U''\) in \(G_n({\mathbb {A}}^\infty )\). If U is of the form \(U' \ltimes M\), then the quasi-isogeny \(i_{A^{{\text {univ}}}}: A^{(m)}_{n,U} \rightarrow (A^{{\text {univ}}})^{m-m'}\) over \(A^{(m')}_{n,U'}\) gives rise to an isomorphism
$$\begin{aligned} {{\text {Hom}}}_F\left( F^{m-m'},\varOmega _{n,U''}\right) \otimes {\mathcal {O}}_{A^{(m)}_{n,U}} \cong \varOmega ^1_{A^{(m)}_{n,U}/A^{(m')}_{n,U'}} \end{aligned}$$
and a canonical embedding
$$\varXi _{n,U''} \otimes {\mathcal {O}}_{A^{(m')}_{n,U'}}\hookrightarrow \varXi _{n,U''}^{\oplus (m-m')} \otimes {\mathcal {O}}_{A^{(m')}_{n,U'}} \hookrightarrow \left( R^1\pi _* \varOmega ^1_{A^{(m)}/A^{(m')}}\right) _{U'},$$
where the first map denotes the diagonal embedding. These maps do not depend on the choice of \(A^{{\text {univ}}}\). They are \(G_n^{(m)}({\mathbb {A}}^\infty )\)-equivariant. The first map is also \(Q_{m,m'}(F)\)-equivariant, where an element \(\gamma \in Q_{m,m'}(F)\) acts on the left hand sides by composition with the inverse of the projection of \(\gamma \) to \(GL_{m-m'}(F)\). This remains true if we do not assume that U has the form \(U' \ltimes M\).
This gives rise to canonical \(G_n^{(m')}({\mathbb {A}}^\infty )\)-equivariant isomorphisms
$${{\text {Hom}}}_F\left( F^{m-m'},\varOmega _{n,U''} \right) \otimes {\mathcal {O}}_{A^{(m')}_{n,U'}} \cong \left( \pi _* \varOmega ^1_{A^{(m)}_n/A^{(m')}_n}\right) _{U'}.$$
Moreover the composite maps
$$\begin{array}{l} {{\text {Hom}}}\left( \left( \pi _* \varOmega ^1_{A^{(m)}_n/A^{(m')}_n}\right) _{U'}, \varXi _{n,U''} \otimes {\mathcal {O}}_{A^{(m')}_{n,U'}}\right) \\ \quad \hookrightarrow {{\text {Hom}}}\left( \left( \pi _* \varOmega ^1_{A_n^{(m)}/A_n^{(m')}}\right) _{U'},\left( R^1\pi _* \varOmega ^1_{A_n^{(m)}/A_n^{(m')}}\right) _{U'}\right) \\ \quad \mathop {\longleftarrow }\limits ^{\sim } {{\text {Hom}}}\left( \left( \pi _* \varOmega ^1_{A_n^{(m)}/A_n^{(m')}}\right) _{U'},\left( \pi _* \varOmega ^1_{A_n^{(m)}/A_n^{(m')}}\right) _{U'} \otimes \left( R^1\pi _* {\mathcal {O}}_{A_n^{(m)}}\right) _{U'}\right) \\ \quad \mathop {\longrightarrow }\limits ^{{{\text {tr}}}} \left( R^1\pi _* {\mathcal {O}}_{A_n^{(m)}}\right) _{U'} \end{array}$$
are \(G_n^{(m')}({\mathbb {A}}^\infty )\)-equivariant isomorphisms.

4.4.6 Higher direct images from Kuga–Sato varieties to Shimura varieties, mixed characteristic case

If \(m \ge m'\) and if \(U^p\) is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), and if \(0 \le N_1 \le N_2\) are integers, then the sheaf
$$R^j\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n}/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n,*} \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}/{\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2)}}$$
depends only on \((U^p)'\) and not on \(U^p\). We will denote it
$$ \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)}.$$
If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(g^{-1}U^p_1(N_{11},N_{12})g \subset U^p_2(N_{21},N_{22})\), then there is a natural map
$$g: (g')^*\left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p_2)'(N_{21},N_{22})} \rightarrow \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p_1)'(N_{11},N_{12})},$$
where \((U_i^p)'\) denotes the image of \(U^p_i\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\) and \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is an isomorphism. Moreover this map only depends on \(g'\), \((U_1^p)'(N_{11},N_{12})\) and \((U^p_2)'(N_{21},N_{22})\) and not on g, \(U_1^p(N_{11},N_{12})\) and \(U_2^p(N_{21},N_{22})\). This gives the system of sheaves
$$\left\{ \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \right\} $$
a left action of \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\gamma : A^{(m)}_{n,U^p(N_1,N_2)} \rightarrow A^{(m)}_{n,\gamma U^p(N_1,N_2)}\) gives a natural isomorphism
$$\gamma : \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \mathop {\longrightarrow }\limits ^{\sim }\left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)},$$
which depends only on \((U^p)'(N_1,N_2)\) and not on \(U^p(N_1,N_2)\). This gives the system of sheaves
$$\left\{ \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \right\} $$
a right action of \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \((U^p_1)'(N_{11},N_{12}) \supset (U_2^p)'(N_{21},N_{22})\) and \(g \in (U^p_1)' (N_{11},N_{12})\) normalizes the subgroup \((U^p_2)'(N_{21},N_{22})\), then on
$$\begin{array}{l} \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p_2)'(N_{21},N_{22})} \\ \quad \cong \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p_1)'(N_{11},N_{12})} \otimes _{{\mathcal {O}}_{{\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p_1)'(N_{11},N_{12})}}} {\mathcal {O}}_{{\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p_2)'(N_{21},N_{22})}} \end{array}$$
the actions of g and \(1 \otimes g\) agree. Moreover if \(U^p\) is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), and if \(0 \le N_1 \le N_2\), then the natural map
$$\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}^* \left( \pi _* \varOmega ^1_{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \longrightarrow \varOmega ^1_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}/{\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2)}}$$
is an isomorphism. These isomorphisms are \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\) equivariant.
The natural maps
$$\begin{array}{l} \wedge ^i \left( \pi _* \varOmega ^1_{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \otimes \wedge ^j \left( R^1\pi _* {\mathcal {O}}_{{\mathcal {A}}_n^{(m),{{\text {ord}}}}}\right) _{(U^p)'(N_1,N_2)} \\ \quad \longrightarrow \left( R^j\pi _* \varOmega ^i_{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\right) _{(U^p)'(N_1,N_2)} \end{array}$$
are \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\) equivariant isomorphisms.
Under the identification
$${\mathcal {X}}^{{\text {ord}}}_{n, (U^p)'(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong X_{n,(U^p)'(N_1,N_2)}$$
the sheaves \(\varOmega ^{{\text {ord}}}_{n, (U^p)'(N_1,N_2)}\) (resp. \(\varXi ^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}\)) are naturally identified with the sheaves \(\varOmega _{n, (U^p)'(N_1,N_2)}\) (resp. \(\varXi _{n,(U^p)'(N_1,N_2)}\)). Moreover, under the identification
$$\begin{aligned} {\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong A^{(m')}_{n,U^p(N_1,N_2)} \end{aligned}$$
the sheaf \((R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}})_{(U^p)'(N_1,N_2)}\) is naturally identified with the sheaf \((R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n})_{(U^p)'(N_1,N_2)}\). These identifications are equivariant for the actions of \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\).

5 Generalized Shimura varieties

We will introduce certain disjoint unions of mixed Shimura varieties, which are associated to \(L_{n,(i),{{\text {lin}}}}\) and \(L_{n,(i)}\) and \(P_{n,(i)}^+/Z(N_{n,(i)})\) and \(P_{n,(i)}^+\); to \(L^{(m)}_{n,(i),{{\text {lin}}}}\) and \(L^{(m)}_{n,(i)}\) and \(P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)})\) and \(P^{(m),+}_{n,(i)}\); and to \(\widetilde{{P}}^{(m),+}_{n,(i)}\). The differences with the last section are purely book keeping. We then describe certain torus embeddings for these generalized Shimura varieties and discuss their completion along the boundary. These completions will serve as formal local models near the boundary of the toroidal compactifications of the \(X_{n,U}\) and the \(A^{(m)}_{n,\widetilde{{U}}}\) to be discussed in the next section.

We remind the reader of our convention that, if U is a subgroup of G and H is a quotient of G, then we will sometimes use U to denote its image in H. We hope that this causes no confusion as we will only do this when the context makes clear we are referring to a subgroup of H.

5.1 Generalized Shimura varieties

If U is a neat open compact subgroup of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) we set
$$\begin{aligned} Y_{n,(i),U}^{(m),+}=\coprod _{L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )/U} {{\text {Spec}}\,}{\mathbb {Q}}. \end{aligned}$$
In the case \(m=0\) we will write simply \(Y_{n,(i),U}^+\). Then \(\{ Y^{(m),+}_{n,(i),U} \}\) is a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\)-action. Each \(Y_{n,(i),U}^{(m),+}\) also has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), and the inverse system has a right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\). If \(\delta \in GL_m(F)\) we get a map
$$\begin{aligned} \delta : Y_{n,(i),U}^{(m)} \longrightarrow Y_{n,(i),\delta (U)}^{(m)} \end{aligned}$$
which sends \(({{\text {Spec}}\,}{\mathbb {Q}})_{hU} \rightarrow ({{\text {Spec}}\,}{\mathbb {Q}})_{\delta (h) \delta (U)}\) via the identity. This gives a left action of \(GL_m(F)\) on the inverse system of the \(Y_{n,(i),U}^{(m)}\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(U'\) denotes the image of U in \(L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) then there is a natural map
$$\begin{aligned} Y_{n,(i),U}^{(m),+} \twoheadrightarrow Y_{n,(i),U'}^+. \end{aligned}$$
These maps are equivariant for
$$\begin{aligned} L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty ) \longrightarrow L_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty ). \end{aligned}$$
The naive quotient
$$\begin{aligned} L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \Big \backslash Y^{(m),+}_{n,(i),U} \end{aligned}$$
makes sense. We will denote this space
$$\begin{aligned} Y^{(m),\natural }_{n,(i),U} \end{aligned}$$
and drop the (m) if \(m=0\). The inverse system of these spaces has a right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\), and a left action of \(GL_m(F)\). The induced map
$$\begin{aligned} Y^{(m),\natural }_{n,(i),U} \longrightarrow Y^{\natural }_{n,(i),U} \end{aligned}$$
is an isomorphism, and \(GL_m(F)\) acts trivially on these spaces. (Use the fact that
$$\begin{aligned} \left. \left( U \cap \left( {{\text {Hom}}}_F(F^m,F^i) \otimes _{\mathbb {Q}}{\mathbb {A}}^\infty \right) \right) + {{\text {Hom}}}_F(F^m,F^i)={{\text {Hom}}}_F(F^m,F^i) \otimes _{\mathbb {Q}}{\mathbb {A}}^\infty . \right) \end{aligned}$$
Similarly if \(U^p\) is a neat open compact subgroup of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\) and if \(N \in {\mathbb {Z}}_{\ge 0}\) we set
$$\begin{aligned} {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}=\coprod _{L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{{\text {ord}}}, \times }/U^p(N)} {{\text {Spec}}\,}{\mathbb {Z}}_{(p)}. \end{aligned}$$
In the case \(m=0\) we drop it from the notation. Each \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and the inverse system of the \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) has a commuting right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). It also has a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )^{{\text {ord}}}\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). There are equivariant maps
$$\begin{aligned} {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} \longrightarrow {\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},+}. \end{aligned}$$
We set
$$\begin{aligned} {\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},\natural }= L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \Big \backslash {\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},+}= L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \Big \backslash {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}. \end{aligned}$$
There are maps
$$\begin{aligned} {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow Y_{n,(i),U^p(N)}^{(m),+} \end{aligned}$$
which are equivariant for the actions of the groups \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(GL_m({\mathcal {O}}_{F,(p)})\). Moreover the maps \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} \rightarrow {\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},+}\) and \(Y_{n,(i),U^p(N)}^{(m),+}\rightarrow Y_{n,(i),U^p(N)}^{+}\) are compatible. The induced maps
$$\begin{aligned} {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},\natural } \times {{\text {Spec}}\,}{\mathbb {Q}}\mathop {\longrightarrow }\limits ^{\sim }Y_{n,(i),U^p(N)}^{(m),\natural } \end{aligned}$$
are isomorphisms.
Suppose now that U is a neat open compact subgroup of \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\). We set
$$\begin{aligned} X_{n,(i),U}^{(m),+}= \left( X_{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )} \times Y^{(m),+}_{n,(i),U \cap L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )}\right) \big /U \end{aligned}$$
In the case \(m=0\) we will write simply \(X^+_{n,(i),U}\). Then \(\{ X^{(m),+}_{n,(i),U} \}\) is a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-action via finite etale maps. Each \(X_{n,(i),U}^{(m),+}\) has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), which commutes with the right \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-action. The system also has a left action of \(GL_m(F)\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(U'\) is an open normal subgroup of U then \(X_{n,(i),U}^{(m),+}\) is identified with \(X_{n,(i),U'}^{(m),+}/U\). Projection to the second factor gives \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant maps
$$\begin{aligned} X_{n,(i),U}^{(m),+} \longrightarrow Y_{n,(i),U}^{(m),+}. \end{aligned}$$
The fibre over \(g \in L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) is simply \(X_{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )}\). If \(U'\) denotes the image of U in \(L_{n,(i)}({\mathbb {A}}^\infty )\) then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant, commutative diagram
$$\begin{aligned} \begin{array}{ccc} X_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{} X_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow \\ Y_{n,(i),U}^{(m),+} &{} \twoheadrightarrow &{} Y_{n,(i),U'}^+. \end{array} \end{aligned}$$
We have
$$X^{(m),+}_{n,(i),U}({\mathbb {C}})= L_{n,(i),{{\text {herm}}}}({\mathbb {Q}}) \Big \backslash \left( L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\Big /U \times {\mathfrak {H}}_{n-i}^\pm \right) $$
and
$$\pi _0\left( X^{(m),+}_{n,(i),U} \times {{\text {Spec}}\,}\overline{{{\mathbb {Q}}}}\right) \cong \left( L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty ) \times \left( C_{n-i}({\mathbb {Q}})\Big \backslash C_{n-i}({\mathbb {A}})\Big /C_{n-i}({\mathbb {R}})^0 \right) \right) \Big /U.$$
The naive quotient
$$\begin{aligned} X^{(m),\natural }_{n,(i),U}=L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \Bigg \backslash X^{(m),+}_{n,(i),U} \end{aligned}$$
makes sense and fibres over \(Y^{(m),\natural }_{n,(i),U}\), the fibre over g being \(X_{n-i,U_1}\), where \(U_1\) denotes the projection to \(G_{n-i}({\mathbb {A}}^\infty )\) of the subgroup \(U_2 \subset U\) consisting of elements whose projection to \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) lies in \(g^{-1}L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})g\). We sometimes write \(X^{\natural }_{n,(i),U}\) for \(X^{(0),\natural }_{n,(i),U}\). If \(U'\) denotes the projection of U to \(L_{n,(i)}({\mathbb {A}}^\infty )\), then the induced map
$$\begin{aligned} X^{(m),\natural }_{n,(i),U} \mathop {\longrightarrow }\limits ^{\sim }X_{n,(i),U'}^{\natural } \end{aligned}$$
is an isomorphism. The action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\) is by finite etale maps and if \(U'\) is an open normal subgroup of U then \(X_{n,(i),U}^{(m),\natural }\) is identified with \(X_{n,(i),U'}^{(m),\natural }/U\). We have
$$\pi _0\left( X^{(m),\natural }_{n,(i),U} \times {{\text {Spec}}\,}\overline{{{\mathbb {Q}}}}\right) \cong \left( F^\times \times C_{n-i}({\mathbb {Q}})\right) \big \backslash \left( {\mathbb {A}}_F^\times \times C_{n-i}({\mathbb {A}})\right) \big / U\left( F_\infty ^\times \times C_{n-i}({\mathbb {R}})^0\right) .$$
We define sheaves \(\varOmega ^{+}_{n,(i),U}\) and \(\varXi ^{+}_{n,(i),U}\) over \(X^+_{n,(i),U}\) as the quotients of
$$\begin{aligned} \varOmega _{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )} / X_{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )} \times Y^{+}_{n,(i),U \cap L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )} \end{aligned}$$
and
$$\begin{aligned} \varXi _{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )} / X_{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )} \times Y^{+}_{n,(i),U \cap L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )} \end{aligned}$$
by U. Then \(\{ \varOmega ^{+}_{n,(i),U} \}\) and \(\{ \varXi ^{+}_{n,(i),U} \}\) are systems of locally free sheaves on \(X^{+}_{n,(i),U}\) with left \(L_{n,(i)}({\mathbb {A}}^\infty )\)-action and commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action.
Let \({\mathcal {E}}_{(i),U}^{+}\) denote the principal \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-bundle on \(X^+_{n,(i),U}\) in the Zariski topology defined by setting, for \(W \subset X^+_{n,(i),U}\) a Zariski open, \({\mathcal {E}}_{(i),U}^{+}(W)\) to be the set of triples \((\xi _0,\xi _{11},\xi _{12})\), where
$$\begin{aligned} \xi _0: \varXi ^+_{n,(i),U}\big |_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\begin{aligned} \xi _{11}: \varOmega ^+_{n,(i),U} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}( V_{n-i}/V_{n-i,(n-i)} , {\mathcal {O}}_W) \end{aligned}$$
and
$$\begin{aligned} \xi _{12}: F^i \otimes _{\mathbb {Q}}{\mathcal {O}}_W \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}\left( V_n/V_{n,(i)}^\perp , {\mathcal {O}}_W\right) . \end{aligned}$$
We define the \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-action on \({\mathcal {E}}_{(i),U}^{+}\) by
$$h(\xi _0,\xi _{11},\xi _{12})=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _{11},(\circ h^{-1}) \circ \xi _{12} \right) .$$
The inverse system \(\{ {\mathcal {E}}_{(i),U}^{+} \}\) has an action of \(L_{n,(i)}({\mathbb {A}}^\infty )\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of the algebraic group \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{+}_{(i),U,\rho }\) over \(X^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{+}_{(i),U,\rho }(W)\) to be the set of \((R_{n,(n),(i)}/N(R_{n,(n),(i)}))({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}^{+}_{(i),U}|_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{+}_{(i),U,\rho } \}\) is a system of locally free sheaves with an \(L_{n,(i)}({\mathbb {A}}^\infty )\)-action and an \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action over the system of schemes \(\{ X^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^+_{(i),U,\rho }\) to \(X_{n-i,hUh^{-1} \cap G_{n-i}({\mathbb {A}}^\infty )}\) can be identified with \({\mathcal {E}}_{hUh^{-1} \cap G_{n-i}({\mathbb {A}}^\infty ),\rho |_{L_{n-i,(n-i)}}}\). However the description of the actions of \(L_{n,(i)}({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) involve \(\rho \) and not just \(\rho |_{L_{n-i,(n-i)}}\). If \(g \in L_{n,(i)}({\mathbb {A}}^\infty )\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U,\rho }^+ \longrightarrow {\mathcal {E}}_{(i),U',\rho }^+ \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{(i),U,\rho }^+ \longrightarrow {\mathcal {E}}_{(i),U',\rho }^+ \end{aligned}$$
are isomorphisms.
We will also write
$$\begin{aligned} \varOmega ^{\natural }_{n,(i),U}=L_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \Big \backslash \varOmega ^{+}_{n,(i),U} \end{aligned}$$
and
$$\begin{aligned} \varXi ^{\natural }_{n,(i),U}=L_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \Big \backslash \varXi ^{+}_{n,(i),U}, \end{aligned}$$
locally free sheaves on \(X^{\natural }_{n,(i),U}\). (If \(\rho \) is trivial on \(L_{n,(i),{{\text {lin}}}}\) then one can also form the quotient of \({\mathcal {E}}^+_{(i),U,\rho }\) by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), but in general this quotient does not make sense.)
If \(U^p\) is a neat open compact subgroup of \(L^{(m)}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and \(N_2 \ge N_1 \ge 0\) we set
$$\begin{aligned} {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}= \left( {\mathcal {X}}^{{\text {ord}}}_{n-i,(U^p \cap G_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)} \times {\mathcal {Y}}^{(m),{{\text {ord}}},+}_{n,(i),(U^p \cap L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p}))(N_1)}\right) \big /U^p. \end{aligned}$$
In the case \(m=0\) we drop it from the notation. Each \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\) and the inverse system has a commuting right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). There is also a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and if
$$\begin{aligned} g: {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \longrightarrow {\mathcal {X}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+}, \end{aligned}$$
then this map is quasi-finite and flat. If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1'\le N_2\) then
$$\begin{aligned} {\mathcal {X}}_{n,(i),(U^p)'(N_1',N_2)}^{(m),{{\text {ord}}},+}\Big /U^p(N_1,N_2) \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}. \end{aligned}$$
There are commutative diagrams
$$\begin{array}{ccc} {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {Y}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{} \twoheadrightarrow &{} {\mathcal {Y}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}. \end{array}$$
We set
$$\begin{aligned} {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural }=L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \backslash {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}, \end{aligned}$$
and write \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural }\) for \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(0),{{\text {ord}}},\natural }\). The system of these spaces has a right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then \(\delta \circ g=\delta (g) \circ \delta \). If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and if
$$\begin{aligned} g: {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural } \longrightarrow {\mathcal {X}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},\natural }, \end{aligned}$$
then this map is quasi-finite and flat. If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1'\le N_2\) then
$$\begin{aligned} {\mathcal {X}}_{n,(i),(U^p)'(N_1',N_2)}^{(m),{{\text {ord}}},\natural }\Big /U^p(N_1,N_2) \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural }. \end{aligned}$$
The natural maps
$$\begin{aligned} {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural } \longrightarrow {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural } \end{aligned}$$
are isomorphisms.
We define sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) and \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) as the quotients of
$$\varOmega _{n-i,\left( U^p \cap G_{n-i}({\mathbb {A}}^{\infty ,p})\right) (N_1,N_2)}^{{{\text {ord}}}} \Big / {\mathcal {X}}^{{\text {ord}}}_{n-i,\left( U^p \cap G_{n-i}({\mathbb {A}}^{\infty ,p})\right) (N_1,N_2)} \times {\mathcal {Y}}^{{{\text {ord}}},+}_{n,(i),\left( U^p \cap L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\right) (N_1)}$$
and
$$\varXi ^{{{\text {ord}}}}_{n-i,\left( U^p \cap G_{n-i}({\mathbb {A}}^{\infty ,p})\right) (N_1,N_2)} \Big / {\mathcal {X}}^{{\text {ord}}}_{n-i,\left( U^p \cap G_{n-i}({\mathbb {A}}^{\infty ,p})\right) (N_1,N_2)} \times {\mathcal {Y}}^{{{\text {ord}}},+}_{n,(i),\left( U^p \cap L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\right) (N_1)}$$
by \(U^p\). Then the systems of sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) and \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) have commuting actions of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(L_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
Let \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) denote the principal \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-bundle for the Zariski topology on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) defined by setting, for \(W \subset {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}(W)\) to be the set of triples \((\xi _0,\xi _{11},\xi _{12})\), where
$$\begin{aligned} \xi _0: \varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\begin{aligned} \xi _{11}: \varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}( \varLambda _{n-i}/\varLambda _{n-i,(n-i)} , {\mathcal {O}}_W) \end{aligned}$$
and
$$\xi _{12}: {\mathcal {O}}_F^i \otimes _{\mathbb {Z}}{\mathcal {O}}_W \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}\left( \varLambda _n/\varLambda _{n,(i)}^\perp ,{\mathcal {O}}_W\right) .$$
We define the \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-action on \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) by
$$\begin{aligned} h(\xi _0,\xi _{11},\xi _{12})=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _{11}, (\circ h^{-1}) \circ \xi _{12}\right) . \end{aligned}$$
The inverse system \(\{ {\mathcal {E}}_{(i),U^p(N_1,N_2)}^{+} \}\) has an action of \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of the algebraic group \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }(W)\) to be the set of \((R_{n,(n),(i)}/N(R_{n,(n),(i)}))({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2)}\big |_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action over the system of schemes \(\{ {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) to \({\mathcal {X}}^{{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}({\mathbb {A}}^{\infty ,p})(N_1,N_2))}\) can be identified with \({\mathcal {E}}^{{{\text {ord}}}}_{(hU^ph^{-1} \cap G_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2),\rho |_{L_{n-i,(n-i)}}}\). However the description of the actions of the groups \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) involves \(\rho \) and not just \(\rho |_{L_{n-i,(n-i)}}\). If \(g \in L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U^p(N_1,N_2),\rho }^{{{\text {ord}}},+} \longrightarrow {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\rho }^{{{\text {ord}}},+} \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{(i),U^p(N_1,N_2),\rho }^{{{\text {ord}}},+} \longrightarrow {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\rho }^{{{\text {ord}}},+} \end{aligned}$$
are isomorphisms.
We will also write
$$\begin{aligned} \varOmega ^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}=L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \Big \backslash \varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \end{aligned}$$
and
$$\begin{aligned} \varXi ^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}=L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \Big \backslash \varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}, \end{aligned}$$
locally free sheaves on \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\).
There are maps
$$\begin{aligned} {\mathcal {X}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow X^{(m),+}_{n,(i),U^p(N_1,N_2)} \end{aligned}$$
which are equivariant for the actions of the groups \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\) and \(GL_m({\mathcal {O}}_{F,(p)})\). Under these maps the sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) (resp. \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), resp. \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\)) correspond to \(\varOmega ^+_{n,(i),U^p(N_1,N_2)}\) (resp. to \(\varXi ^+_{n,(i),U^p(N_1,N_2)}\), resp. to \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho \otimes {\mathbb {Q}}}\)). The induced maps
$$\begin{aligned} {\mathcal {X}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\longrightarrow X^{(m),\natural }_{n,(i),U^p(N_1,N_2)} \end{aligned}$$
are isomorphisms.

5.2 Generalized Kuga–Sato varieties

Now suppose that U is a neat open compact subgroup of
$$\left( P^{(m),+}_{n,(i)}\Big /Z\left( N^{(m)}_{n,(i)}\right) \right) ({\mathbb {A}}^\infty )= \left( \widetilde{{P}}^{(m),+}_{n,(i)}\Big /Z\left( \widetilde{{N}}^{(m)}_{n,(i)}\right) \right) ({\mathbb {A}}^\infty ).$$
We set
$$A_{n,(i),U}^{(m),+}= \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )/U} A^{(i+m)}_{n-i, h U h^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) }.$$
In the case \(m=0\) we will write simply \(A^+_{n,(i),U}\).
If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) and \(g^{-1} U g \subset U'\), then we define a finite etale map
$$\begin{aligned} g: A_{n,(i),U}^{(m),+} \longrightarrow A_{n,(i),U'}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} g': A^{(i+m)}_{n-i, h U h^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) } \longrightarrow A^{(i+m)}_{n-i, h' U' (h')^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) }, \end{aligned}$$
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(g' \in G^{(i+m)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(hg=g' h'\). This makes \(\{ A^{(m),+}_{n,(i),U} \}\) a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right action of the group \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\). If \(U'\) is an open normal subgroup of U then \(A_{n,(i),U}^{(m),+}\) is identified with \(A_{n,(i),U'}^{(m),+}/U\).
If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
$$\begin{aligned} \gamma : A_{n,(i),U}^{(m),+} \longrightarrow A_{n,(i),U}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \gamma : A^{(i+m)}_{n-i, h U h^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) } \longrightarrow A^{(i+m)}_{n-i, (\gamma h)U (\gamma h)^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) }. \end{aligned}$$
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(A_{n,(i),U}^{(m),+}\), which commutes with the action of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\).
If \(\delta \in GL_m(F)\) define a map
$$\begin{aligned} \delta : A_{n,(i),U}^{(m),+} \longrightarrow A_{n,(i),\delta (U)}^{(m),+} \end{aligned}$$
as the coproduct of the maps
$$\begin{aligned} \delta : A^{(i+m)}_{n-i, h U h^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) } \longrightarrow A^{(i+m)}_{n-i, \delta (hUh^{-1}) \cap G_{n-i}^{(i+m)}({\mathbb {A}}^\infty ) }. \end{aligned}$$
This gives a left \(GL_m(F)\)-action on the system of the \(A_{n,(i),U}^{(m),+}\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \).
There are natural maps
$$\begin{aligned} A_{n,(i),U}^{(m),+} \longrightarrow X_{n,(i),U}^{(m),+}, \end{aligned}$$
which are equivariant for the actions of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(GL_m(F)\).
If \(U'\) denotes the image of U in \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\) then there is a natural commutative diagram:
$$\begin{aligned} \begin{array}{ccc} A_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{} A_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow \\ X_{n,(i),U}^{(m),+} &{} \twoheadrightarrow &{} X_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow \\ Y_{n,(i),U}^{(m),+} &{} \twoheadrightarrow &{} Y_{n,(i),U'}^+, \end{array} \end{aligned}$$
which is \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) equivariant.
We have
$$A_{n,(i),U}^{(m),+}({\mathbb {C}})=\left( P_{n,(i)}^{(m)}/Z \left( N_{n,(i)}^{(m)}\right) \right) ({\mathbb {Q}}) \Big \backslash \left( P_{n,(i)}^{(m),+}/Z\left( N_{n,(i)}^{(m)}\right) \right) ({\mathbb {A}})\Big / \left( UU_{n-i,\infty }^0A_{n-i}({\mathbb {R}})^0\right) .$$
Note that it does not make sense to divide \(A_{n,(i),U}^{(m),+}\) by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\), so we don’t do so.
We define a semi-abelian scheme \(\widetilde{{G}}^{{\text {univ}}}/A^+_{n,(i),U}\) by requiring that over the open and closed subscheme \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) it restricts to \(G^{{\text {univ}}}\). It is unique up to unique quasi-isogeny. We also define a sheaf \(\widetilde{\varOmega }^{+}_{n,(i),U}\) (resp. \(\widetilde{\varXi }^+_{n,(i),U}\)) over \(A^+_{n,(i),U}\) to be the unique sheaf which, for each h, restricts to \(\varOmega ^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) (resp. \(\varXi _{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\)) on \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\). Thus \(\widetilde{\varOmega }^+_{n,(i),U}\) is the pull-back by the identity section of \(\varOmega ^1_{\widetilde{{G}}^{{\text {univ}}}/A^+_{n,(i),U}}\). Then \(\{ \widetilde{\varOmega }^{+}_{n,(i),U} \}\) (resp. \(\{ \widetilde{\varXi }^{+}_{n,(i),U} \}\)) is a system of locally free sheaves on \(A^{+}_{n,(i),U}\) with a left \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\)-action and a commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action. There are equivariant exact sequences
$$\begin{aligned} (0) \longrightarrow \pi ^* \varOmega ^+_{n,(i),U} \longrightarrow {\widetilde{\varOmega }}^+_{n,(i),U} \longrightarrow F^i \otimes _{\mathbb {Q}}{\mathcal {O}}_{A^+_{n,(i),U}} \longrightarrow (0), \end{aligned}$$
where \(\pi \) denotes the map \(A^+_{n,(i),U} \rightarrow X^+_{n,(i),U}\).
Let \({\widetilde{{\mathcal {E}}}}_{(i),U}^{+}\) denote the principal \(R_{n,(n),(i)}\)-bundle on \(A^+_{n,(i),U}\) in the Zariski topology defined by setting, for \(W \subset A^+_{n,(i),U}\) a Zariski open, \({\widetilde{{\mathcal {E}}}}_{(i),U}^{+}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi ^+_{n,(i),U}\big |_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\xi _1: \widetilde{\varOmega }^+_{n,(i),U} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}\left( V_{n-i}/V_{n-i,(n-i)}\oplus {{\text {Hom}}}_{\mathbb {Q}}(F^i,{\mathbb {Q}}) , {\mathcal {O}}_W\right) $$
satisfies
$$\begin{aligned} \xi _1: \varOmega ^+_{n,(i),U} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Q}}\left( V_{n-i}/V_{n-i,(n-i)}, {\mathcal {O}}_W\right) . \end{aligned}$$
We define the \(R_{n,(n),(i)}\)-action on \({\mathcal {E}}_{(i),U}^{+}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\widetilde{{\mathcal {E}}}}_{(i),U}^{+} \}\) has an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{+}_{(i),U,\rho }\) over \(A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{+}_{(i),U,\rho }(W)\) to be the set of \(R_{n,(n),(i)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\widetilde{{\mathcal {E}}}}^{+}_{(i),U}|_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{+}_{(i),U,\rho } \}\) is a system of locally free sheaves with both a \(P_{n,(i)}^+({\mathbb {A}}^\infty )\)-action and an \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action over the system of schemes \(\{ A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^+_{(i),U,\rho }\) to \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) can be identified with \({\mathcal {E}}^{(i)}_{hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty ),\rho |_{R_{n-i,(n-i)}^{(i)}}}\). However the description of the actions of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) involve \(\rho \) and not just \(\rho |_{R^{(i)}_{n-i,(n-i)}}\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U,\rho }^+ \longrightarrow {\mathcal {E}}_{(i),U',\rho }^+ \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{(i),U,\rho }^+ \longrightarrow {\mathcal {E}}_{(i),U',\rho }^+ \end{aligned}$$
are isomorphisms. If \(\rho \) factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) then \({\mathcal {E}}^{+}_{(i),U,\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^+_{(i),U,\rho }\) from \(X^+_{n,(i),U}\). In general \(W_\rho \) has a filtration by \(R_{n,(n),(i)}\)-invariant local direct summands such that the action of \(R_{n,(n),(i)}\) on each graded piece factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(A_{n,(i),{{\text {lin}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{+}_{(i),U,\rho }\) has a \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^+_{(i),U,\rho '}\) from \(X^+_{n,(i),U}\).
Similarly if \(U^p\) is a neat open compact subgroup of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty ,p})=(\widetilde{{P}}^{(m),+}_{n,(i)}/Z(\widetilde{{N}}^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty ,p})\) we set
$$\begin{aligned} {\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}= \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }/U^p(N_1)} {\mathcal {A}}^{(i+m),{{\text {ord}}}}_{n-i, (h U^p h^{-1} \cap G_{n-i}^{(i+m)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}. \end{aligned}$$
In the case \(m=0\) we will write simply \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). The inverse system of the \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) has a right action of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a commuting left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then the map
$$\begin{aligned} g: {\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \longrightarrow {\mathcal {A}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+}, \end{aligned}$$
is quasi-finite and flat. If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1' \le N_2\) then \({\mathcal {A}}_{n,(i),(U^p)'(N_1',N_2)}^{(m),{{\text {ord}}},+}/U^p(N_1,N_2)\) is identified with \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). Further there is a left action of \(GL_m({\mathcal {O}}_{F,(p)})\) such that if \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\), \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). There are natural equivariant maps
$$\begin{aligned} {\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \longrightarrow {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}. \end{aligned}$$
If \((U^p)'\) denotes the image of \(U^p\) in \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^{\infty ,p})\) then there is a natural equivariant, commutative diagram:
$$\begin{array}{ccc} {\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {A}}_{n,(i),(U^p)'(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{} \twoheadrightarrow &{} {\mathcal {X}}_{n,(i),(U^p)'(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {Y}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{} \twoheadrightarrow &{} {\mathcal {Y}}_{n,(i),(U^p)'(N_1,N_2)}^{{{\text {ord}}},+}. \end{array}$$
There are equivariant embeddings
$${\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow A_{n,(i),U^p(N_1,N_2)}^{(m),+}.$$
We define a semi-abelian scheme \({\widetilde{{\mathcal {G}}}}^{{\text {univ}}}/{\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) over \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) by requiring that over \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) it restricts to \({\mathcal {G}}^{{\text {univ}}}\). It is unique up to unique prime-to-p quasi-isogeny. We define a locally free sheaf \(\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) (resp. \(\widetilde{\varXi }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)) over the scheme \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to be the sheaf which, for each h, restricts to the sheaf
$$\begin{aligned} \varOmega ^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)} \end{aligned}$$
(resp. \(\varXi ^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\)) on the subscheme \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\). Then \(\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) is the pull-back by the identity section of \(\varOmega ^1_{{\widetilde{{\mathcal {G}}}}^{{\text {univ}}}/{\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}}\). The collection \(\{\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\) (resp. \(\{ \widetilde{\varXi }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\)) is a system of locally free sheaves on \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) with a left \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action and a commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action. Also there are equivariant exact sequences
$$\begin{aligned} (0) \longrightarrow \pi ^* \varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow \widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow {\mathcal {O}}_F^i \otimes _{\mathbb {Z}}{\mathcal {O}}_{{\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}} \longrightarrow (0), \end{aligned}$$
where \(\pi \) denotes the map \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U} \rightarrow {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U}\).
Let \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) denote the principal \(R_{n,(n),(i)}\)-bundle on the scheme \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) a Zariski open, \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
$$\begin{aligned} \xi _0: \varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\big |_W \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {O}}_W \end{aligned}$$
and
$$\xi _1: \widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Z}}\left( \varLambda _{n-i}/\varLambda _{n-i,(n-i)}\oplus {{\text {Hom}}}_{\mathbb {Z}}\left( {\mathcal {O}}_F^i,{\mathbb {Z}}\right) , {\mathcal {O}}_W\right) $$
satisfies
$$\xi _1: \varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \mathop {\longrightarrow }\limits ^{\sim }{{\text {Hom}}}_{\mathbb {Z}}\left( \varLambda _{n-i}/\varLambda _{n-i,(n-i)}, {\mathcal {O}}_W\right) .$$
We define the \(R_{n,(n),(i)}\)-action on \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) by
$$\begin{aligned} h(\xi _0,\xi _1)=\left( \nu (h)^{-1}\xi _0, (\circ h^{-1}) \circ \xi _1\right) . \end{aligned}$$
The inverse system \(\{ {\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+} \}\) has an action both of the groups \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) over \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }(W)\) to be the set of \(R_{n,(n),(i)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
$$\begin{aligned} {\widetilde{{\mathcal {E}}}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2)}\big |_W \longrightarrow W_\rho \otimes _{R_0} {\mathcal {O}}_W. \end{aligned}$$
Then \(\{ {\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action over the system of schemes \(\{ {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) to \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) can be identified with
$$\begin{aligned} {\mathcal {E}}^{(i),{{\text {ord}}}}_{(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2),\rho |_{R_{n-i,(n-i)}^{(i)}}}. \end{aligned}$$
However the description of the actions of the groups \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) involves \(\rho \) and not just \(\rho |_{R^{(i)}_{n-i,(n-i)}}\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then the natural maps
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U^p(N_1,N_2),\rho }^{{{\text {ord}}},+} \longrightarrow {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\rho }^{{{\text {ord}}},+} \end{aligned}$$
and
$$\begin{aligned} \gamma ^*{\mathcal {E}}_{(i),U^p(N_1,N_2),\rho }^{{{\text {ord}}},+} \longrightarrow {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\rho }^{{{\text {ord}}},+} \end{aligned}$$
are isomorphisms. If \(\rho \) factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) then \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) from \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). In general \(W_\rho \) has a filtration by \(R_{n,(n),(i)}\)-invariant local direct summands such that the action of \(R_{n,(n),(i)}\) on each graded piece factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(A_{n,(i),{{\text {lin}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) has a \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho '}\) from \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\).

The next lemma follows from the discussion in Sect. 3.4.

Lemma 4.1

If \(U'\) is the image of U (resp. \(U^p\)) and if \(\pi \) denotes the map \(A_{n,(i),U}^{(m),+} \rightarrow A_{n,(i),U'}^+\) then there are \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant, \((P^{+}_{n,(i)}/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\)-equivariant and \(GL_m(F)\)-equivariant isomorphisms
$$\begin{array}{l} R^j\pi _* \varOmega ^k_{A_{n,(i),U}^{(m),+}/ A_{n,(i),U'}^+} \\ \quad \cong \left( \wedge ^k (F^m \otimes _F \varOmega ^{+}_{n,(i),U'})\right) \otimes \left( \wedge ^j \left( F^m\otimes _F {{\text {Hom}}}\left( \varOmega ^{+}_{n,(i),U'},\varXi ^{+}_{n,(i),U'}\right) \right) \right) . \end{array}$$

5.3 Generalized mixed Shimura varieties

Next suppose \(\widetilde{{U}}\) is an open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). We define a split torus \(\widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}/Y_{n,(i),\widetilde{{U}}}^{(m),+}\) as
$$\begin{aligned} \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )/\widetilde{{U}}} S^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} . \end{aligned}$$
Thus \(X_*(\widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}})_{\mathbb {Q}}\) is a constant sheaf:
$$X_*\left( \widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}\right) _{\mathbb {Q}}\cong {{\text {Herm}}}^{(m+i)}_{\mathbb {Q}}\cong Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {Q}}).$$
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}\widetilde{{g}}\subset \widetilde{{U}}'\), then we define
$$\begin{aligned} \widetilde{{g}}: \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \longrightarrow \widetilde{{S}}_{n,(i),\widetilde{{U}}'}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \widetilde{{g}}': S^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} \longrightarrow S^{(m+i)}_{n-i,h' \widetilde{{U}}' (h')^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )}, \end{aligned}$$
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}} \}\) a system of relative tori with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
$$\begin{aligned} \gamma : \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \longrightarrow \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \gamma : S^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} \longrightarrow S^{(m+i)}_{n-i,(\gamma h) \widetilde{{U}}(\gamma h)^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )}. \end{aligned}$$
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\).
Similarly suppose \(\widetilde{{U}}^p\) is an open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and that N is a non-negative integer. We define a split torus \(\widetilde{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)}/{\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\) as
$$\begin{aligned} \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }/\widetilde{{U}}^p(N)} {\mathcal {S}}^{(m+i)}_{n-i,(h \widetilde{{U}}^p h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))} \end{aligned}$$
for any \(N'\ge N\). Thus \(X_*(\widetilde{{S}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)})_{{\mathbb {Z}}_{(p)}}\) is a constant sheaf:
$$X_*\left( \widetilde{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)}\right) _{{\mathbb {Z}}_{(p)}} \cong {{\text {Herm}}}^{(m+i)}_{{\mathbb {Z}}_{(p)}} \cong Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {Z}}_{(p)}).$$
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}^p(N) \widetilde{{g}}\subset (\widetilde{{U}}^p)'(N')\), then we define
$$\begin{aligned} \widetilde{{g}}: \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+} \longrightarrow \widetilde{{{\mathcal {S}}}}_{n,(i),(\widetilde{{U}}^p)'(N')}^{(m),{{\text {ord}}},+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \widetilde{{g}}': {\mathcal {S}}^{(m+i)}_{n-i,(h \widetilde{{U}}^p h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))} \longrightarrow {\mathcal {S}}^{(m+i)}_{n-i,(h' (\widetilde{{U}}^p)' (h')^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))}, \end{aligned}$$
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )^{{\text {ord}}}\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{{\mathcal {S}}}}^{(m),+}_{n,(i),\widetilde{{U}}^p(N)} \}\) a system of relative tori with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action. If \(\gamma \) is an element of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then we define
$$\begin{aligned} \gamma : \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),+} \longrightarrow \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \gamma : {\mathcal {S}}^{(m+i)}_{n-i,(h \widetilde{{U}}^p h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))} \longrightarrow {\mathcal {S}}^{(m+i)}_{n-i,((\gamma h) \widetilde{{U}}^p (\gamma h)^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))}. \end{aligned}$$
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) on each \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).

The sheaves \(X^*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\) and \(X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\) have actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\). The sheaves \(X^*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\) and \(X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\) have actions of the group \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The systems of sheaves \(\{ X^*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}) \}\) and \(\{ X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}) \}\) (resp. \(\{ X^*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\}\) and \(\{X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\}\)) have actions of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)).

The sheaf
$$\left( X_*\left( \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\right) \cap {{\text {Herm}}}_{F^m}\right) =\coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )/\widetilde{{U}}} \left( X_*\left( \widetilde{{S}}^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )}\right) \cap {{\text {Herm}}}_{F^m}\right) $$
is a subsheaf of \(X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\). (Recall the embedding
$$\left. {{\text {Herm}}}^{(m)} \cong \ker \left( Z\left( \widetilde{{N}}^{(m)}_{n,(i)}\right) \rightarrow Z\left( N^{(m)}_{n,(i)}\right) \right) \subset {{\text {Herm}}}^{(i+m)}.\right) $$
It is invariant by the actions of the groups \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). We define a split torus
$$\begin{aligned} \widehat{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}/Y_{n,(i),\widetilde{{U}}}^{(m),+} \end{aligned}$$
by
$$X_*\left( \widehat{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\right) =X_* \left( \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\right) \cap {{\text {Herm}}}_{F^m}.$$
If U denote the image of \(\widetilde{{U}}\) in \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\), then we will write
$$\begin{aligned} S_{n,(i),U}^{(m),+}=\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\Big /\widehat{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}. \end{aligned}$$
It depends only on U and not on the choice of \(\widetilde{{U}}\) mapping onto U. The sheaf \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {Q}}\) is constant:
$$X_*\left( S_{n,(i),U}^{(m),+}\right) _{\mathbb {Q}}\cong Z \left( N_{n,(i)}^{(m)}\right) ({\mathbb {Q}}).$$
In the case \(m=0\) we will write simply \(S^+_{n,(i),U}\). The tori \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\) and \(S_{n,(i),\widetilde{{U}}}^{(m),+}\) inherit a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and a right action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). In the case of \(S_{n,(i),\widetilde{{U}}}^{(m),+}\) the latter factors through \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). If \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) with image U in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and image \(U'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\), then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant, commutative diagram:
$$\begin{array}{ccccc} \widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} &{}\twoheadrightarrow &{} S_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{} S_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ Y_{n,(i),\widetilde{{U}}}^{(m),+} &{}=&{} Y_{n,(i),U}^{(m),+} &{} \twoheadrightarrow &{} Y_{n,(i),U'}^+. \end{array}$$
Similarly the sheaf
$$\begin{array}{l} \left( X_*\left( \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\right) \cap {{\text {Herm}}}_{F^m}\right) \\ \quad =\coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/\widetilde{{U}}^p(N)} \left( X_*\left( \widetilde{{{\mathcal {S}}}}^{(m+i),{{\text {ord}}}}_{n-i,(h \widetilde{{U}}^p h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))}\right) \cap {{\text {Herm}}}_{F^m}\right) \end{array}$$
is a subsheaf of \(X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\). It is invariant by the actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). We define a split torus
$$\begin{aligned} \widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\Big /{\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+} \end{aligned}$$
by
$$X_*\left( \widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\right) =X_* \left( \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\right) \cap {{\text {Herm}}}_{F^m}.$$
If \(U^p\) denotes the image of \(\widetilde{{U}}^p\) in \(P_{n,(i)}^{(m)}({\mathbb {A}}^{\infty ,p})\), then we will write
$${\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}=\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+} \Big /\widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}.$$
It depends only on \(U^p\) and not the \(\widetilde{{U}}^p\) mapping to \(U^p\). The sheaf \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{{\mathbb {Z}}_{(p)}}\) is constant:
$$X_*\left( {\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\right) _{{\mathbb {Z}}_{(p)}} \cong Z\left( N_{n,(i)}^{(m)}\right) ({\mathbb {Z}}_{(p)}).$$
In the case \(m=0\) we will write simply \({\mathcal {S}}^+_{n,(i),U^p(N)}\). The tori \(\widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\) and \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) inherit a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and a right action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). In the case of \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) the latter factors through \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). If \(\widetilde{{U}}^p\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) with image \(U^p\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and image \((U^p)'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\), then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant, commutative diagram:
$$\begin{array}{ccccc} \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {S}}_{n,(i),(U^p)'(N)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ {\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+} &{}=&{} {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} &{} \twoheadrightarrow &{} {\mathcal {Y}}_{n,(i),(U^p)'(N)}^{{{\text {ord}}},+}. \end{array}$$
There are natural equivariant embeddings
$$\begin{aligned} \widetilde{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow \widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}^p(N)} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow S^{(m),+}_{n,(i),U^p(N)} \end{aligned}$$
and
$$\begin{aligned} \widehat{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),(U^p)'(N)} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow \widehat{{S}}^{(m),+}_{n,(i),(U^p)'(N)}. \end{aligned}$$
We write \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), resp. \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\)) for the subsheaves (of monoids) of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\) corresponding to the subset \({\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\) (resp. to the subset \({\mathfrak {C}}^{(m),> 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\), resp. to the subset \({\mathfrak {C}}^{(m),\ge 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\)).

We will also write \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0}\)) for the subsheaves (of monoids) of \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\) (resp. \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\), resp. \(X^*(S_{n,(i),U}^{(m),+})\), resp. \(X^*(S_{n,(i),U}^{(m),+})\)) consisting of sections that have non-negative (resp. strictly positive, resp. non-negative, resp. strictly positive) pairing with each nonzero section of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\). All these sheaves have (compatible) actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\). The system of sheaves \(\{ X_*(S_{n,(i),U}^{(m),+}) \}\) has an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\), and the same is true for all the other systems of sheaves we are considering in this paragraph.

We may take the quotients of the sheaves \(X^*(S_{n,(i),U}^{(m),+})\) (resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0}\)) by \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) to give sheaves of sets on \(Y_{n,(i),U}^{(m),\natural }\), which we will denote \(X^*(S_{n,(i),U}^{(m),+})^\natural \) (resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0,\natural }\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0,\natural }\)). If \(y=h U\) lies in \(Y^{(m),+}_{n,(i),U}\) above \(y^\natural \in Y^{(m),\natural }_{n,(i),U}\) then the stalk of \(X^*(S_{n,(i),U}^{(m),+})^\natural \) at \(y^\natural \) equals
$$\left\{ \gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}): \gamma y=y\right\} \Big \backslash X^*\left( S^{(m+i)}_{n-i,h U h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )}\right) .$$
We will write \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\), resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\)) for the subsheaves (of monoids) of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) corresponding to
$$\begin{aligned} {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)}) \subset Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {R}}) \end{aligned}$$
(resp.
$$\begin{aligned} {\mathfrak {C}}^{(m),> 0}(V_{n,(i)}) \subset Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {R}}), \end{aligned}$$
resp.
$$\begin{aligned} {\mathfrak {C}}^{(m),\ge 0}(V_{n,(i)}) \subset Z\left( N^{(m)}_{n,(i)}\right) ({\mathbb {R}})\Big ). \end{aligned}$$
Again we will write \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0}\)) for the subsheaves (of monoids) of \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\)) consisting of sections that have non-negative pairing with each section of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\). We will also write \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0}\)) for the subsheaves (of monoids) of the sheaves \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\)) consisting of sections that have strictly positive pairing with each nonzero section of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\). All these sheaves have actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The system of sheaves \(\{ X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}) \}\) has an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), and the same is true for all the other systems of sheaves we are considering in this paragraph.

We may take the quotients of the sheaves \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0}\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0}\) by \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) to give sheaves of sets on \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},\natural }\), which we will denote \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^\natural \) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0,\natural }\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0,\natural }\).

Suppose again that \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and set
$$\begin{aligned} \widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}= \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )/\widetilde{{U}}} T^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} . \end{aligned}$$
It is an \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\)-torsor over \(A_{n,(i),\widetilde{{U}}}^{(m),+}\). If U denotes the image of \(\widetilde{{U}}\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) then the push-out of \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\) under \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \twoheadrightarrow S_{n,(i),U}^{(m),+}\) is an \(S_{n,(i),U}^{(m),+}\)-torsor over \(A_{n,(i),U}^{(m),+}=A_{n,(i),\widetilde{{U}}}^{(m),+}\), which only depends on U (and not \(\widetilde{{U}}\)), and which we will denote \(T_{n,(i),U}^{(m),+}\). In the case \(m=0\) we will write simply \(T^+_{n,(i),U}\). Note that \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\) is an \(\widehat{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}\)-torsor over \(T_{n,(i),\widetilde{{U}}}^{(m),+}\).
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}\widetilde{{g}}\subset \widetilde{{U}}'\), then we define
$$\begin{aligned} \widetilde{{g}}: \widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+} \longrightarrow \widetilde{{T}}_{n,(i),\widetilde{{U}}'}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \widetilde{{g}}': \widetilde{{T}}^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} \longrightarrow \widetilde{{T}}^{(m+i)}_{n-i,h' \widetilde{{U}}' (h')^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} , \end{aligned}$$
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{T}}^{(m),+}_{n,(i),\widetilde{{U}}} \}\) a system of \(\{\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \} \)-torsors over \(\{ A_{n,(i),\widetilde{{U}}}^{(m),+}\} \) with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. It also induces an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) on \(\{ T^{(m),+}_{n,(i),U} \}\), which makes \(\{ T^{(m),+}_{n,(i),U} \}\) a system of \(\{S_{n,(i),U}^{(m),+} \} \)-torsors over \(\{ A_{n,(i),U}^{(m),+}\} \) with right \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
$$\begin{aligned} \gamma : \widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+} \longrightarrow \widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+} \end{aligned}$$
to be the coproduct of the maps
$$\begin{aligned} \gamma : T^{(m+i)}_{n-i,h \widetilde{{U}}h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} \longrightarrow T^{(m+i)}_{n-i,(\gamma h) \widetilde{{U}}(\gamma h)^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )} . \end{aligned}$$
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). It induces a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(T_{n,(i),U}^{(m),+}\), which commutes with the action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). Suppose that \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) with image U in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and image \(U'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\). Then there is a commutative diagram
$$\begin{array}{ccccc} \widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+} &{}\twoheadrightarrow &{} T_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{}T_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ A_{n,(i),U}^{(m),+} &{}=&{} A_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{} A_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ X_{n,(i),U}^{(m),+} &{}=&{} X_{n,(i),U}^{(m),+} &{}\twoheadrightarrow &{} X_{n,(i),U'}^+ \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ Y_{n,(i),U}^{(m),+} &{}=&{} Y_{n,(i),U}^{(m),+} &{} \twoheadrightarrow &{} Y_{n,(i),U'}^+. \end{array}$$
This diagram is \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant. We have
$$T_{n,(i),U}^{(m),+}({\mathbb {C}})= P^{(m)}_{n,(i)}({\mathbb {Q}})\Big \backslash \left( P^{(m),+}_{n,(i)}({\mathbb {A}}) Z\left( N_{n,(i)}^{(m)}\right) ({\mathbb {C}})\right) \Big / \left( U U_{n-i,\infty }^0 A_{n-i}({\mathbb {R}})^0\right) .$$
Similarly if \(\widetilde{{U}}^p\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and \(0 \le N_1 \le N_2\) we set
$$\begin{aligned} {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}= \coprod _{h \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }/\widetilde{{U}}^p(N_1,N_2)} {\mathcal {T}}^{(m+i),{{\text {ord}}}}_{n-i,(h \widetilde{{U}}^p h^{-1} \cap \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)} . \end{aligned}$$
It is an \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+}\)-torsor over \({\mathcal {A}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). If \(U^p\) denotes the image of \(\widetilde{{U}}^p\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) then the push-out of \({\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) under \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} \twoheadrightarrow {\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+}\) is an \({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+}\)-torsor over \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), which only depends on \(U^p\) (and not \(\widetilde{{U}}^p\)) and \(N_1,N_2\), and which we will denote \({\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). In the case \(m=0\) we will write simply \({\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). Note that \({\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) is a \(\widehat{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N_1)}\)-torsor over \({\mathcal {T}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\).
As above the system \(\{ {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\) has a right action of \(\widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a commuting left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). If \(g \in \widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then the map g is finite etale. The map
$$\begin{aligned} \varsigma _p: {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2-1)}^{(m),{{\text {ord}}},+} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$
equals absolute Frobenius composed with the forgetful map. If \(N_2>1\) then the map
$$\begin{aligned} \varsigma _p: {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \longrightarrow {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2-1)}^{(m),{{\text {ord}}},+} \end{aligned}$$
is finite flat. Further there is a left action of \(GL_m({\mathcal {O}}_{F,(p)})\) such that if \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in \widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\), then \(\gamma \) followed by \(\delta \) equals \(\delta \) followed by \(\delta \gamma \delta ^{-1}\), and g followed by \(\delta \) equals \(\delta \) followed by \(\delta g \delta ^{-1}\). These actions are also all compatible with the actions on \(\{ \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} \}\). There are induced actions of the groups \(GL_m({\mathcal {O}}_{F,(p)})\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) on \(\{ {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\), which are compatible with the actions on \(\{ {\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+} \}\). There is an equivariant commutative diagram
$$\begin{array}{ccccc} {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {T}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{}{\mathcal {T}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ {\mathcal {A}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}=&{} {\mathcal {A}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {A}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ {\mathcal {X}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}=&{} {\mathcal {X}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} &{}\twoheadrightarrow &{} {\mathcal {X}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{{{\text {ord}}},+} \\ \downarrow &{}&{} \downarrow &{}&{} \downarrow \\ {\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} &{}=&{} {\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} &{} \twoheadrightarrow &{} {\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N_1)}^{{{\text {ord}}},+}. \end{array}$$
There are natural equivariant embeddings
$$\begin{aligned} {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow \widetilde{{T}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),+} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \times {{\text {Spec}}\,}{\mathbb {Q}}\hookrightarrow T_{n,(i),U^p(N_1,N_2)}^{(m),+}. \end{aligned}$$
If a is a section of \(X^*(S^{(m),+}_{n,(i),U})\) over \(W\subset Y^{(m)}_{n,(i),U}\) then we can associate to it a line bundle
$$\begin{aligned} {\mathcal {L}}_{U}^+(a) \end{aligned}$$
over \(A^{(m),+}_{n,(i),U}|_W\) as in Sect. 2.1. There are natural isomorphisms
$$\begin{aligned} {\mathcal {L}}_{U}^+(a)\otimes {\mathcal {L}}_{U}^+(a') \cong {\mathcal {L}}_{U}^+(a+a'). \end{aligned}$$
Suppose that \(R_0\) is a noetherian \({\mathbb {Q}}\)-algebra. Suppose also that U is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). If a is a section in \(X^*(S^{+}_{n,(i),U})^{>0}(W)\) then \({\mathcal {L}}_{U}^+(a)\) is relatively ample for \(A^{+}_{n,(i),U}|_W/X^{+}_{n,(i),U}|_W\). If \(\pi ^+\) denotes the map
$$\begin{aligned} A^+_{n,(i),U}|_W \times {{\text {Spec}}\,}R_0 \longrightarrow X^+_{n,(i),U}\big |_W \times {{\text {Spec}}\,}R_0, \end{aligned}$$
then we see that
$$\begin{aligned} R^i\pi ^+_* {\mathcal {L}}_{U}^+(a)=(0) \end{aligned}$$
for \(i>0\). (Because \(A_{n,(i),U}^+|_W/X_{n,(i),U}^+|_W\) is a torsor for an abelian scheme and \({\mathcal {L}}_U^+(a)\) is relatively ample for this morphism.) We will denote by \((\pi _{A^+/X^+,*} {\mathcal {L}})^+_U(a)\) the image \(\pi ^+_* {\mathcal {L}}_U^+(a)\). Suppose further that \({\mathcal {F}}\) is a locally free sheaf on \(X^+_{n,(i),U} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action. If \(a^\natural \) is a section of \(X^*(S^{+}_{n,(i),U})^{>0,\natural }\) we will define
$$\begin{aligned} (\pi _{A^+/X^\natural ,*} {\mathcal {L}}\otimes {\mathcal {F}})^+_U(a^\natural ) \end{aligned}$$
as follows: Over a point \(y^\natural \) of \(Y_{n,(i),U}^\natural \) we take the sheaf
$$\begin{aligned} \prod _{y,a} (\pi _{A^+/X^+,*} {\mathcal {L}})^+_{U}(a)_y \otimes {\mathcal {F}}_y \end{aligned}$$
over \(X^\natural _{n,(i),U,y^\natural } \times {{\text {Spec}}\,}R_0\), where y runs over points of \(Y_{n,(i),U}^+ \) above \(y^\natural \) and a runs over sections of \(X^*(S_{n,(i),U}^+)_y\) above \(a^\natural \). It is a sheaf with an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).

Lemma 4.2

Keep the notation and assumptions of the previous paragraph.
  1. (1)
    $$\left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural ) \cong {{\text {Ind}}}_{\{1\}}^{L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})} \left( \left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural )^{L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})} \right) $$
    as a sheaf on \(X^{\natural }_{n,(i),U} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action.
     
  2. (2)
    If
    $$\pi : A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0 \longrightarrow X^{\natural }_{n,(i),U} \times {{\text {Spec}}\,}R_0$$
    then
    $$\begin{array}{l} R^i\pi _* \prod _{a \in X^*\left( S^{+}_{n,(i),U}\right) ^{>0}} \left( {\mathcal {L}}^+_{U}(a)\otimes {\mathcal {F}}\right) \\ \quad \cong \left\{ \begin{array}{ll} \prod _{a^\natural \in X^*(S^{+}_{n,(i),U})^{>0,\natural }} \left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural )&{} \quad \mathrm{if}\,\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise}. \end{array} \right. \end{array}$$
     

Proof

For the first part note that if y in \(Y^+_{n,(i),U}\) and if \(a \in X^*(S^+_{n,(i),U})^{>0}_y\) then the stabilizer of a in \(\{ \gamma \in L_{n,(i)}({\mathbb {Q}}): \gamma y=y\}\) is finite, and that if U is neat then it is trivial. The second part follows from the observations of the previous paragraph together with proposition 0.13.3.1 of [23]. \(\square \)

Similarly if a is a section of \(X^*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1)})\) over \(W \subset {\mathcal {Y}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1)}\) then we can associate to it a line bundle
$$\begin{aligned} {\mathcal {L}}_{U^p(N_1,N_2)}^+(a) \end{aligned}$$
over \({\mathcal {A}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\). There are natural isomorphisms
$$\begin{aligned} {\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\otimes {\mathcal {L}}_{U^p(N_1,N_2)}^+(a') \cong {\mathcal {L}}_{U^p(N_1,N_2)}^+(a+a'). \end{aligned}$$
Suppose that \(R_0\) is a noetherian \({\mathbb {Z}}_{(p)}\)-algebra. Suppose that \(U^p\) is a neat open compact subgroup of \(P^+_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and that \(0 \le N_1 \le N_2\). If a is a section in \(X^*({\mathcal {S}}^{+}_{n,(i),U^p(N_1)})^{>0}(W)\) then \({\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\) is relatively ample for \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\). If \(\pi ^+\) denotes the map
$$\begin{aligned} {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W \times {{\text {Spec}}\,}R_0 \longrightarrow {\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}|_W \times {{\text {Spec}}\,}R_0 \end{aligned}$$
then we see that
$$\begin{aligned} R^i\pi _*^+ {\mathcal {L}}_{U^p(N_1,N_2)}^+(a)=(0) \end{aligned}$$
for \(i>0\). (Again because \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}|_W/{\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}|_W\) is a torsor for an abelian scheme and \({\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\) is relatively ample for this morphism.) We will denote by \((\pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},+},*} {\mathcal {L}})^+_{U^p(N_1,N_2)}(a)\) the image \(\pi ^+_* {\mathcal {L}}^+(a)\). Suppose further that \({\mathcal {F}}\) is a locally free sheaf on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action. If \(a^\natural \) is a section of \(X^*({\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)})^{>0,\natural }\) we define a sheaf
$$\begin{aligned} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural ) \end{aligned}$$
as follows: Over a point \(y^\natural \) of \({\mathcal {Y}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural }\) we take the sheaf
$$\prod _{y,a} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},+},*} {\mathcal {L}}\right) ^+_{U^p(N_1,N_2)}(a)_y \otimes {\mathcal {F}}_y$$
over \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2),y^\natural } \times {{\text {Spec}}\,}R_0\), where y runs over points of \({\mathcal {Y}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\) above \(y^\natural \) and a runs over sections of \(X^*({\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)})_y\) above \(a^\natural \). It is a sheaf with an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). As above we have the following lemma.

Lemma 4.3

Keep the notation and assumptions of the previous paragraph.
  1. (1)
    $$\begin{array}{l} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural ) \\ \quad \cong {{\text {Ind}}}_{\{1\}}^{L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})} \left( \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},+},*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural )^{L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})} \right) \end{array}$$
    as a sheaf on \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action.
     
  2. (2)
    If
    $$\begin{aligned} \pi : {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\longrightarrow {\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0 \end{aligned}$$
    then
    $$R^i\pi _* \prod _{a \in X^*\left( {\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)}\right) ^{>0}} \left( {\mathcal {L}}^+_{U^p(N_1,N_2)}(a) \otimes {\mathcal {F}}\right) $$
    is isomorphic to
    $$\prod _{a^\natural \in X^*\left( {\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)}\right) ^{>0,\natural }} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural )$$
    if \(i=0\), and otherwise is (0).
     

5.4 Partial compactifications

We will now turn to the partial compactification of the generalized Shimura varieties, \(T^{(m)}_{n,(i),U}\), we discussed in the last section. These will serve as models for the full compactification of the \(A^{(m)}_{n,U}\), which near the boundary can be formally modelled on the partial compactifications of the \(T^{(m)}_{n,(i),U}\).

Suppose that U (resp. \(U^p\)) is a neat open compact subgroup of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^{\infty ,p})\)) and that N is a non-negative integer. By an admissible cone decomposition \(\varSigma _0\) for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\)) we shall mean a partial fan \(\varSigma _0\) in \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}\)) such that
  • \(|\varSigma _0|=X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\));

  • \(|\varSigma _0|^0=X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{> 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{> 0}\));

  • \(\varSigma _0\) is invariant under the left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\));

  • \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \backslash \varSigma _0\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \backslash \varSigma _0\)) is a finite set;

  • if \(\sigma \in \varSigma _0\) and \(1 \ne \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) then
    $$\begin{aligned} \sigma \cap \gamma \sigma \not \in \varSigma _0. \end{aligned}$$
(Many authors would not include the last condition in the definition of an ‘admissible cone decomposition’.) In concrete terms \(\varSigma _0\) consists of a partial fan \(\varSigma _{g,0}\) in \(Z(N_{n,(i)}^{(m)})({\mathbb {R}})\) for each \(g \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)), such that
  • \(\varSigma _{\gamma g u,0}=\gamma \varSigma _{g,0}\) for all \(\gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) and \(u \in U\) (resp. \(U^p(N)\));

  • \(|\varSigma _{g,0}|={\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\) and \(|\varSigma _{g,0}|^0={\mathfrak {C}}^{(m),>0}(V_{n,(i)})\) for each g;

  • \((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap gUg^{-1})\backslash \varSigma _{g,0}\) (resp. \((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \cap gU^p(N)g^{-1})\backslash \varSigma _g\)) is finite for all g;

  • for each g and each \(\sigma \in \varSigma _{g,0}\), if \(1 \ne \gamma \in (L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap gUg^{-1})\) (resp. \((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \cap gU^p(N)g^{-1})\)) then
    $$\begin{aligned} \sigma \cap \gamma \sigma \not \in \varSigma _{g,0}. \end{aligned}$$
Note that an admissible cone decomposition \(\varSigma _0\) for \(X_*(S^{(m),+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\) induces (by restriction) one, which we will denote \(\varSigma _0^{{\text {ord}}}\), for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\). This sets up a bijection between admissible cone decompositions for \(X_*(S^{(m),+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\) and for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\).

Lemma 4.4

Suppose that U (resp. \(U^p\)) is a neat open compact subgroup of the group \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\)) and that N is a non-negative integer. Suppose also that \(\varSigma _0\) is an admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\)). Also suppose that \(\tau \subset |\varSigma _0|\) is a rational polyhedral cone. Then the set
$$\begin{aligned} \left\{ \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}): \gamma \tau \cap \tau \cap |\varSigma _0|^0 \ne \emptyset \right\} \end{aligned}$$
(resp.
$$\begin{aligned} \left. \left\{ \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}): \gamma \tau \cap \tau \cap |\varSigma _0|^0 \ne \emptyset \right\} \right) \end{aligned}$$
is finite.

Proof

We treat the case of \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\), the other being exactly similar. Suppose that \(\tau \) has support \(y=hU\) and set \(\varGamma =L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap hUh^{-1}\) a discrete subgroup of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\). We certainly have
$$\left\{ \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}): \gamma \tau \cap \tau \cap |\varSigma _0|^0 \ne \emptyset \right\} =\left\{ \gamma \in \varGamma : \gamma \tau \cap \tau \cap |\varSigma _0|^0(y) \ne \emptyset \right\} .$$
That this set is finite follows from theorem II.4.6 and the remark (ii) at the end of section II.4.1 of [3]. \(\square \)

Corollary 4.5

If \(\varSigma _0\) is an admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) or for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\), then \(\varSigma _0\) is locally finite.

Proof

Let \(\tau \subset |\varSigma _0|\) be a rational polyhedral cone. Let \(\sigma _1,\ldots ,\sigma _r\) be representatives for \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\backslash \varSigma _0\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \backslash \varSigma _0\)); and suppose they are chosen with the same support as \(\tau \) whenever possible. Let \(\tau '\) be the rational polyhedral cone spanned by \(\tau \) and those \(\sigma _i\) with the same support as \(\tau \). If \(\gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) and
$$\begin{aligned} \gamma \sigma _i \cap \tau \cap |\varSigma _0|^0 \ne \emptyset , \end{aligned}$$
then
$$\begin{aligned} \gamma \tau ' \cap \tau ' \cap |\varSigma _0|^0 \ne \emptyset \end{aligned}$$
and so by the previous lemma \(\gamma \) lies in a finite set. The corollary follows. \(\square \)

If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\), if \(U' \supset g^{-1} U g\) are neat open compact subgroups of the group \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and if \(\varSigma '_0\) is a \(U'\)-admissible cone decomposition for \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\), then \(\varSigma '_0 g^{-1}\) is a U-admissible cone decomposition for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\). We will call a U-admissible cone decomposition \(\varSigma _0\) for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) compatible with \(\varSigma _0'\) with respect to g if \(\varSigma _0\) refines \(\varSigma '_0 g^{-1}\). Similarly if \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\), if \((U^p)' (N')\supset (g^{-1} U^p g)(N)\) and if \(\varSigma '_0\) is a \((U^p)'(N')\)-admissible cone decomposition for \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\), then \((\varSigma ' g^{-1},\varSigma '_0 g^{-1})\) is an admissible cone decomposition for \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\). We will call a \(U^p(N)\)-admissible cone decomposition \(\varSigma _0\) for \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) compatible with \(\varSigma _0'\) with respect to g if \(\varSigma _0\) refines \(\varSigma '_0 g^{-1}\).

If \(U'\) is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) which contains the image of U, we will call an admissible cone decomposition \(\varSigma _0\) of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and an admissible cone decomposition \(\varDelta _0\) of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) compatible if, under the natural map
$$X_*\left( S_{n,(i),U}^{(m),+}\right) _{\mathbb {R}}^{\succ 0} \twoheadrightarrow X_*\left( S_{n,(i),U'}^{+}\right) _{\mathbb {R}}^{\succ 0},$$
the image of each \(\sigma \in \varSigma _0\) is contained in some element of \(\varDelta _0\). Similarly if \((U^p)'\) is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\) which contains the image of \(U^p\) and if \(N' \ge N\), we will call an admissible cone decomposition \(\varSigma _0\) of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and an admissible cone decomposition \(\varDelta _0\) of \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N')}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) compatible if, under the natural map
$$X_*\left( {\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\right) _{\mathbb {R}}^{\succ 0} \twoheadrightarrow X_*\left( {\mathcal {S}}_{n,(i),(U^p)'(N')}^{{{\text {ord}}},+}\right) _{\mathbb {R}}^{\succ 0},$$
the image of each \(\sigma \in \varSigma _0\) is contained in some element of \(\varDelta _0\).
If \(\varSigma _0\) is a smooth admissible cone decomposition for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2))}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the log smooth, log scheme
$$\begin{aligned} \left( T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \end{aligned}$$
(resp. the log smooth, log scheme
$$\begin{aligned} \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \right) \end{aligned}$$
has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) extending that on \(T^{(m),+}_{n,(i),U}\) (resp. \({\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)). (Recall the definition of \(\widetilde{\varSigma }_0\) from Sect. 2.5.) If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g then the map
$$\begin{aligned} g: T^{(m),+}_{n,(i),U} \longrightarrow T^{(m),+}_{n,(i),U'} \end{aligned}$$
(resp.
$$\left. g: {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),(U^p)'(N_1',N_2')}\right) $$
uniquely extends to an \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) log etale map
$$g: \left( T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \longrightarrow \left( T^{(m),+}_{n,(i),U',\widetilde{\varSigma }_0'},{\mathcal {M}}_{\widetilde{\varSigma }_0'}\right) $$
(resp.
$$g: \left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \longrightarrow \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),(U^p)'(N'_1,N_2'),\widetilde{\varSigma }_0'},{\mathcal {M}}_{\widetilde{\varSigma }_0'}\right) \right) .$$
This makes \(\{ (T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}) \}\) (resp. \(\{ ( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0})\}\)) a system of log schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action). There are equivariant embeddings
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0^{{\text {ord}}}} \times {{\text {Spec}}\,}{\mathbb {Q}},{\mathcal {M}}_{\widetilde{\varSigma }_0^{{\text {ord}}}}\right) \hookrightarrow \left( T^{(m),+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) .$$
We have
$$\begin{array}{l} \left| {\mathcal {S}}\left( \partial T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0}\right) \right| -\left| {\mathcal {S}}\left( \partial T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0-\varSigma _0}\right) \right| \\ \quad =\left( L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty ) \times \left( C_{n-i}({\mathbb {Q}}) \backslash C_{n-i}({\mathbb {A}})/ C_{n-i}({\mathbb {R}})^0\right) \right) /U \times \left( {\mathfrak {C}}^{(m),> 0}_{(i)}\Big /{\mathbb {R}}^\times _{>0}\right) . \end{array}$$
If \(U'\) (resp. \((U')^p\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U')^p(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the map
$$\begin{aligned} T^{(m),+}_{n,(i),U} \longrightarrow T^{+}_{n,(i),U'} \end{aligned}$$
(resp.
$$\left. {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),(U')^p(N_1,N_2)}\right) $$
extends to an \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) log smooth map
$$\left( T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \longrightarrow \left( T^{+}_{n,(i),U',\widetilde{\varDelta }_0},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \longrightarrow \left. \left( {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),(U')^p(N_1,N_2),\widetilde{\varDelta }_0},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) \right) .$$
This gives rise to a \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) map of systems of log schemes
$$\left\{ \left( T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \right\} \longrightarrow \left\{ \left( T^{+}_{n,(i),U',\widetilde{\varDelta }_0},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) \right\} $$
(resp.
$$\left\{ \left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \right\} \longrightarrow \left. \left\{ \left( {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),(U')^p(N'_1,N_2'),\widetilde{\varDelta }_0},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) \right\} \right) .$$
These maps are compatible with the embeddings
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0} \times {{\text {Spec}}\,}{\mathbb {Q}},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) \hookrightarrow \left( T^{(m),+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}\right) $$
and
$$\left( {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varDelta }_0} \times {{\text {Spec}}\,}{\mathbb {Q}},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) \hookrightarrow \left( T^{+}_{n,(i),U^p(N_1,N_2),\widetilde{\varDelta }_0},{\mathcal {M}}_{\widetilde{\varDelta }_0}\right) .$$

5.5 Completions

If \(\varSigma _0\) denotes a smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. of \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2))}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the associated log formal scheme \((T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\) (resp. \(({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\)) inherits a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)). If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g, then there is an induced \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) map
$$g: \left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{(m),+,\wedge }_{n,(i),U',\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) $$
(resp.
$$g: \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}, {\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N'_1,N_2'),\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) \right) .$$
This makes \(\{ (T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\)) a system of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
Similarly the schemes \(\partial _{\varSigma _0} T^{(m),+}_{n,(i),U}\) (resp. \(\partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)) inherit a left action of the group \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)). If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g, then there is an induced \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) map
$$\begin{aligned} g: \partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \longrightarrow \partial _{\varSigma _0'} T^{(m),+}_{n,(i),U'} \end{aligned}$$
(resp.
$$\begin{aligned} \left. g: \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow \partial _{\varSigma _0'} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),(U^p)'(N_1',N_2')}\right) . \end{aligned}$$
This makes \(\{ \partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \}\) (resp. \(\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\)) a system of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
If \(U'\) (resp. \((U^p)'\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then there are induced maps
$$\left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{+,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) $$
and
$$\partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \longrightarrow \partial _{\varDelta _0} T^{+}_{n,(i),U'}$$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) $$
and
$$\left. \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \longrightarrow \partial _{\varDelta _0} {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),(U^p)'(N_1,N_2)} \right) ,$$
which are \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant). This gives rise to \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) maps of systems of log formal schemes
$$\left\{ \left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left( T^{+,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} $$
and of systems of schemes
$$\left\{ \partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \right\} \longrightarrow \left\{ \partial _{\varDelta _0} T^{+}_{n,(i),U'} \right\} $$
(resp.
$$\left\{ \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left( {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(U')^p(N'_1,N_2'),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} $$
and
$$\left\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\right\} \longrightarrow \left. \left\{ \partial _{\varDelta _0} {\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),(U^p)'(N_1,N_2)} \right\} \right) .$$
If \(\sigma \in \varSigma _0\) and if \(1 \ne \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) then \(\sigma \cap \gamma \sigma \not \in \varSigma _0\). Thus
$$\left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0}\right) _\sigma \cap \left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0}\right) _{\gamma \sigma }=\emptyset $$
and
$$\left( \partial _{\varSigma _0} T^{(m),+}_{n,(i),U}\right) _\sigma \cap \left( \partial _{\varSigma _0} T^{(m),+}_{n,(i),U}\right) _{\gamma \sigma }=\emptyset $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}\right) _\sigma \cap \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}\right) _{\gamma \sigma }=\emptyset $$
and
$$\left( \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\right) _\sigma \cap \left. \left( \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\right) _{\gamma \sigma }=\emptyset \right) .$$
It follows we can form log formal schemes
$$\left( T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) =L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \Big \backslash \left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) =L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \Big \backslash \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right) $$
and
$$\left( T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) ={{\text {Hom}}}_F(F^m,F^i) \Big \backslash \left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) ={{\text {Hom}}}_{{\mathcal {O}}_{F,(p)}}\left( {\mathcal {O}}_{F,(p)}^m,{\mathcal {O}}_{F,(p)}^i\right) \Big \backslash \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right) .$$
We can also form schemes
$$\partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U}=L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \Big \backslash \partial _{\varSigma _0} T^{(m),+}_{n,(i),U}$$
(resp.
$$\partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}=L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \Big \backslash \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \Big ).$$
The quotient maps
$$\left( T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \twoheadrightarrow \left( T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \twoheadrightarrow \left( T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) $$
and
$$\partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \twoheadrightarrow \partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U}$$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \!\twoheadrightarrow \! \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \twoheadrightarrow \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) $$
and
$$\left. \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \twoheadrightarrow \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \right) $$
are Zariski locally isomorphisms. The log formal scheme \((T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\) (resp. \(({\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\)) inherits an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)).
If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g then there are induced maps
$$g: \left( T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) $$
(resp.
$$g: \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}, {\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N'_1,N_2'),\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) \right) $$
and
$$g: \left( T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{(m),\natural +,\wedge }_{n,(i),U',\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) $$
(resp.
$$g: \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}, {\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left. \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),(U^p)'(N'_1,N_2'),\varSigma _0'},{\mathcal {M}}_{\varSigma _0'}^\wedge \right) \right) $$
and
$$g: \partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U} \longrightarrow \partial _{\varSigma _0'} T^{(m),\natural }_{n,(i),U'}$$
(resp.
$$\left. g: \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \longrightarrow \partial _{\varSigma _0'} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),(U^p)'(N_1',N_2')} \right) .$$
This makes the collections \(\{ (T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\}\)) and \(\{ (T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\)) systems of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action). It also makes the collections \(\{ \partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U} \}\) (resp. \(\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \}\)) systems of schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
If \(U'\) (resp. \((U^p)'\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then there are induced maps
$$\left( T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{\natural ,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left. \left( {\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right) $$
and
$$\left( T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left( T^{+,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) $$
(resp.
$$\left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \longrightarrow \left. \left( {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right) $$
and
$$\partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U} \longrightarrow \partial _{\varDelta _0} T^{\natural }_{n,(i),U'}$$
(resp.
$$\left. \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \longrightarrow \partial _{\varDelta _0} {\mathcal {T}}^{{{\text {ord}}},\natural }_{n,(i),(U^p)'(N_1,N_2)} \right) .$$
These give rise to \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) maps of systems of log formal schemes
$$\left\{ \left( T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left( T^{\natural ,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} $$
(resp.
$$\left\{ \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left. \left( {\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N'_1,N_2'),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} \right) $$
and
$$\left\{ \left( T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left( T^{+,\wedge }_{n,(i),U',\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} $$
(resp.
$$\left\{ \left( {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge \right) \right\} \longrightarrow \left\{ \left. \left( {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N'_1,N_2'),\varDelta _0},{\mathcal {M}}_{\varDelta _0}^\wedge \right) \right\} \right) .$$
They also give rise to a \(P^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m)}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) map of systems of schemes
$$\left\{ \partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U} \right\} \longrightarrow \left\{ \partial _{\varDelta _0} T^{\natural }_{n,(i),U'} \right\} $$
(resp.
$$\left. \left\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \right\} \longrightarrow \left\{ \partial _{\varDelta _0} {\mathcal {T}}^{{{\text {ord}}},\natural }_{n,(i),(U^p)'(N_1,N_2)}\right\} \right) .$$
We will write
$$\partial _{\varSigma _0} \overline{{T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N)}=\partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p.$$
It is independent of \(N_2\).
We also get a commutative diagram
$$\begin{array}{ccc} T^{(m),+,\wedge }_{n,(i),U,\varSigma _0} &{}&{} \\ \downarrow &{}&{} \\ T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0} &{} \longrightarrow &{} T^{+,\wedge }_{n,(i),U',\varDelta _0} \\ \downarrow &{}&{} \downarrow \\ T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0} &{} \longrightarrow &{} T^{\natural ,\wedge }_{n,(i),U',\varDelta _0} \\ \downarrow &{}&{} \downarrow \\ X^{(m),\natural }_{n,(i),U} &{}=&{} X^{\natural }_{n,(i),U'} \\ \downarrow &{}&{} \downarrow \\ Y^{(m),\natural }_{n,(i),U} &{}=&{} Y^{\natural }_{n,(i),U'} \end{array}$$
(resp.
$$\begin{array}{ccc} {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} &{}&{} \\ \downarrow &{}&{} \\ {\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} &{} \longrightarrow &{} {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} &{} \longrightarrow &{} {\mathcal {T}}^{\natural ,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {X}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} &{}=&{} {\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),(U')^p(N_1,N_2)} \\ \downarrow &{}&{} \downarrow \\ {\mathcal {Y}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} &{}=&{} {\mathcal {Y}}^{{{\text {ord}}},\natural }_{n,(i),(U')^p(N_1,N_2)}\Big ). \end{array}$$
We will let \({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote the formal completion of the ideal sheaf defining
$$\begin{aligned} \partial T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0} \subset T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0}. \end{aligned}$$
We will let \({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote its quotient by \({{\text {Hom}}}_F(F^m,F^i)\) and \({\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote its quotient by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\). Similarly we will let \({\mathcal {I}}^{(m),{{\text {ord}}},+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote the formal completion of the ideal sheaf defining
$$\begin{aligned} \partial {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0} \subset {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0}. \end{aligned}$$
We will let \({\mathcal {I}}^{(m),{{\text {ord}}},\natural +,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote its quotient by \({{\text {Hom}}}_{{\mathcal {O}}_{F,(p)}}({\mathcal {O}}_{F,(p)}^m,{\mathcal {O}}_{F,(p)}^i)\) and \({\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote its quotient by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\).
There are \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) equivariant maps
$$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0^{{\text {ord}}}} \times {{\text {Spf}}\,}{\mathbb {Q}}\hookrightarrow T^{(m),+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} , \end{aligned}$$
if \(\varSigma _0^{{\text {ord}}}\) and \(\varSigma _0\) correspond under the bijection of Sect. 4.4. These embeddings are compatible with the maps
$$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0^{{\text {ord}}}} \longrightarrow {\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0^{{\text {ord}}}} \end{aligned}$$
and
$$\begin{aligned} T^{(m),+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} \longrightarrow T^{+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}. \end{aligned}$$
Moreover they are also compatible with the log structures and with the sheaves \({\mathcal {I}}^{(m),{{\text {ord}}},+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0^{{\text {ord}}}}\) and \({\mathcal {I}}^{(m),+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\). They induce isomorphisms
$$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0^{{\text {ord}}}} \times {{\text {Spf}}\,}{\mathbb {Q}}\mathop {\longrightarrow }\limits ^{\sim }T^{(m),\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} . \end{aligned}$$

Lemma 4.6

Suppose that \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology. Suppose also that \(U \supset U'\) (resp. \(U^p \supset (U^p)'\)) are neat open compact subgroups of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2' \ge N_1' \ge 0\) and \(N_2 \ge N_1 \ge 0\) are integers with \(N_2' \ge N_2\) and \(N_1' \ge N_1\) and that \(\varSigma _0\) and \(\varSigma '_0\) are compatible smooth admissible cone decompositions for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)). Let \(\pi _{(U',\varSigma _0'),(U,\varSigma _0)}\) (resp. \(\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0)}\)) denote the map
$$\begin{aligned} 1_*: T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \rightarrow T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0} \end{aligned}$$
(resp.
$$\begin{aligned} \left. 1_*: {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1',N_2'),\varSigma _0'} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}\right) . \end{aligned}$$
  1. (1)
    If \(i>0\) then
    $$R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U',\varSigma _0'} {\widehat{\otimes }}R_0\right) = R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {O}}_{ T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0}=(0)$$
    (resp.
    $$R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),(U^p)'(N_1',N_2') ,\varSigma _0'} {\widehat{\otimes }}R_0\right) =(0)$$
    and
    $$\begin{aligned} R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {O}}_{ {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1',N_2'),\varSigma _0'} \times {{\text {Spf}}\,}R_0}=(0) \big ). \end{aligned}$$
     
  2. (2)
    Suppose further that \(U'\) (resp. \((U^p)'\)) is a normal subgroup of U (resp. \(U^p\)) and that \(\varSigma _0'\) is U-invariant (resp. \(U^p(N_1,N_2)\)-invariant). Then the natural maps
    $${\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0} \times {{\text {Spf}}\,}R_0} \longrightarrow \left( \pi _{(U',\varSigma '_0),(U,\varSigma _0),*} {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0}\right) ^{U}$$
    (resp.
    $$\begin{array}{l} {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} \times {{\text {Spf}}\,}R_0} \\ \quad \left. \longrightarrow \left( \pi _{(((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_2,N_2),\varSigma _0'} \times {{\text {Spf}}\,}R_0}\right) ^{U^p(N_1,N_2)} \right) \end{array}$$
    and
    $${\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0}{\widehat{\otimes }}R_0 \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U',\varSigma _0'} {\widehat{\otimes }}R_0\right) \right) ^{U}$$
    (resp.
    $$\begin{array}{l} {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2), \varSigma _0} {\widehat{\otimes }}R_0 \\ \left. \quad \longrightarrow \left( \pi _{((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_2,N_2),\varSigma _0'}{\widehat{\otimes }}R_0\right) \right) ^{U^p(N_1,N_2)} \right) \end{array}$$
    are isomorphisms.
     
The same statements are true with \(\natural \) replaced by \(+\) or by \(\natural +\).

Proof

It suffices to treat the case of \(+\). We treat the case of \(T^{(m),+,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0\), the case of \({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varSigma _0'} \times {{\text {Spf}}\,}R_0\) being exactly similar.

Let \(U''\) denote the open compact subgroup of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) generated by \(U'\) and \(U \cap Z(N_{n,(i)}^{(m)})({\mathbb {A}}^\infty )\). Then \(\varSigma _0\) is a \(U''\) admissible smooth cone decomposition of \(X_*(S^{(m),+}_{n,(i),U''})_{\mathbb {R}}^{\succ 0}\). Moreover
$$\begin{aligned} T^{(m),+,\wedge }_{n,(i),U'',\widetilde{\varSigma }_0}\times {{\text {Spf}}\,}R_0 \longrightarrow T^{(m),+,\wedge }_{n,(i),U,\widetilde{\varSigma }_0} \times {{\text {Spf}}\,}R_0 \end{aligned}$$
is finite etale, and if \(U'\) is normal in U then it is Galois with group \(U/U''\). Thus we may replace U by \(U''\) and reduce to the case that U and \(U'\) have the same projection to \((P^{(m),+}_{n,(i)}/Z (N_{n,(i)}^{(m)}))({\mathbb {A}}^\infty )\). In this case the result follows from Lemma 2.15. \(\square \)

Define \(\widetilde{\varOmega }_{n,(i),U,\varDelta _0}^\natural \) on \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) of the pull-back of \(\widetilde{\varOmega }_{n,(i),U}^+\) from \(A^+_{n,(i),U}\) to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\). Also define \(\widetilde{\varOmega }_{n,(i),U^p(N_1,N_2),\varDelta _0}^{{{\text {ord}}},\natural }\) on \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) of the pull-back of the sheaf \(\widetilde{\varOmega }_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) from \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\).

Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Then we define a locally free sheaf \({\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \) on \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) of the pull-back of \({\mathcal {E}}^+_{(i),U,\rho }\) from \(A^+_{n,(i),U}\) to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\). Then the system of sheaves \(\{ {\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \}\) over \(\{ T^{\natural ,\wedge }_{n,(i),U,\varDelta _0} \}\) has an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )\), then the natural map
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \longrightarrow {\mathcal {E}}_{(i),U',\varDelta '_0,\rho }^\natural \end{aligned}$$
is an isomorphism. The sheaves \({\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \) have \(P_{n,(i)}^+({\mathbb {A}}^\infty )\)-invariant filtrations by local direct summands whose graded pieces pull-backed to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\) are equivariantly isomorphic to the pull-backs of sheaves of the form \({\mathcal {E}}_{(i),U,\rho '}^+\) on \(X^+_{n,(i),U}\).
Similarly in the case of mixed characteristic suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Then we define a locally free sheaf \({\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\) on \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) of the pull-back of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) from \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\). Then the collection \(\{ {\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\}\) is a system of sheaves over \(\{ {\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \}\) with an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), then the natural map
$$\begin{aligned} g^*{\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural } \longrightarrow {\mathcal {E}}_{(i),(U^p)'(N_1,N_2),\varDelta '_0,\rho }^{{{\text {ord}}},\natural } \end{aligned}$$
is an isomorphism. The sheaves \({\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\) have \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-invariant filtrations by local direct summands whose graded pieces pull-backed to the formal scheme \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) are equivariantly isomorphic to the pull-backs of sheaves of the form \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho '}\) on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\).

Corollary 4.7

Suppose that \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology. Let \(\rho \) be a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Suppose also that \(U \supset U'\) (resp. \(U^p \supset (U^p)'\)) are neat open compact subgroups of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2' \ge N_1' \ge 0\) and \(N_2 \ge N_1 \ge 0\) are integers with \(N_2' \ge N_2\) and \(N_1' \ge N_1\) and that \(\varSigma _0\) and \(\varSigma '_0\) are compatible smooth admissible cone decompositions for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)).   Let \(\pi _{(U',\varSigma _0'),(U,\varSigma _0)}\) (resp. \(\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0)}\)) denote the map
$$\begin{aligned} 1_*: T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \rightarrow T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0} \end{aligned}$$
(resp.
$$\begin{aligned} \left. 1_*: {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1',N_2'),\varSigma _0'} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}\right) . \end{aligned}$$
  1. (1)
    If \(i>0\) then
    $$R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U',\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural \right) = R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural =(0)$$
    (resp.
    $$R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),(U^p)'(N_1',N_2') ,\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) =(0)$$
    and
    $$\begin{aligned} R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\varSigma _0',\rho }^{{{\text {ord}}},\natural }=(0) \big ). \end{aligned}$$
     
  2. (2)
    Suppose further that \(U'\) (resp. \((U^p)'\)) is a normal subgroup of U (resp. \(U^p\)) and that \(\varSigma _0'\) is U-invariant (resp. \(U^p(N_1,N_2)\)-invariant). Then the natural maps
    $${\mathcal {E}}_{(i),U,\varSigma _0,\rho }^\natural \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural \right) ^{U}$$
    (resp.
    $$\begin{array}{l} {\mathcal {E}}_{(i),U^p(N_1,N_2),\varSigma _0,\rho }^{{{\text {ord}}},\natural } \\ \quad \longrightarrow \left. \left( \pi _{(((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {E}}_{(i),(U^p)'(N_2,N_2),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) ^{U^p(N_1,N_2)} \right) \end{array}$$
    and
    $${\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0}{\widehat{\otimes }}{\mathcal {E}}_{(i),U,\varSigma _0,\rho }^\natural \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} ({\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U',\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),U',\varSigma '_0,\rho }^\natural ) \right) ^{U}$$
    (resp.
    $$\begin{array}{l} {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2), \varSigma _0} {\widehat{\otimes }}{\mathcal {E}}_{(i),U^p(N_1,N_2),\varSigma _0,\rho }^{{{\text {ord}}},\natural } \longrightarrow \\ \quad \left. \left( \pi _{((U^p)'(N_2,N_2),\varSigma '),(U^p(N_1,N_2),\varSigma ),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_2,N_2),\varSigma _0'}{\widehat{\otimes }}{\mathcal {E}}_{(i),(U^p)'(N_2,N_2),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) \right) ^{U^p(N_1,N_2)} \right) \end{array}$$
    are isomorphisms.
     

Lemma 4.8

Suppose that U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and let \(U'\) denote the image of U in \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). Let \(\varDelta _0\) be a smooth admissible cone decomposition for \(X_*(S^+_{n,(i),U'})\) and let \(\varSigma _0\) be a compatible smooth admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})\). Let \(\pi ^+=\pi _{(U,\varSigma _0),(U',\varDelta _0)}^+\) denote the map
$$\begin{aligned} T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0} \longrightarrow T^{+,\wedge }_{n,(i),U',\varDelta _0} \end{aligned}$$
and let \(\pi ^\natural =\pi _{(U,\varSigma _0),(U',\varDelta _0)}^\natural \) denote the map