The adic, cuspidal, Hilbert eigenvarieties

Remark 1.1

In [2] and also in this paper, we work with toroidal compactifications \(\overline{M}(\mu _N, \mathfrak {c})\) of the integral models of the Shimura varieties associated with \(G^*\) defined by Deligne and Pappas [9], completing previous work of Rapoport [16]. These models are singular at primes dividing the discriminant of F. One could use one of the splitting models \(\widetilde{\overline{M}}(\mu _N, \mathfrak {c}) \rightarrow \overline{M}(\mu _N, \mathfrak {c})\) of \(\overline{M}(\mu _N, \mathfrak {c})\) introduced by Pappas and Rapoport [15]. Such models depend on some auxiliary choices, namely an ordering of the embeddings of F in an algebraic closure, but they have the advantage of being smooth. The given map is an isomorphism over an open dense subscheme (the Rapoport locus, see Sect. 3); in particular, the two moduli spaces differ only at primes p dividing the discriminant of F.

Recall that the complement of the Rapoport locus is characterized by the fact that the sheaf \(\omega _G\) of invariant differentials of the versal semi-abelian scheme G over \(\overline{M}(\mu _N, \mathfrak {c})\) is not locally free as \(\mathcal {O}_F\otimes \mathcal {O}_{{\overline{M}}(\mu _N, \mathfrak {c})}\)-module. On the other hand on a splitting model, \(\omega _G\) admits a filtration by invertible \(\mathcal {O}_{\widetilde{\overline{M}}(\mu _N, \mathfrak {c})}\)-modules, stable for the action of \(\mathcal {O}_F\). These invertible sheaves allow to define Hilbert modular forms of non-parallel weight over the whole of \(\widetilde{\overline{M}}(\mu _N, \mathfrak {c})\).

For our purposes, i.e., the construction of modular sheaves, it makes no difference which model we choose and we prefer to work with the minimal (and more canonical) one, the Deligne–Pappas model. The main reason is the fact that the key ingredient in the construction of the modular sheaves is the introduction of a different integral structure \(\mathcal {F}\) of \(\omega _G\) which is locally free as \(\mathcal {O}_F\otimes \mathcal {O}_{\mathfrak {IG}_{n,r,I}}\)-module, e.g., even on the complement of the Rapoport locus in the formal scheme \(\mathfrak {IG}_{n,r,I}\) (see [2], Proposition 4.1, or Sect. 4.1 of the present article).

Remark 2.1

In [2], the weight space has been defined over the ring of integers of a finite extension K of \(\mathbb {Q}_p\) splitting F. The reason is that the classical weights are defined over K. Here we prefer to work over \(\mathbb {Z}_p\). As a consequence, it will turn out that the characteristic series of the \(U_p\) operator will have coefficients in the Iwasawa algebra \(\varLambda _F^G\) defined in Theorem 8.8, with no need to extend scalars.

Lemma 2.2

The morphism \(t:\widetilde{\mathcal {W}}_F\rightarrow \mathcal {W}_F\) is an isomorphism of adic spaces.

Proof

For all \(\alpha \in \mathfrak {m}\), the subset \(\{ x \in \mathcal {W}_F, 0 \ne \vert \alpha \vert _x \ge \vert \beta \vert _x, \forall \beta \in \mathfrak {m}\}\) of \(\mathcal {W}_F\) equals \(\mathcal {W}_\alpha \) by definition. Moreover, \(\mathcal {W}_F\) is covered by the \(\mathcal {W}_\alpha \). The conclusion follows. \(\square \)

Remark 2.3

Definition 3.1

We also denote by \(\mathfrak {Y}_{{r}}^\mathrm{R} \subset \mathfrak {Y}_{{r}}\) the open formal subscheme where the Rapoport condition holds (see Sect. 3).

Proposition 3.2

Assume that \(p\in \alpha ^{p^k} A_0\). Then for every integer \(1\le n \le r +k\) one has a canonical subgroup scheme \(H_{n}\) of \(G[p^n]\) over \(\mathfrak {Y}_{{r}}\) and \(H_{n}\) modulo \(p \mathrm{Hdg}^{-\frac{p^n-1}{p-1}}\) lifts the kernel of the nth power of Frobenius. Moreover, \(H_{n}\) is finite flat and locally of rank \(p^{ng}\), it is stable under the action of \(\mathcal {O}_F\), and the Cartier dual \(H_{n}^D\) is étale locally over \(A_0[\alpha ^{-1}]\) isomorphic to \(\mathcal {O}_F/p^n\) (as \(\mathcal {O}_F\)-module).

Proof

All claims follow from [3, Appendix A]. \(\square \)

Proposition 3.3

For every \({r}\in \mathbb {Z}_{\ge 2}\), the isogeny given by dividing by the canonical subgroup \(H_{1}\) of level 1 defines a finite morphism \( \phi :\mathfrak {Y}_{{r}} \rightarrow \mathfrak {Y}_{r-1}\). The restriction to the Rapoport locus \( \phi :\mathfrak {Y}_{{r}}^\mathrm{R} \rightarrow \mathfrak {Y}_{r-1}^\mathrm{R}\) is finite and flat of degree \(p^{g}\).

Proof

This is the content of [3, Cor. A.2] which is written for general p-divisible groups. The last claim follows as relative Frobenius is finite and it is flat over the (smooth) Rapoport locus. \(\square \)

Proposition 4.1

Proof

This is a variant of [2, Prop. 3.4]. Let \(U:=\hbox {Spf}~R\subset \mathfrak {IG}_{n,r,I}\) be an open formal affine subscheme such that \(\omega _G\vert _U\) is free of rank g as an R-module. Write \(\overline{M}\in \mathrm{M}_{n\times n}(R{/}p^{n}\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}}R)\) for the matrix of the linearization of the map \(\mathrm {HT}\) over U. Thanks to [3, prop. A.3] it has determinant ideal equal to \(\mathrm {Hdg}^{\frac{1}{p-1}} \). In particular, \(\mathrm {Hdg}^{\frac{1}{p-1}} \omega _{G}{/}p^{n}\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}}\omega _{G}\) lies in the span of \(\mathrm {HT}(P)\).

Let \(M\in \mathrm{M}_{n\times n}(R)\) be any lift of \(\overline{M}\). Its determinant \(\delta \) is \(\mathrm {Hdg}^{\frac{1}{p-1}} \) (up to unit). Let \(\mathcal {S}\subset \omega _G\vert _U\) be the submodule spanned by the columns of M. Then \(\delta \omega _{G}\subset \mathcal {S}\). Since \(p^{n}\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}}\omega _{G}=p^{n}\mathrm {Hdg}^{-\frac{p^{n}}{p-1}} \cdot \delta \omega _{G}\subset \mathcal {S}\) one deduces that \(\mathcal {F}\vert _U\) coincides with the \(\mathcal {S}\). In particular, it is a free R-module of rank g. By definition it is stable for the action of \(\mathcal {O}_F\).

For every \(x\in \mathcal {O}_F/p^n\mathcal {O}_F\), the image \(y\in \mathrm {HT}(\psi _n(x))\) lies by construction in the image of \(\mathcal {S}\) in \(\omega _{G}/p^{n}\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}}\omega _{G}\). As \(\mathrm {Hdg}^{-\frac{p^{n}}{p-1}}=\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}} \cdot \mathrm {Hdg}^{\frac{1}{p-1}}\) and as \(\omega _G{/}\mathcal {F}\) is annihilated by \(\mathrm {Hdg}^{\frac{1}{p-1}} \), it follows that any two lifts \(y^{\prime }\) and \(y^{\prime \prime }\) in \(\mathcal {S}\) differ by an element lying in \(\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}} \cdot \mathrm {Hdg}^{\frac{1}{p-1}} \omega _G=\mathrm {Hdg}^{-\frac{p^{n}-1}{p-1}} \cdot \mathcal {S}\). We then get a well-defined map \(\mathcal {O}_F{/}p^n\mathcal {O}_F\rightarrow \mathcal {F}{/}p^{n}\mathrm {Hdg}^{-\frac{p^{n}}{p-1}}\mathcal {F}\) inducing \(\mathrm {HT} \circ \psi _n\) when composed with the projection to \(\omega _G/p^{n}\mathrm {Hdg}^{-\frac{p^{n}}{p-1}}\mathcal {F}\omega _G\). This provides the \(\mathrm {HT}^{\prime }\). By construction, its restriction to U is a surjective map of free \(R{/}p^{n}\mathrm {Hdg}^{-\frac{p^{n}}{p-1}}R\)-modules of rank g and hence it is an isomorphism. It follows that \(\mathcal {S}=\mathcal {F}\vert _U\) is a free \(\mathcal {O}_F\otimes R\)-module of rank 1 concluding the proof of the proposition. \(\square \)

Lemma 4.2

We have \(p\mathrm {Hdg}^{-p^{n}} \subset \mathcal {O}_{\mathfrak {IG}_{n,r,I}}\). In particular, \(p^{n}\mathrm {Hdg}^{-\frac{p^{n}}{p-1}} \subset p^{k^{\prime }+1} \mathcal {O}_{\mathfrak {IG}_{n,r,I}}\) (resp. \(\subset p^{k^{\prime }+3} \mathcal {O}_{\mathfrak {IG}_{n,r,I}}\) if \(p=2\)).

Proof

The claim is local on \(\mathfrak {IG}_{n,r,I}\). Let \(\alpha \in \mathfrak {m}{\setminus } \mathfrak {m}^2\). We prove the claim over an open \(U=\hbox {Spf}~R\subset \mathfrak {IG}_{n,r, \alpha , I}\) over the open \(\mathfrak {W}^0_{\alpha ,I}=\hbox {Spf}~B^0_{\alpha ,I}\) of \(\widetilde{\mathfrak {W}}^0_F\). By construction, we have \(p/\alpha ^{p^k}\in B^0_{\alpha ,I}\) and \(\alpha \mathrm {Hdg}^{-p^{r+1}} \subset R\). Hence, \(p\mathrm {Hdg}^{-p^{r+k+1}} \subset R\). In particular, \(p \mathrm {Hdg}^{-p^n} \subset R\) and the second claim follows. \(\square \)

Proposition 4.3

The sheaf \(\mathfrak {w}_{n,r,I}\) is an invertible \(\mathcal {O}_{\mathfrak {X}_{r,I}}\)-module of rank 1.

Lemma 4.4

Proof

We deal with the case \(p\ne 2\) leaving to the reader the case \(p=2\). The claim is local on \(\mathfrak {X}_{r,I}\). We restrict ourselves to an open formal affine subscheme \(U=\hbox {Spf}~R\) mapping to the open \(\mathfrak {W}_\alpha ^0\) of \(\widetilde{\mathfrak {W}}^0_F\) defined by an element \(\alpha \in \mathfrak {m}{\setminus } \mathfrak {m}^2\). By construction, \(\kappa ((\mathcal {O}_F \otimes \mathbb {Z}_p)^*)-1 \subset \alpha B_{\alpha }^0\) and since \(\alpha \mathrm {Hdg}^{-1} \subset (\mathcal {O}_{\mathfrak {X}_{r,I}})^{00}\), we can conclude that the first point holds. Using Lemma 2.5, we see that for \(\ell \ge 2\), we have that \(\kappa (1+p^{\ell -1}\mathcal {O}_F\otimes \mathbb {Z}_p)-1 \subset (\alpha ^{p^{\ell -2}},p) B_\alpha ^0\). Arguing as in Lemma 4.2 we deduce from the assumption that \(\ell \le r+k\) that \({p}{{\mathrm {Hdg}}^{-p^\ell }}\in (\mathcal {O}_{\mathfrak {X}_{r,I}})^{00}\). On the other hand as \(r\ge 1\), then \(\alpha \in \mathrm {Hdg}^{p^2} \mathcal {O}_{\mathfrak {X}_{r,I}}\) so that \(\alpha ^{p^{\ell -2}} \in \mathrm {Hdg}^{p^{\ell }} \mathcal {O}_{\mathfrak {X}_{r,I}}\). As \(\frac{p^\ell -1}{p-1} < p^\ell \), it follows that \(\alpha ^{p^{\ell -2}} {{\mathrm {Hdg}}^{-\frac{p^\ell -1}{p-1}}} \subset (\mathcal {O}_{\mathfrak {X}_{r,I}})^{00}\). \(\square \)

Lemma 4.5

Proof

Consider an open formal affine subscheme \(U=\hbox {Spf}~R\subset \mathfrak {IG}_{n,r,I}\) mapping to the open formal subscheme \(\mathfrak {W}_\alpha ^0\) of \(\widetilde{\mathfrak {W}}^0_F\) defined by some \(\alpha \in \mathfrak {m}{\setminus }\mathfrak {m}^2\). Assume that \(\omega _G\vert _U\) is free. The choice of an element \(\tilde{s}\in \mathcal {F}\vert _U\) lifting \(s:=\mathrm {HT}^{\prime }(1)\) defines a section of the morphism \(\mathfrak {F}_{n,r,I}\vert _U \cong \mathfrak {IG}_{n,r,I}\vert _U\) and hence an isomorphism \(f_{\tilde{s}}:\mathfrak {w}_{n,r,I}^1\vert _U \rightarrow \mathcal {O}_{\mathfrak {IG}_{n,r,I}}\vert _U\) given by evaluating the functions at \(\tilde{s}\).

Two different lifts \(\tilde{s}\) and \(\tilde{s^{\prime }}\) differ by an element of \(1 + p^{n}\mathrm {Hdg}^{-\frac{p^n}{p-1}} \mathcal {O}_F\otimes R\) thanks to Proposition 4.1. Proposition 4.2 implies that \(1+p^{n}\mathrm {Hdg}^{-\frac{p^n}{p-1}}\mathcal {O}_F \otimes R \subset 1+p^{k^{\prime }+1 +n^{\prime }}\mathcal {O}_F \otimes R \) (resp. \(1+p^{k^{\prime }+3 +n^{\prime }}\mathcal {O}_F \otimes R \) if \(p=2\)). As \(I=[p^k,p^{k^{\prime }}]\) we conclude that \(\kappa (1+p^{k^{\prime }+1 +n^{\prime }}\mathcal {O}_F \otimes R) \subset 1 + p^{n^{\prime }+1} R\) (and similarly \(\kappa (1+p^{k^{\prime }+3+n^{\prime }}\mathcal {O}_F \otimes R) \subset 1 + q p^{n^{\prime }} R\) for \(p=2\)). Thus \(f_{\tilde{s}}\equiv f_{\tilde{s^{\prime }}}\) modulo \(qp^{n^{\prime }}\). This provides the inverse to the isomorphism in the lemma. \(\square \)

Lemma 4.6

Let \(s \in g_{n,*} \mathfrak {w}^1_{n,r,I}(R)\) be an element such that \(s \equiv 1 \;\mod \;p\) (in the sense of Lemma 4.5). Then \(e_{c_n}(s) \in \mathfrak {w}_{n,r,I}(R)\) and \(\mathfrak {w}_{n,r,I}(R)\) is the free R-module generated by \(e_{c_n}(s)\).

Proof

Proposition 5.1

We have \(\mathrm {Hdg}_s \mathcal {O}_{\mathfrak {IG}_{n-1,\infty ,I}} \subset \mathrm {Tr}_{\mathfrak {IG}} \bigl (\mathcal {O}_{\mathfrak {IG}_{n,\infty ,I}})\) for every \(s\ge 1\).

Proof

Thanks to Proposition 3.4, we have \(\mathrm {Hdg}^{p^{n-1}}_s \mathcal {O}_{\mathfrak {IG}_{n-1,s,I}}\subset \mathrm {Tr}_{\mathfrak {IG}}(h_*\mathcal {O}_{ \mathfrak {IG}_{n,s,I}})\). It follows from [3, Cor. A.2] that \(\mathrm {Hdg}_{s+1}^p=\mathrm {Hdg}_s\). Since \(\mathrm {Hdg}^{p^{n-1}}_s \mathcal {O}_{\mathfrak {IG}_{n-1,\infty ,I }} \subset \mathrm {Tr}_{\mathfrak {IG}} (\mathcal {O}_{\mathfrak {IG}_{n,\infty ,I}})\) and s is arbitrary, the claim follows. \(\square \)

Proposition 5.2

Proof

The proof follows closely the proof of [3, Proposition 6.2]. We first provide the analog of [3, §6.3.2 and §6.3.3] which reduces the proof to [3, Lemme 6.1].

Lemma 5.4

Proof

(1) Let \(U:=\hbox {Spf}~R\) be an open formal subscheme of \(\mathfrak {X}_{\infty ,\alpha ,I}\). For s large enough, it is the inverse image of an open formal subscheme \(\hbox {Spf}~R_s\) of \(\mathfrak {X}_{s,\alpha ,I}\). Let \(f \in R[1/\alpha ]\). For s large enough, we have \(f = f_s + h\) with \(f_s \in R_s[1/\alpha ]\) and \(h \in R\) so that we may assume that \(f \in R_s[1/\alpha ]\). Let \(f^n + f^{n-1} a_{n-1} + \cdots + a_0=0\) with \(a_0\), \(\ldots \), \(a_{n-1}\in R\) be an integral relation.

Applying \(\mathrm {Tr}_{s}\) one gets \(f^n + f^{n-1} \mathrm {Tr}_{h}( a_{n-1}) + \cdots + \mathrm {Tr}_{h}(a_0)=0\) and as \(a_i \in R\) for s large enough we have \(\mathrm {Tr}_{s}(a_i) \in R \cap \alpha ^{-1} R_{s}\). For \(h\ge 0\) the morphism \(R_{s} \rightarrow R_{s+h}\) is a finite dominant morphism of normal rings. Hence \(R_{s}/\alpha \rightarrow R_{s+h}/\alpha \) is injective so that \(R_{s}/\alpha \rightarrow R/\alpha \) is injective as well. We deduce that \(\alpha \mathrm {Tr}_{s}(a_i) \in \alpha R_{s}\) and hence that \(\mathrm {Tr}_{s}(a_i) \in R_{s}\). Thus f is integral over \(R_{s}\) and, hence, \(f \in R_{s}\) proving the first claim of the lemma.

(2) The inverse image of \(\hbox {Spf}~R\) in \({\mathfrak {IG}_{n,\infty ,\alpha ,I}}\) is equal to \(\hbox {Spf}~R_n\) with \(R_n\) integral over R and \(R[\alpha ^{-1}] \subset R_n[\alpha ^{-1}]\) finite and étale. In particular, \((R_n [\alpha ^{-1}])^{(\mathcal {O}_F/p^n\mathcal {O}_F)^*}=R[\alpha ^{-1}]\) so that \(R_n^{(\mathcal {O}_F/p^n\mathcal {O}_F)^*}\) contains R and is integral over R and, hence by the first claim, it must be equal to R. Let \(R_\infty \) be the inverse image of \(\hbox {Spf}~R\) in \({\mathfrak {IG}_{\infty ,\infty ,\alpha ,I}}\). Consider an element \(x\in R_\infty \) fixed by \((\mathcal {O}_F\otimes \mathbb {Z}_p)^*\). There exists n large enough and \(x_n\in R_n\) such that \(x-x_n=\alpha x^{\prime }\) for some \(x^{\prime }\in R_\infty \). In particular, \(x^{\prime }\) is fixed by \(1+p^n \mathcal {O}_F\otimes \mathbb {Z}_p\). Thanks to Proposition 5.1 for every \(n' \ge n\) there exists an element \(c_{n'} \in R_n'\) such that \(\mathrm {Tr}_{R_{n'}/R_n}(c_{n'}) =\alpha \). In particular, the higher cohomology groups of \((1+p^{n}\mathcal {O}_F/p^{n'}\mathcal {O}_F)\) acting on \(R_{n'}\) are annihilated by \(\alpha \).

For every s, there exists \(n(s) \ge n\) such that \(x' \in (R_{n(s)}{/}\alpha ^s)^{1+p^n \mathcal {O}_F\otimes \mathbb {Z}_p}\) and hence there exists \(y_s \in R_n\) such that \(y_s\equiv \alpha x^{\prime }\) modulo \(\alpha ^s\). We deduce that \(y_s\) converges to an element y for \(s \rightarrow \infty \) such that \(\alpha x^{\prime } =y\). Hence \(x \in R_n^{(\mathcal {O}_F/p^n\mathcal {O}_F)^*}\) which is R by the first part of the argument. \(\square \)

Proposition 5.5

The sheaf \(\mathfrak {w}_{I}^\mathrm{perf}\) is an invertible \(\mathcal {O}_{\mathfrak {X}_{\infty ,I}}\)-module. Moreover, for every subinterval \(J\subset I\) the pullback of \(\mathfrak {w}_{I}^\mathrm{perf}\) via the natural morphism \( \iota _{J,I} :\mathfrak {X}_{\infty ,J} \rightarrow \mathfrak {X}_{\infty , I} \) coincides with \(\mathfrak {w}_{J}^\mathrm{perf}\).

Proof

We prove the first claim. Let \(U:=\hbox {Spf}~R\) be an open formal subscheme of \(\mathfrak {X}_{\infty ,\alpha ,I}\). Suppose that \(\mathrm {Hdg}_1\) is a principal ideal over \(\hbox {Spf}~R\) with generator \(\tilde{\mathrm {Ha}}_1\). Let \(\hbox {Spf}~R_n\) (resp. \(\hbox {Spf}~R_\infty \)) be the inverse image of U in \(\mathfrak {IG}_{n,\infty ,\alpha ,I}\) (resp. in \(\mathfrak {IG}_{\infty ,\infty ,\alpha ,I}\)). Due to Corollary 5.4 it suffices to exhibit an invertible element \(x\in R_\infty \) such that \(\sigma (x)=\kappa ^{-1}(\sigma ) x\) for every \(\sigma \in (\mathcal {O}_F\otimes \mathbb {Z}_p)^*\).

Proposition 5.1 implies that there exist elements \(c_n \in \tilde{\mathrm {Ha}}_1^{-1} R_n\) such that \(\mathrm {Tr}_{R_n/R_{n-1}} (c_n) = c_{n-1}\) and \(c_0=1\). Define \(b_n:=\sum _{\sigma \in (\mathcal {O}_F/p^n\mathcal {O}_F)^*} {\kappa }(\tilde{\sigma }) {\sigma }(c_n)\in \tilde{\mathrm {Ha}}^{-1}_1 R_n\) for \(n\ge 1\). Here \(\tilde{\sigma }\in \mathbb {T}(\mathbb {Z}_p)\) is a lift of \(\sigma \). It follows from Lemma 2.6 that \(b_n\) converges to an element \(b_\infty \in R_\infty \) such that \(\sigma (b_\infty )=\kappa ^{-1}(\sigma ) b_\infty \) for every \(\sigma \in (\mathcal {O}_F\otimes \mathbb {Z}_p)^*\) and \(b_\infty \equiv 1\) modulo \(\frac{\alpha }{\tilde{\mathrm {Ha}}_1} R_\infty \) so that \(b_\infty \)is invertible in \(R_\infty \) as claimed.

The last claim can be proved as in Sect. 4.3. \(\square \)

Proposition 6.1

There exists a canonical isomorphism \(\mathfrak {w}_{I}^\mathrm{perf} \simeq h_{r}^*\mathfrak {w}_{I}\).

Proof

Over \(\mathfrak {X}_{\infty ,\alpha ,I}\) we have a chain of isogenies

$$\begin{aligned} \cdots G_{n+r} \mathop {\rightarrow }\limits ^{F}G_{n + r-1} \rightarrow \cdots G_r \end{aligned}$$

where \(G_{s}\) is the versal semi-abelian scheme over \(\mathfrak {X}_{s,I}\). Denote by \(C_{n,r} \hookrightarrow G_r[p^n]\) the kernel of \((F^n)^D :G_r[p^n]^D \rightarrow G_{r+n}[p^n]^D)\). Clearly \(C_{n,r} = H_{n}(G_{r+n})^D\). The isogeny \(F :G_{r+1} \rightarrow G_{r}\) induces a morphism \(C_{n,r+1} \rightarrow C_{n,r}\) which is generically an isomorphism. Over \(\mathfrak {IG}_{n,\infty ,I}\) we have a universal morphism \(\mathcal {O}_F/p^n\mathcal {O}_F \rightarrow H_n(G_s)^D\) for every \(s \ge n -k\). The map \(C_{n,s} \rightarrow G_s[p^n]/H_n(G_s) \simeq H_n(G_s)^D\) is generically an isomorphism as both group schemes are generically étale. Composing we then get an \(\mathcal {O}_F\)-equivariant map \(\mathcal {O}_F/p^n\mathcal {O}_F \rightarrow C_{n,s}\) for every \(s \ge n-k\). Using the morphisms \(C_{n,r+1} \rightarrow C_{n,r}\), we get an \(\mathcal {O}_F\)-equivariant morphism \(\mathcal {O}_F/p^n\mathcal {O}_F \rightarrow C_{n,r}\) for every n which is generically an isomorphism. Passing to the projective limit, we get a \(\mathcal {O}_F\)-equivariant map \(\mathcal {O}_F\otimes \mathbb {Z}_p \rightarrow \lim \nolimits _n C_{n,r}\). Let \(\mathrm {HT}^{un}\) be the image of 1 in \(\lim \nolimits _n C_{n,r}\) via the Hodge–Tate map \( \lim \nolimits _n G_r[p^n] \rightarrow \omega _{G_r}\). Then \(\mathrm {HT}^{un}\) defines an \((\mathcal {O}_F\otimes \mathbb {Z}_p)^*\)-equivariant map:

$$\begin{aligned} \mathfrak {IG}_{\infty , \infty , I} \rightarrow \mathfrak {F}_{n,r,I} \end{aligned}$$

fitting in the commutative diagram:

We then get an injective homomorphism \(h_{r}^*\mathfrak {w}_{I} \rightarrow \mathfrak {w}_{I}^\mathrm{perf} \). Moreover,

$$\begin{aligned} \mathfrak {w}_{I}^\mathrm{perf} \otimes h_{r}^*\mathfrak {w}_{I}^{-1} \subset \left( \mathcal {O}_{\mathfrak {IG}_{\infty ,\infty ,I}}\right) ^{(\mathcal {O}_F \otimes \mathbb {Z}_p)^*} = \mathcal {O}_{\mathfrak {X}_{\infty , I}} \end{aligned}$$

thanks to Corollary 5.4. \(\square \)

Corollary 6.2

We have a Tate trace map \(\mathrm {Tr}_r :h_{r,*} \mathfrak {w}_{I}^\mathrm{perf} \rightarrow \alpha ^{-1}\mathfrak {w}_{I}\), which is functorial in I.

Lemma 6.3

Proof

We prove the first statement. The second follows arguing as in [3, Lemme 6.6] using the Tate traces constructed in Proposition 5.2.

Let \(U:=\hbox {Spf}~A\) be a formal open affine subscheme of \(\overline{\mathfrak {M}}(\mu _N, \mathfrak {c})\) over which \(\omega _G\) is trivial. Let \(\tilde{\mathrm {Ha}}\) be a lift of \(\mathrm {Ha}\). Arguing as in the proof of Lemma 3.7, one deduces that the inverse image of U in \(\mathfrak {X}_{r,\alpha ,I}\) is \(\hbox {Spf}~R\) with \(R:=A \otimes _{\mathbb {Z}_p} B_\alpha \langle u,w\rangle /(w\tilde{\mathrm {Ha}}^{r}-\alpha , \alpha u-p)\) and that for integers \(1\le h <k\) the inverse image of U in \(\mathfrak {X}_{r,\alpha ,[p^h,p^{k}]}\) is \(\hbox {Spf}~R_{h,k}\) with \(R_{h,k}:=R \langle u_h,v_k\rangle /(\alpha ^{p^h} u_h -p, u_h v_k-\alpha ^{p^k-p^h})\).

There are maps \(R_{h,k} \rightarrow R_{h',k'}\) for \(h \le h '\le k \le k' \) given by \(u_h\mapsto \alpha ^{p^{h'}-p^h} u_{h'}\) and \(v_k\mapsto \alpha ^{p^{k'}-p^k} v_{k'}\). The argument in [3, Lemme 6.4] shows that the map \(R\rightarrow \lim _k R_{1,k} \) is an isomorphism. This proves the claim. \(\square \)

Theorem 6.4

The sheaf \(\mathfrak {w}_{I}^\mathrm{perf}\) descends to an invertible sheaf \(\mathfrak {w}_{I}\) over \(\mathfrak {X}_{r,\alpha ,I}\) for \(r\ge \sup \{ 4, 1 + \log _p{2g+1}\}\) if \(p \ge 3\) and \(r\ge \sup \{ 6, 1 + \log _p{2g+1}\}\) if \(p=2\). More precisely, \(\mathfrak {w}_{I}\) is the subsheaf of \(\mathcal {O}_{\mathfrak {X}_{r,\alpha ,I}}\)-modules of \(\mathfrak {w}_{I}^\mathrm{perf}\) characterized by the fact that for every interval \(J=[p^k,p^{k'}]\) with \(k' \ge k \ge 0\) integers and denoting \(\iota _{J,I} :\mathfrak {X}_{r,\alpha ,J} \rightarrow \mathfrak {X}_{r,\alpha ,I}\) the natural morphism, then \( \iota _{J,I}^*\mathfrak {w}_{I}\) is the sheaf \(\mathfrak {w}_{J}\) of Proposition 4.3 compatibly with the identification of Proposition 6.1.

Moreover, \(\mathfrak {w}_{I}\) is free of rank 1 over every formal affine subscheme \(U\subset \overline{\mathfrak {M}} (\mu _N, \mathfrak {c})\) such that \(\omega _G\vert _U\) is trivial.

Proof

We prove the claim for the minimal r possible, i.e., \(r=4\) if \(p \ge 3\) and \(r= 6\) for if \(p=2\). Let \(\tilde{\mathrm {Ha}}_r\) be a lift of the Hasse invariant over U. Thanks to Corollary 3.5 and Proposition 5.1, we can find elements \(c_n \in \tilde{\mathrm {Ha}}_r^{-\frac{p^{n}-1}{p-1}}\mathcal {O}_{\mathfrak {IG}_{n,r,\alpha ,I}}(W)\) for integers \(n\le r\) and elements \(c_n \in \tilde{\mathrm {Ha}}_r^{-p^{r}}\mathcal {O}_{\mathfrak {IG}_{n,\infty ,\alpha ,I}}(W)\) for general \(r\le n\) so that \(c_0=1\) and \(\mathrm {Tr}_{\mathfrak {IG}}(c_n) = c_{n-1}\). Define \(b_n = \sum _{\sigma \in (\mathcal {O}_F/p^n\mathcal {O}_F)^*}\kappa (\tilde{\sigma }) \sigma (c_n)\) where \(\tilde{\sigma }\) is a lift of \(\sigma \) in \((\mathcal {O}_F\otimes \mathbb {Z}_p)^*\).

Using Lemma 2.6, we deduce that:

• The sequence \(b_n\) converges to an element \(b_\infty \),

• \(b_\infty = 1\;\mod \;{\mathrm {Ha}}^{-p^r}\alpha \), and in particular, \(b_\infty \) generates \(\mathfrak {w}_{I}^\mathrm{perf}(W)\),

• \(b_\infty = b_r \;\mod \; \mathfrak {m}_{r-1} \tilde{\mathrm {Ha}}^{-p^r}\) (resp. \(\mathfrak {m}_{r-2} \tilde{\mathrm {Ha}}^{-p^r}\) if \(p=2\)).

Consider an interval \(J = [p^k,p^{k+1}]\) and the commutative diagram:

Let \(T:=\hbox {Spf}~C\) be the inverse image of W in \(\mathfrak {X}_{r,\alpha ,J}\). Then \(\mathfrak {w}_{J}\vert _T\) admits a generator f as C-module constructed as follows. For every \(0 \le n \le r+k\) take \(c'_n \in \tilde{\mathrm {Ha}}_r^{-\frac{p^n-1}{p-1}}\mathcal {O}_{\mathfrak {IG}_{n,r ,\alpha ,J}}(T)\) such that \(c_0'=1\), \(\mathrm {Tr}_{\mathfrak {IG}}(c'_n) = c'_{n-1}\) and \(c'_n = c_n\) if \(n \le r\). Take a section \(s \in \mathcal {O}_{\mathfrak {F}_{k+r,r,\alpha ,J}}(T)\) which is \(1 \mod p^2\) and which generates \(\mathfrak {w}_{r+k, r, J}(T)\) (see Lemma 4.5, and note that \( 1 = k+r-(k+1)-2\) if \(p \ne 2\) and \(1 = k+r - (k+1)-4\) if \(p=2\)). Let \(f:=\sum \nolimits _{\sigma \in ( \mathcal {O}_F/p^{k+r}\mathcal {O}_F)^*} \kappa (\tilde{\sigma }) \sigma (c'_{r+k} s)\).

Proposition 6.5

Proof

Due to Theorem 6.4, the sheaf \(\mathfrak {w}_{I}\) over \(\mathfrak {X}_{r,\alpha ,I}\) is canonically determined by the sheaves \(\mathfrak {w}_{I}^\mathrm{perf}\) and the sheaves \(\mathfrak {w}_{J}\) for \(J=[p^k,p^{k'}]\). These glue for varying \(\alpha \) and have compatible Frobenius morphisms by Sect. 4.3.2. The claim follows. \(\square \)

Theorem 6.6

Proof

Let \(U:=\hbox {Spf}~A\) be a formal open affine subscheme of \(\overline{\mathfrak {M}}(\mu _N, \mathfrak {c})\) where \(\omega _G\) is trivial. Let \(W:=\hbox {Spf}~B\) be the inverse image of U in \(\mathfrak {Z}_{r}\). In particular, we have elements \(\eta _p,\eta _1,\ldots ,\eta _g\) such that \(B=A\llbracket T_1, \ldots , T_g\rrbracket \langle \eta _p,\eta _1,\ldots ,\eta _g\rangle /(\mathrm {Ha}^{p^{r+1}} \eta _p - p,\mathrm {Ha}^{p^{r+1}} \eta _1 - T_1, \ldots ,\mathrm {Ha}^{p^{r+1}} \eta _g - T_g )\). Arguing as in Lemma 3.7 we deduce that B is normal.

(1) The map g is the base change of the morphism \(\widetilde{\mathfrak {W}}_F^0 \rightarrow \mathfrak {W}_F^0\) which is the \(\mathfrak {m}\)-adic completion of a blowup; therefore, it is proper. Thus \(C:=g_\star \mathcal {O}_{\mathfrak {X}_r}(W)\) is a finite B-module. The map \(B\rightarrow C\) is an isomorphism over the ordinary locus of U and, hence, it is generically an isomorphism. As \(\mathfrak {X}_r\) is also normal, it follows that \(B=C\) as claimed.

(2) Let \(Z:=g^{-1} W\). Thanks to (1) it suffices to prove that \(\mathfrak {w}_{I}\vert _Z\) is a free \(\mathcal {O}_Z\)-module of rank 1. Let \(Z^\mathrm{perf}\) be the inverse image of Z in \(\mathfrak {X}_{\infty }\). We will actually prove that \(\mathfrak {w}^\mathrm{perf}_{I}\vert _{Z^\mathrm{perf}}\) is a free \(\mathcal {O}_{Z^\mathrm{perf}}\)-module of rank 1 and find a generator \(b_\infty \) whose trace will be a generator of \(\mathfrak {w}_{I}\vert _Z\).

We apply the construction of Sect. 3.3 with \(A_0 = \mathbb {Z}_p\). We thus obtain formal schemes \(\mathfrak {Y}_s\) together with partial Igusa towers \(\mathfrak {IGY}_{n,s} \rightarrow \mathfrak {Y}_s\) for \(n \le s\).

Passing to the limit over Frobenius, we obtain \(\mathfrak {IGY}_{n, \infty } \rightarrow \mathfrak {Y}_\infty \). There is an obvious commutative diagram:

As in Corollary 3.5, one deduces from Proposition 3.4 the existence of elements

$$\begin{aligned} c'_n \in \tilde{\mathrm {Ha}}_r^{-\frac{p^{n}-1}{p-1}}\mathcal {O}_{\mathfrak {IGY}_{n,r}}(W) \end{aligned}$$

for integers \(n\le r\) and elements \(c_n' \in \tilde{\mathrm {Ha}}_r^{-p^{r}}\mathcal {O}_{\mathfrak {IGY}_{n,\infty }}(W)\) for general \(r\le n\) so that \(c'_0=1\) and \(\mathrm {Tr}_{\mathfrak {IG}}(c'_n) = c'_{n-1}\). We call \(c_n\) the pull back of \(c'_n\) in \(\mathcal {O}_{\mathfrak {IG}_{n,\infty }}(Z)\). We can now repeat the proof of Theorem 6.4 using these elements \(c_n\) to obtain a trivialization \(b_\infty \) of \(\mathfrak {w}^\mathrm{perf}_{I}\vert _{Z^\mathrm{perf}}\) whose trace gives the trivialization of \(\mathfrak {w}_{I}\vert _Z\). \(\square \)

Theorem 6.7

Lemma 7.1

For every normal \(\mathbb {F}_p\)-algebra R and every R-valued point \(x\in \overline{M}(\mu _N, \mathfrak {c})(R)\) such that the ordinary locus \((\hbox {Spec}~R)_\mathrm{ord}\) is dense in \(\hbox {Spec}~R\), the R-valued points of \(\overline{\mathrm {IG}}_{n}\) over x consist of the \(\mathcal {O}_F\)-equivariant morphisms \(\mathcal {O}_F/p^n\mathcal {O}_F\rightarrow H_{n,x}^D\), which are isomorphisms over \((\hbox {Spec}~R)_\mathrm{ord}\). Here \(H_{n,x}\) is the pull back of the canonical subgroup to \(\hbox {Spec}~R\) via x.

Lemma 7.2

For all \(n \ge 0\), we have \(\mathrm {Hdg}^{p^{n}} \subset \mathrm {Tr}_\mathrm{IG} \left( (h_{n+1})_\star \mathcal {O}_{ \overline{\mathrm{IG}}_{n}}\right) \).

Proof

Similar to the proof of Proposition 3.4. \(\square \)

Corollary 7.3

Let \(\hbox {Spec}~A\) be an open subset of \(\overline{M}(\mu _N, \mathfrak {c})_{\mathbb {F}_p}\) such that the sheaf \(\mathrm {Hdg}\) is trivial. Let us identify \(\mathrm {Ha}\) with a generator of \(\mathrm Hdg\). There is a sequence of elements \(c_0 = 1\), \(c_n \in \mathrm {Ha}^{-\frac{p^{n}-1}{p-1}}\mathcal {O}_{\overline{\mathrm{IG}}_n}(\hbox {Spec}~A)\) for \(n \ge 1\) such that \(\mathrm {Tr}_\mathrm{IG} (c_n) = c_{n-1}\).

Lemma 7.4

Proof

Easy and left to the reader. \(\square \)

Proposition 7.5

Theorem 7.6

Assume that \(r\ge 2\) (resp. \(r\ge 3\) if \(p = 2\)). Then the sheaf \(\mathfrak {w}_{\{\infty \}}\) is an invertible sheaf of \(\mathcal {O}_{\mathfrak {Z}_{r,\{\infty \}}}\)-modules. Its restriction to \(\mathfrak {Z}_{\mathrm{ord}, \{\infty \}}\) is the sheaf defined in Sect. 7.3.

Proof

The proof is local on \(\overline{M}(\mu _N, \mathfrak {c})_{\mathbb {F}_p}\). Let \(\hbox {Spec}~A\) be an open subset of \(\overline{M}(\mu _N, \mathfrak {c})_{\mathbb {F}_p}\) where the Hodge ideal is trivial. We denote abusively \(\mathrm {Ha}\) a generator. Let \(\hbox {Spf}~R\) be the inverse image of \(\hbox {Spec}~A\) in \(\mathfrak {Z}_{r,\{\infty \}}\), \(\hbox {Spf}~R_n\) the inverse image in \(\mathfrak {IGU}_{n,r,\{\infty \}}\) and \(\hbox {Spf}~R_\infty \) the inverse image in \(\mathfrak {IGZ}_{\infty ,r,\{\infty \}}\). By Lemma 7.4, \(R_\infty ^{(\mathcal {O}_F\otimes \mathbb {Z}_p)^*}=R\). Thus to prove the theorem, it suffices to show that there exists an invertible element \(x\in R_\infty \) such that \(\sigma (x)=\overline{\kappa }^{-1}(\sigma ) x\) for every \(\sigma \in (\mathcal {O}_F\otimes \mathbb {Z}_p)^*\).

Proposition 7.7

Proof

Fix an integer \(k_0\ge 1\). Let \(\hbox {Spf}~B\) be an open affine of \(\mathfrak {X}_{r, \alpha , [p^{k_0}, \infty ]}\). Assume that \(\mathrm {Hdg}\) is trivial on \(\hbox {Spf}~B\), generated by \(\tilde{\mathrm {Ha}}\). Fix elements \(c_0=1\) and, for \(1 \le n \le k_0+r\), \(c_n \in \tilde{\mathrm {Ha}}^{-\frac{p^n-1}{p-1}}\mathcal {O}_{\mathfrak {IG}_{n, r,\alpha , [p^{k_0}, \infty ]}}(\hbox {Spf}~B)\) such that \(\mathrm {Tr}_{\mathfrak {IG}}(c_n) = c_{n-1}\). Complete the sequence by choosing, for \(n \ge r+k_0+1\), elements \(c_n \in \tilde{\mathrm {Ha}}^{-p^{r+k_0}}\mathcal {O}_{\mathfrak {IG}_{n, \infty ,\alpha , [p^{k_0}, \infty ]}}(\hbox {Spf}~B)\) satisfying \(\mathrm {Tr}_{\mathfrak {IG}}(c_n) = c_{n-1}\). Set \(b_n = \sum _{\sigma \in (\mathcal {O}_F/p^n\mathcal {O}_F)^*} \kappa (\tilde{\sigma }) \sigma . c_n\), where \(\tilde{\sigma }\) is a lift of \(\sigma \) in \(\mathbb {T}(\mathbb {Z}_p)\). The sequence \(b_n\) converges in \(\mathcal {O}_{\mathfrak {IG}_{\infty , \infty , [p^{k_0}, \infty ]}}\) to a generator \(b_\infty \) of the sheaf \(\mathfrak {w}_{I}^\mathrm{perf}\). By Lemma 2.6, for all \(n \ge r+k_0\), \(b_{n} = b_{r+k_0} \mod \mathfrak {m}_{r+k_0-1} \tilde{\mathrm {Ha}}^{-p^{r+k_0}} \) (resp. \(\mod ~\mathfrak {m}_{r+k_0-2} \tilde{\mathrm {Ha}}^{-p^{r+k_0}} \) if \(p = 2\)). It follows that \(b_\infty = b_{r+k_0} \mod \alpha ^{p^{k_0}-p^{k_0-1}}\).

Fix now an interval \([p^{k}, p^{k+1}]\) with \(k \ge k_0\). Let \(\hbox {Spf}~C\) be the inverse image of \(\hbox {Spf}~B\) in \(\mathfrak {X}_{r,\alpha , [p^{k}, p^{k+1}]}\). Fix elements \(c'_{n} \in \tilde{\mathrm {Ha}}^{-\frac{p^n-1}{p-1}} \mathcal {O}_{\mathfrak {IG}_{n, r, \alpha , [p^{k}, p^{k+1}]}}(\hbox {Spf}~C)\) for \(r+k_0+1 \le n \le r+k\) satisfying \(\mathrm {Tr}_{\mathfrak {IG}} (c_n') = c'_{n-1}\) for \(n \ge r+k_0 + 2\) and \(\mathrm {Tr}_{\mathfrak {IG}} (c_{r+k_0+1}') = c_{r+k_0}\). There is a generator f of the sheaf \(\mathfrak {w}_I\) over \(\hbox {Spf}~C\) such that \(f = \sum \nolimits _{\sigma \in (\mathcal {O}/p^{r+k}\mathcal {O})^*} \kappa (\tilde{\sigma }) \sigma \cdot c'_{r+k}~\mod p \tilde{\mathrm {Ha}}^{-p^{r+k}}\). As in C we have \(\alpha ^{p^k} \mid p\) and \(\tilde{\mathrm {Ha}}^{-p^{r+k}} \mid \alpha ^{p^{k-1}}\) it follows that \(f = \sum \nolimits _{\sigma \in (\mathcal {O}/p^{r+k}\mathcal {O})^*} \kappa (\tilde{\sigma }) \sigma \cdot c'_{r+k} \mod \alpha ^{p^{k_0}-p^{k_0-1}}\). Using Lemma 2.6 one more time, we deduce that \(f = b_{r+k_0} \mod \alpha ^{p^{k_0}-p^{k_0-1}}\). As a consequence, in \(\mathcal {O}_{\mathfrak {IG}_{\infty , \infty , \alpha , [p^k,p^{k+1}]}}(\hbox {Spf}~C)\) we have \( f = b_\infty = b_{r+k_0} \mod \alpha ^{p^{k_0}-p^{k_0-1}}\). Using Tate’s traces, we get \(f = \mathrm {Tr}_r(b_\infty ) = b_\infty ~\mod ~\alpha ^{p^{k_0}-p^{k_0-1}-1}\mathfrak {w}_{[p^{k}, p^{k+1}]}^{perf}(\hbox {Spf}~C)\). As this relation holds on all intervals \([p^k, p^{k+1}]\), it follows that \(\mathrm {Tr}_{r}(b_\infty ) = b_\infty \mod \alpha ^{p^{k_0}-p^{k_0-1}-1}\mathfrak {w}_{[p^{k_0}, \infty ]}^{\mathrm{perf}}(\hbox {Spf}~B)\). As a result, \(\mathrm {Tr}_{r}(b_\infty ) = b_{r+k_0} \mod \alpha ^{p^{k_0}-p^{k_0-1}-1} \mathcal {O}_{\mathfrak {IG}_{\infty , \infty ,\alpha , [p^{k_0}, \infty ]}}(\hbox {Spf}~B)\).

It follows that there is a morphism \( \mathfrak {w}_{[p^{k_0}, \infty ]} \rightarrow \mathcal {O}_{\mathfrak {IG}_{r+k_0, r, \alpha , [p^{k_0}, \infty ]}}{/}\alpha ^{p^{k_0}-p^{k_0-1}-1}\) which on \(\hbox {Spf}~B\) sends \(\mathrm {Tr}_r(b_\infty )\) on \(b_{r+k_0}\). It factors the map \(\mathfrak {w}_{[p^{k_0}, \infty ]} \rightarrow \mathcal {O}_{\mathfrak {IG}_{\infty , \infty ,\alpha , [p^{k_0}, \infty ]}}{/} \alpha ^{p^{k_0}-p^{k_0-1}-1}\). Comparing the definition of \(b_{r+k_0}\) and the construction of the sheaf \(\mathfrak {w}_{\{\infty \}}\), we obtain that the restriction of \(\mathfrak {w}_{[1, \infty ]}\) to \(\mathfrak {X}_{r,\{\infty \}}\) is \(\mathfrak {w}_{\{\infty \}}\). \(\square \)

Proposition 7.8

Proof

Take \((R,R^+)\) as in the proposition. Without loss of generality, we can assume that R has a noetherian ring of definition. Using (5), we observe that the rule f defines compatible sections of \(\mathrm {H}^0(\hbox {Spf}~R_{i,0}, \mathfrak {w}_{\{\infty \}})\otimes _{R_{i,0}} R\) which glue, by the sheaf property, to a section of \(\omega ^{\overline{\kappa }^{un}}\) on \(\hbox {Spa}(R, R^+)\). \(\square \)

Lemma 8.1

The group \(\varDelta \) acts freely on \(\overline{M}(\mu _N, \mathfrak {c})\). One can form the quotient \(\overline{M}(\mu _N, \mathfrak {c})_G: = \overline{M}(\mu _N, \mathfrak {c}){/}\varDelta \). The quotient map \(\overline{M}(\mu _N, \mathfrak {c})\rightarrow \overline{M}(\mu _N, \mathfrak {c})_G\) is finite étale with group \(\varDelta \).

Proof

Since \(\overline{M}(\mu _N, \mathfrak {c})\) is a projective scheme, it can be covered by open affine subschemes fixed by the action of \(\varDelta \). Thus we can form the quotient \(\overline{M}(\mu _N, \mathfrak {c})_G\). We now show that\(\varDelta \) acts freely on \(\overline{M}(\mu _N, \mathfrak {c})\). This can be proved over an algebraically closed field k where the freeness of the action amounts to proving the following:

Consider an abelian variety with real multiplication \((A,\iota ,\varPsi ,\lambda )\) over k as in Sect. 3 and a totally positive unit \(\epsilon \in \mathcal {O}_F^{+,*}\). Let \(\alpha :A\rightarrow A\) be an automorphism commuting with the \(\mathcal {O}_F\)-action, the level N structure \(\varPsi \) and such that \(\lambda \circ \alpha =\epsilon \alpha ^\vee \circ \lambda \). Then \(\alpha \in U_N\) is a totally positive unit, congruent to 1 modulo N.

As \(\alpha \) respects the level N structure, it suffices to prove that \(\alpha \) is an endomorphism lying in \(\mathcal {O}_F\). Suppose this is not the case. Then \(E:=F[\alpha ]\subset \mathrm{End}^0(A)\) would be a commutative algebra of dimension at least 2g. It must be a field, else A would decompose as a product of at least two abelian varieties of dimension less than g, with real multiplication by F which is impossible. As a maximal commutative subalgebra of \(\mathrm{End}^0(A)\) has dimension \(\le \)2g, it follows that E is a CM field of degree 2 over F. Moreover, the Rosati involution associated with any \(\mathcal {O}_F\)-invariant polarization induces complex conjugation on E. As the rank of the group of units in \(\mathcal {O}_E\) is equal to the rank of the group of units of \(\mathcal {O}_F\) by Dirichlet’s unit theorem, it follows that there exists an integer \(n\ge 2\) such that \(\alpha ^n\in \mathcal {O}_F^*\) and \(\alpha ^{n-1}\not \in \mathcal {O}_F^*\). Hence \(\zeta =(\alpha /\bar{\alpha })\) is a primitive n-root of unity in \(\mathcal {O}_E\). It preserves every \(\mathcal {O}_F\)-equivariant polarization \(\lambda \) as \(\lambda ^{-1}\circ \zeta ^\vee \circ \lambda \circ \zeta = \overline{\zeta } \zeta =1\). As \(N\ge 4\) it follows from Serre’s lemma that \(\zeta =1\) leading to a contradiction. We are left to show that \(\varDelta \) acts freely on the boundary \( D: = \overline{M}(\mu _N, \mathfrak {c})_G {\setminus } {M}(\mu _N, \mathfrak {c})_G\). Recall that the boundary is the union of its connected components parametrized by the cusps of the minimal compactification. Each connected component of the boundary is stratified. More precisely, for each connected component, there is a polyhedral decomposition \(\varSigma \) of the cone of totally positive elements \(M^{+}\) inside a fractional ideal \(M \subset F\) determined by the cusp. To every cone \(\sigma \in \varSigma \) corresponds a stratum \(S_\sigma \subset D\). By construction of the toroidal compactification, if \(\epsilon \in U_N\), then \(S_{\epsilon ^2 \sigma } = S_{\sigma }\). We now claim that the action of \(\varDelta \) on the set of all strata in D is free. This follows from the fact that the stabilizer of \(\sigma \in \varSigma \) is a finite subgroup of \({\mathcal {O}_F^{+,*}}\) and thus trivial. This concludes the proof. \(\square \)

Theorem 8.2

We have \(\mathrm {R}^ih_\star \mathfrak {w}^{\kappa _G^\mathrm{un}}(-D) = 0\) for all \(i > 0\).

Proof

This is a variant of [2, Theorem 3.17]. Recall that \(\overline{{M}}(\mu _N, \mathfrak {c})_{G}\) is the quotient of \(\overline{{M}}(\mu _N, \mathfrak {c})\) by \(\varDelta \). Let \({M}^*(\mu _N, \mathfrak {c})\) be the minimal compactification and let \({M}^*(\mu _N, \mathfrak {c})_{G}\) be its quotient by \(\varDelta \). We have a map \(h': \overline{{M}}(\mu _N, \mathfrak {c})_{G} \rightarrow {M}^*(\mu _N, \mathfrak {c})_{G}\). Let \(\mathcal {L}\) be a torsion invertible sheaf on \(\overline{{M}}(\mu _N, \mathfrak {c})_{G}\). Then we claim that \(\mathrm{{ R}}^i h'_\star \mathcal {L}(-D) = 0 \) for \(i >0\). This follows from [2, Prop. 6.4]. Note that in that reference the proposition is stated for the trivial sheaf, but the proof works without any change for a torsion sheaf.

The map \(h:\mathfrak {M}_{r,G} \rightarrow \mathfrak {M}^*_{r,G}\) is an isomorphism away from the cusps and in particular away from the ordinary locus, so we are left to prove the statement for the map \( h_\mathrm{{{\mathrm{\mathrm{ord}}}}}: \mathfrak {M}_{\mathrm{ord}, \mathrm{G}} \rightarrow \mathfrak {M}^*_{\mathrm{ord},\mathrm{G}}\) over the ordinary locus. In this case, the sheaf \(\mathfrak {w}^{\kappa _G^\mathrm{un}}(-D)\) is invertible. Recall that the ring \(\varLambda _F^G\) is semi-local and complete. Let \(\mathfrak {n}\) be a maximal ideal of \(\Lambda _F^G\) corresponding to a character \(\eta : (\mathcal {O}_F/ \mathfrak {p} \mathcal {O}_F)^*\times (\mathbb {F}_p)^\times \rightarrow \mathbb {F}_q^\times \) where \(\mathbb {F}_q\) is a finite extension of \(\mathbb {F}_p\). We are left to prove the vanishing for the sheaf \(\mathcal {F}: = \mathfrak {w}^{\kappa _G^\mathrm{un}}(-D)/\mathfrak {n}\) over \(\mathfrak {M}_{\mathrm{ord}, \mathrm{G}}\). This is an invertible sheaf on its support \(\overline{M}_\mathrm{ord}(\mu _N, \mathfrak {c})_{G,\mathbb {F}_q} \hookrightarrow \mathfrak {M}_{\mathrm{ord}, \mathrm{G}}\). Moreover, \(\mathcal {F}^{\otimes q-1} = \mathcal {O}_{\overline{M}_\mathrm{ord}(\mu _N, \mathfrak {c})_{G,\mathbb {F}_q}}\) because the order of the character \(\eta \) divides \(q-1\), and we can conclude. \(\square \)

Corollary 8.3

Proof

For the last point, fix some open \(\hbox {Spa}(R_i, R_i^+) \subset \mathcal {W}^G_F\). Since the sheaf \(\omega ^{\kappa _G^\mathrm{un}}(-D)\) is invertible, there is a finite covering \(\cup \hbox {Spa}(S_i, S_j^+)\) of \(\mathcal {M}_{r,G} \vert _{ \hbox {Spa}(R_i, R_i^+)}\) such that the sheaf \(\omega ^{\kappa _G^\mathrm{un}}(-D)\) is trivial on every open of this covering. Arguing as in [3, Prop. 6.9], one proves that each \(S_i\) is a projective Banach \(R_i\)-module. We can form the Chech complex associated with the covering \(\cup \hbox {Spa}(S_i, S_j^+)\), which, by (1), is a resolution of \(g_\star \omega ^{\kappa _G^\mathrm{un}}(-D) (\hbox {Spa}(R_i, R_i^+))\). Moreover, each term appearing in the complex is a projective Banach module and thus \(g_\star \omega ^{\kappa _G^\mathrm{un}}(-D) (\hbox {Spa}(R_i, R_i^+))\) is a projective Banach module. \(\square \)

Proposition 8.4

The natural morphisms \(g_*^o \omega ^{\kappa ^\mathrm{un}} \rightarrow g_*\omega ^{\kappa ^\mathrm{un}}\) and \(g_*^o \omega ^{\kappa ^\mathrm{un}_G} \rightarrow g_*\omega ^{\kappa ^\mathrm{un}_G}\) are isomorphisms.

Proof

Since the complement of \({M}(\mu _N)\) in \(\overline{{M}}(\mu _N)\) is contained in the ordinary locus, one may restrict to proving the two assertions on ordinary loci. This is classical: Unraveling the constructions one is reduced to prove the claims for the structural sheaves on the adic Igusa tower. This can be proven at the level of formal schemes and then, by devissage, reducing to the Igusa tower \(\overline{IG}_{n}^{\mathrm{ord},o}\subset \overline{IG}_{n}^\mathrm{ord}\) modulo p over \(\overline{M}(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}\).

Let \(f_n:\overline{IG}_{n}^\mathrm{ord}\rightarrow \overline{M}(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}\) be the structural map. Notice that \(\overline{M}(\mu _N,)^\mathrm{ord}_{\mathbb {F}_p}\) is smooth, the minimal compactification \(\overline{{M}}^*(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}\) is normal and the map \(\gamma :\overline{M}(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}\rightarrow \overline{{M}}^*(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}\) is proper birational and it is an isomorphism outside a codimension 2 locus. Then \(\gamma _*f_{n,*} \mathcal {O}_{\overline{IG}_{n}^\mathrm{ord}}\) is a coherent \(\mathcal {O}_{\overline{{M}}^*(\mu _N)^\mathrm{ord}_{\mathbb {F}_p}}\)-module of normal rings and \(\gamma _*f_{n,*} \mathcal {O}_{\overline{IG}_{n}^{\mathrm{ord},o}}\) is the complement of a codimension two loci. Thus the map on global sections \(\gamma _*f_{n,*} \mathcal {O}_{\overline{IG}_{n,r,\alpha ,I}^\mathrm{ord}}\rightarrow \gamma _*f_{n,*} \mathcal {O}_{\overline{IG}_{n,r,\alpha ,I}^{\mathrm{ord},o}}\) is an isomorphism concluding the proof of the Proposition. \(\square \)

Lemma 8.5

The universal isogeny \(\pi _\ell \) induces an isomorphism \(\pi _\ell ^*:p_1^*\mathfrak {w}_{I} \rightarrow p_2^*\mathfrak {w}_{I}\), of the pullbacks via \(p_1\) and \(p_2\), respectively, of the invertible sheaves \(\mathfrak {w}_{I}\) on \(\mathfrak {X}_{r}^o(\mathfrak {c})\) and \(\mathfrak {X}_{r}^o(\ell \mathfrak {c})\), respectively, defined in Theorem 6.4.

Proof

Over \(\mathcal {W}_{F,[0,1]}\) this is the content of [2, Cor. 3.25] using the compatibility given in Theorem 6.7.

We deal with the general case. It follows from [3, Rmk A.1] and our assumptions on \(\pi _\ell \) that the isogeny \(\pi _\ell :G\rightarrow G'\) induces an isomorphism from the canonical subgroup of level n of G to the canonical subgroup of level n of \(G'\). This implies that \(\pi _\ell \) defines an isomorphism of the pullback to \(\mathfrak {S}_{\ell ,r}\) via \(p_1\) and \(p_2\) of the Igusa towers. By the functoriality of the Hodge–Tate map we also get an isomorphism \(p_2^*\mathcal {F} \rightarrow p_1^*\mathcal {F}\) of the modified integral structures \(\mathcal {F}\) of \(\omega _{G'}\) and \(\omega _G\) of Proposition 4.1. Thus we get an isomorphism \(p_2^*\mathfrak {F}_{n,r,I} \rightarrow p_1^*\mathfrak {F}_{n,r,I}\) of the pullback to \(\mathfrak {S}_{\ell ,r}\) of the torsors \(\mathfrak {F}_{n,r,I}\) of Sect. 4.1. Due to the normality of \(\mathfrak {S}_{\ell ,r}\), we get an isomorphism of the pull backs of the sheaves \(p_1^*\mathfrak {w}_{n,r,I}\rightarrow p_2^*\mathfrak {w}_{n,r,I}\) of overconvergent forms defined in Proposition 4.3.

Similarly, \(\pi _\ell \) defines an isomorphism between the pull backs of \(\mathfrak {IG}_{\infty ,\infty ,I}\) via \(p_1\) and \(p_2\) respectively (see Sect. 5.2 for the definition). The analog of Lemma 5.4 over the base \(\mathfrak {S}_{\ell ,r}\) provides an isomorphism between the pullbacks via \(p_1\) and \(p_2\), respectively, of the sheaves \(\mathfrak {w}_{I}^\mathrm{perf}\) of perfect overconvergent forms defined in Proposition 5.5. Thanks to the normality of \(\mathfrak {S}_{\ell ,r}\) we deduce the sought for isomorphism of the pullbacks via \(p_1\) and \(p_2\) of the sheaves \(\mathfrak {w}_{I}\). \(\square \)

Corollary 8.6

Proof

We have a morphism \(g:\mathfrak {S}_{\ell ,r}\rightarrow \mathfrak {T}_{\ell ,r}\) compatible with the natural projections \(\mathfrak {Z}_{r}^o(\mathfrak {c})\rightarrow \mathfrak {X}_{r}^o(\mathfrak {c})\) and \(\mathfrak {X}_{r}^o(\mathfrak {c})\rightarrow \mathfrak {Z}_{r}^o(\mathfrak {c})\). As in Theorem 6.6(1) and using the normality of \(\mathfrak {T}_{\ell ,r}\), one obtains that the natural morphism \(\mathcal {O}_{\mathfrak {T}_{\ell ,r}} \rightarrow g_*\mathcal {O}_{\mathfrak {S}_{\ell ,r}}\) is an isomorphism. This implies that the isomorphism of Lemma 8.5 induces an isomorphism of the pull backs via \(p_1\) and \(p_2\), respectively, of the invertible sheaves \(\mathfrak {w}^\kappa \) defined on \(\mathfrak {Z}_{r}^o(\mathfrak {c})\) and \(\mathfrak {Z}_{r}^o(\ell \mathfrak {c})\), respectively.

Taking adic analytic fibers and twisting by the invertible sheaf \(\mathfrak {w}^{\chi }\) as in Sect. 6.4 we get the claimed isomorphism. \(\square \)

Lemma 8.7

Proof

The first statement is clear. We prove the second statement: The group scheme \(\mathrm {Ker} \pi _\ell \cong G[\ell ]/H_1[\ell ]\) is finite étale at the level of analytic fibers due to [3, Cor. A.2]. Since it is isomorphic to \(H_1[\ell ]^\vee \) it follows from Proposition 3.2 that it is in fact étale locally on \(\mathcal {M}_{r}^o(\mathfrak {c})\) isomorphic to \(\mathcal {O}_F/\ell \mathcal {O}_F\). Since \(p_1\) is represented over \(\mathcal {M}_{r}^o(\mathfrak {c})\) by the \(\mathcal {O}_F\) splittings of the exact sequence of group schemes (with \(\mathcal {O}_F\)-action) \(0\rightarrow H_1[\ell ] \rightarrow G[\ell ] \rightarrow G[\ell ]/H_1[\ell ] \rightarrow 0\), it follows that \(p_1\) is a torsor under the finite and flat group scheme \(H_1[\ell ]\). This proves the claim. \(\square \)

Theorem 8.8

Acknowledgements

We thank the referee for the careful reading of the manuscript and his/her remarks which hopefully led to improvements. The Remark 1.1 was written at his/her request. During part of the work on this project, Vincent Pilloni was supported by ANR PerCoLaTor.

During part of the work on this article Fabrizio Andreatta was supported by the Italian grant Prin 2010/2011.