A theta operator on Picard modular forms modulo an inert prime
 Ehud de Shalit^{1}Email author and
 Eyal Z. Goren^{2}
https://doi.org/10.1186/s4068701600758
© The Author(s) 2016
Received: 5 July 2015
Accepted: 27 May 2016
Published: 1 September 2016
Over \(\mathbb {C}\), this operator has been considered already by Ramanujan, where it fails to preserve modularity “by a multiple of \(E_{2}\)”. Maass modified it so that modularity is preserved, sacrificing holomorphicity. Shimura studied Maass’ differential operators on more general symmetric domains, as well as their iterations. They have become known as Maass–Shimura operators and play an important role in the theory of automorphic forms [37, chapter III].
At the same time, Serre’s padic operator has been studied in relation to mod p Galois representations, congruences between modular forms, padic families of modular forms and padic Lfunctions. As an example, we cite Coleman’s celebrated classicality theorem, asserting that “overconvergent modular forms of small slope are classical” [6]. A key step in Coleman’s original proof of that theorem was the observation that, although the padic theta operator did not preserve the space of overconvergent modular forms, for any \(k\ge 0,\,\theta ^{k+1}\) mapped overconvergent forms of weight \(k\) to overconvergent forms of weight \(k+2\).
Underlying the padic theory is Katz’ geometric approach to the theta operator, via the Gauss–Manin connection on the de Rham cohomology of the universal elliptic curve [20, 21]. Broadly speaking, Katz’ starting point is the unitroot splitting of the Hodge filtration in this cohomology over the ordinary locus. It is supposed to replace the Hodge decomposition over \(\mathbb {C}\), which can be used to make a geometric theory of the \(C^{\infty }\) operators of Maass–Shimura, thereby explaining their arithmetic significance. This approach has been adapted successfully to other Shimura varieties of PEL type, as long as they admit a nonempty ordinary locus in their characteristic p fiber. For unitary Shimura varieties, this has been done by Eischen [9, 10], if p splits in the quadratic imaginary field [and the signature is (n, n)]. Böcherer and Nagaoka [3] defined theta operators on Siegel modular forms by studying their qexpansions.
The assumption that the ordinary locus is nonempty may nevertheless fail. This is the case, for example, for Picard modular surfaces [associated with the group U(2, 1)] modulo a prime p which is inert in the underlying quadratic imaginary field. In this case, the abelian varieties parametrized by the open dense \(\mu \)ordinary stratum [30] are not ordinary. More generally, this happens for Shimura varieties associated with U(n, m) if \(n\ne m\), and p is inert ([16], Lemma 8.10). Another complication present in these examples is the fact that modular forms on U(n, m) admit Fourier–Jacobi (FJ) expansions at the cusps, which are qexpansions with theta functions as coefficients.
The absence of the unitroot splitting from the abovementioned construction can be “explained” by the use we made of the Igusa curve, which lies over the ordinary stratum. In the case of Picard modular surfaces at an inert prime p, it is nevertheless possible to construct an “Igusa surface” lying over the \(\mu \)ordinary part, even though the ordinary stratum (in the usual sense) is empty. Our construction of the theta operator is based on the same procedure, but there are now two automorphic vector bundles to consider, a line bundle \(\mathcal {L}\) and a plane bundle \(\mathcal {P}\). The Verschiebung homomorphism allows us to project the analogue of \(\text {KS}^{1}(\eta _{f})\) (which is a section of \(\mathcal {P}\otimes \mathcal {L}\)) to an appropriate onedimensional piece.
The resulting operator \(\varTheta \) enjoys all the desired properties. It has the right effect on Fourier–Jacobi expansions, extends holomorphically across the 1dimensional supersingular locus, and compares well with the theta operators on embedded modular curves. The theory of “theta cycles” [19] even presents a surprise (see 4.1).
Section 1 is a rather thorough introduction to Picard modular surfaces and modular forms that will serve us also in future work. Occasionally (e.g. when we compute the Gauss–Manin connection in the complex model), we could not find a reference for the results in the form that was needed. We preferred to work them out from scratch, rather than embark on a tedious translation of notation. This section benefitted in several places from the excellent exposition in Bellaïche’s thesis [2].
It is curious to note that in the case of modular curves, \(a^{k}\cdot \text {KS}^{1}(\eta _{f})\) was of weight \(k+2\), but had poles at the supersingular points, and only \(\theta (f)=h\cdot a^{k}\cdot \text {KS}^{1}(\eta _{f})\) extended holomorphically to a weight \(k+p+1\) modular form. Here, the projection \(V_{\mathcal {P}}\) takes care of the shift by \(p+1\) in the weight, and at the same time reduces the order of the pole along \(Ig_\mathrm{ss}={\bar{I}}g\backslash {\bar{I}}g_{\mu }\), so that \(\varTheta (f)\) becomes holomorphic over the whole surface.
The ultimate justification for our construction comes when we compute the effect of \(\varTheta \) on Fourier–Jacobi expansions, which is essentially a “Tate twist”. The computation uses both padic and complex formalisms. It may be possible to perform it entirely on the “MumfordTate object” (see Section 4.5 of [9, 26]), but we believe that our approach has its own didactical merit.
In Sect. 4 we compare our theta operator with theta operators on embedded modular curves. We also discuss theta cycles and filtrations on modular forms mod p.
Section 5 brings up padic modular forms in the style of Serre and Katz. The study of overconvergent forms, intimately connected with the study of the canonical subgroup and Coleman’s classicality theorem, will be the subject of another paper.
Many of the results of this paper, including the construction of the theta operator, generalize to unitary Shimura varieties associated with \(U(n1,1)\) for general n. Another direction in which the setup could be generalized is to replace \({\mathcal {K}}\) by an arbitrary CM field. This seems to require substantial additional work, apart from a heavy load of notation, even if the general layout would be the same. We refer the reader to [18] for a detailed discussion of some of the topics treated here over general CM fields (albeit for a split prime p).
1 Background
1.1 The unitary group and its symmetric space
1.1.1 Notation

\(d_{{\mathcal {K}}}\)—the squarefree integer such that \({\mathcal {K}}=\mathbb {Q}(\sqrt{d_{{\mathcal {K}}}})\).

\(D_{{\mathcal {K}}}\)—the discriminant of \({\mathcal {K}}\), equal to \(d_{{\mathcal {K}}}\) if \(d_{{\mathcal {K}}}\equiv 1\mod 4\) and \(4d_{{\mathcal {K}}}\) if \(d_{{\mathcal {K}}}\equiv 2,3\mod 4\).

\(\delta _{{\mathcal {K}}}=\sqrt{D_{{\mathcal {K}}}}\)—the square root with positive imaginary part, a generator of the different of \({\mathcal {K}}\), sometimes simply denoted \(\delta \).

\(\omega _{{\mathcal {K}}}=(1+\sqrt{d_{{\mathcal {K}}}})/2\) if \(d_{{\mathcal {K}}}\equiv 1\mod 4\), otherwise \(\omega _{{\mathcal {K}}}=\sqrt{d_{{\mathcal {K}}}}\), so that \({\mathcal {O}}_{{\mathcal {K}}}=\mathbb {Z}+\mathbb {Z}\omega _{{\mathcal {K}}}. \)

\({\bar{a}}\)—the complex conjugate of \(a\in {\mathcal {K}}\).

\(\text {Im}_{\delta }(a)=(a{\bar{a}})/\delta \), for \(a\in {\mathcal {K}}\).
1.1.2 The unitary group
1.1.3 The hermitian symmetric domain
If we let \(K_{\infty }\) be the stabilizer of \(x_{0}\) in \(G_{\infty }\), then \(K_{\infty }\) is compact modulo center (\(K_{\infty }\cap \mathbf {U}(\mathbb {R})\) is compact and isomorphic to \(U(2)\times U(1)\)). Since \(G_{\infty }\) acts transitively on \(\mathfrak {X}\), we may identify \(\mathfrak {X}\) with \(G_{\infty }/K_{\infty }\).
The usual upper half plane embeds in \(\mathfrak {X}\) as the set of points where \(u=0\).
1.1.4 The cusps of \(\mathfrak {X}\)
1.2 Picard modular surfaces over \(\mathbb {C}\)
1.2.1 Lattices and their arithmetic groups
If \(g\in G\) and \(\mu (g)=1\) (i.e. \(g\in U\)), the lattice gL is another lattice of the same sort and the discrete group corresponding to it is \(g\varGamma g^{1}\). Since U acts transitively on the cusps, this reduces the study of \(\varGamma \backslash \mathfrak {X}\) near a cusp to the study of a neighborhood of the standard cusp \(c_{\infty }\) (at the price of changing L and \(\varGamma \)).
Lemma 1.1
([28], p. 25) For any lattice L as above there exists a \(g\in U\) such that \(gL=L_{0}\) or \(gL=L_{1}\). If \(D_{{\mathcal {K}}}\) is odd, \(L_{0}\) and \(L_{1}\) are equivalent. If \(D_{{\mathcal {K}}} \) is even, they are inequivalent.
Indeed, if \(D_{{\mathcal {K}}}\) is even, \(L_{0}\otimes \mathbb {Q}_{p}\) and \(L_{1}\otimes \mathbb {Q}_{p}\) are \(U_{p}\)equivalent for every \(p\ne 2\), but not for \(p=2\).
1.2.2 Picard modular surfaces and the Baily–Borel compactification
Theorem 1.2
(Satake, Baily–Borel) \(X_{\varGamma }^{*}\) is projective, and the singularities at the cusps are normal. In other words, there exists a normal complex projective surface \(S_{\varGamma }^{*}\) and a homeomorphism \(\iota :S_{\varGamma }^{*}(\mathbb {C})\simeq X_{\varGamma }^{*}\), which on \(S_{\varGamma }(\mathbb {C})=\iota ^{1}(X_{\varGamma })\) is an isomorphism of complex manifolds. \(S_{\varGamma }^{*}\) is uniquely determined up to isomorphism.
1.2.3 The universal abelian variety over \(X_{\varGamma }\)
 (1)
\(A_{x}\) is a 3dimensional complex abelian variety,
 (2)
\(\lambda _{x}\) is a principal polarization on \(A_{x}\) (i.e. an isomorphism \(A_{x}\simeq A_{x}^{t}\) with its dual abelian variety induced by an ample line bundle),
 (3)
\(\iota _{x}:{\mathcal {O}}_{{\mathcal {K}}}\hookrightarrow End(A_{x})\) is an embedding of CM type (2, 1) (i.e. the action of \(\iota (a)\) on the tangent space of \(A_{x}\) at the origin induces the representation \(2\varSigma +{\bar{\varSigma }}\)) such that the Rosati involution induced by \(\lambda _{x}\) preserves \(\iota ({\mathcal {O}}_{{\mathcal {K}}})\) and is given by \(\iota (a)\mapsto \iota ({\bar{a}})\),
 (4)\(\alpha _{x}:N^{1}L/L\simeq A_{x}[N]\) is a full level N structure, compatible with the \({\mathcal {O}}_{{\mathcal {K}}}\)action and the polarization. The latter condition means that if we denote by \(\left\langle ,\right\rangle _{\lambda }\) the Weil “\(e_{N}\)pairing” on \(A_{x}[N]\) induced by \(\lambda _{x}\), then for \(l,l^{\prime }\in N^{1}L\)$$\begin{aligned} \left\langle \alpha _{x}(l),\alpha _{x}(l^{\prime })\right\rangle _{\lambda }=e^{2\pi iN\left\langle l,l^{\prime }\right\rangle }. \end{aligned}$$(1.21)
1.2.4 A “moving lattice” model for the universal abelian variety
We want to assemble the individual \(A_{x}\) into an abelian variety A over \(\mathfrak {X}\). In other words, we want to construct a 5dimensional complex manifold A, together with a holomorphic map \(A\rightarrow \mathfrak {X}\) whose fiber over x is identified with \(A_{x}\). For that, as well as for the computation of the Gauss–Manin connection below, it is convenient to introduce another model, in which the complex structure on \(\mathbb {C}^{3}\) is fixed, but the lattice varies.
For simplicity, we assume from now on that \(L=L_{0}\) is spanned over \({\mathcal {O}}_{{\mathcal {K}}}\) by \(\delta e_{1},e_{2}\) and \(e_{3}\). The case of \(L_{1}\) can be handled similarly.
The model \(A^{\prime }\) has another advantage that will become clear when we examine the degeneration of the universal abelian variety at the cusp \(c_{\infty }\). It suffices to note at this point that the first two of the three generating vectors of \(L_{x}^{\prime }\) depend only on u.
1.3 The Picard moduli scheme
1.3.1 The moduli problem
 (1)
A / R is an abelian scheme of relative dimension 3
 (2)
\(\lambda :A\simeq A^{t}\) is a principal polarization
 (3)
\(\iota :\) \({\mathcal {O}}_{{\mathcal {K}}}\rightarrow End(A/R)\) is a homomorphism such that (1) \(\iota \) makes Lie(A / R) a locally free Rmodule of type (2, 1), (2) the Rosati involution induced on \(\iota ({\mathcal {O}}_{{\mathcal {K}}})\) by \(\lambda \) is \(\iota (a)\mapsto \iota ({\bar{a}})\).
 (4)\(\alpha :N^{1}L/L\simeq A[N]\) is an isomorphism of \({\mathcal {O}}_{{\mathcal {K}}}\)group schemes over R which is compatible with the polarization in the sense that there exists an isomorphism \(\nu _{N}:\mathbb {Z}/N\mathbb {Z}\simeq \mu _{N}\) of group schemes over R such thatIn addition we require that for every multiple \(N^{\prime }\) of N, locally étale over Spec(R), there exists a similar level \(N^{\prime }\)structure \(\alpha ^{\prime }\), restricting to \(\alpha \) on \(N^{1}L/L\). One says that \(\alpha \) is locally étale symplectic liftable ([26], 1.3.6.2).$$\begin{aligned} \left\langle \alpha \left( \frac{l}{N}\right) ,\alpha \left( \frac{l^{\prime }}{N}\right) \right\rangle _{\lambda }=\nu _{N}\left( \left\langle l,l^{\prime }\right\rangle \mod N\right) . \end{aligned}$$(1.28)

For any geometric point \(\eta :R\rightarrow k\) (k algebraically closed field, necessarily of characteristic different from 2), the \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {Z}_{2}\) polarized module \((T_{2}A_{\eta },\left\langle ,\right\rangle _{\lambda })\) is isomorphic to \((L\otimes \mathbb {Z}_{2},\left\langle ,\right\rangle )\) under a suitable identification of \(\lim _{\leftarrow }\mu _{2^{n}}(k)\) with \(\mathbb {Z}_{2}\).
A level N structure \(\alpha \) can exist only if the group schemes \(\mathbb {Z}/N\mathbb {Z}\) and \(\mu _{N}\) become isomorphic over R, but the isomorphism \(\nu _{N}\) is then determined by \(\alpha \).
\(\mathcal {M}\) becomes a functor on the category of \(R_{0}\)algebras (and more generally, on the category of \(R_{0}\)schemes) in the obvious way. The following theorem is of fundamental importance ([26], I.4.1.11).
Theorem 1.3
The functor \(R\mapsto \mathcal {M}(R)\) is represented by a smooth quasiprojective scheme S over \(\mathrm{Spec}(R_{0})\), of relative dimension 2.
We call S the (open) Picard modular surface of level N. It comes equipped with a universal structure \(({\mathcal {A}},\lambda ,\iota ,\alpha )\) of the above type over S. We call \({\mathcal {A}}\) the universal abelian scheme over S. For every \(R_{0}\)algebra R and PEL structure in \(\mathcal {M}(R)\), there exists a unique Rpoint of S such that the given PEL structure is obtained from the universal one by base change.
We refer to [26], 1.4.3 for the relation between the given formulation of the moduli problem and other formulations due, e.g. to Kottwitz.
1.3.2 The Shimura variety \(\text {Sh}_{K}\)
Theorem 1.4
The canonical model of \({\text {Sh}}_{K}\) is the generic fiber \(S_{{\mathcal {K}}}\) of S.
Since \(g_{f}\) commutes with the \({\mathcal {K}}\)structure on \(V_{\mathbb {A}}\), \(L_{g}\) is still an \({\mathcal {O}}_{{\mathcal {K}}}\)lattice, hence \(\iota _{g}\) is defined.
Let now \(\gamma \in \mathbf {G}(\mathbb {Q})\). Then the action of \(\gamma \) on V induces an isomorphism between the tuples \(\underline{A}_{g}\) and \(\underline{A}_{\gamma g}\). Indeed, \(\gamma :V_{\mathbb {R}}\rightarrow V_{\mathbb {R}}\) intertwines the complex structures \(x_{g}\) and \(x_{\gamma g}\), and carries \(L_{g}\) to \(L_{\gamma g}\), so induces an isomorphism of the abelian varieties, which clearly commutes with the PEL structures.
This shows that \(\underline{A}_{g}\) depends solely on the double coset of g in \(\mathbf {G(}\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K\). One is left now with two tasks which we leave out: (i) proving that if \(\underline{A}_{g}\simeq \underline{A}_{g^{\prime }}\), then g and \(g^{\prime }\) belong to the same double coset, and that every \(\underline{A}\in \mathcal {M}(\mathbb {C})\) is obtained in this way, (ii) identifying the canonical model of \({\text {Sh}}_{K}\) over \({\mathcal {K}}\) with \(S_{{\mathcal {K}}}\).
1.3.3 The connected components of \({\text {Sh}}_{K}\)
1.3.4 The cl and \(\nu _{N}\) invariants of a connected component
Proposition 1.5
 (i)
cl([g]) is the Steinitz class of the lattice \(L_{g}=g_{f}({\hat{L}})\cap V\) in \(Cl_{{\mathcal {K}}}\).
 (ii)
\(\nu _{N}([g])\) is (essentially) the \(\nu _{N,g}\) that appears in the definition of \(\alpha _{g}\) (see 1.3.1).
Proof
 (i)
cl([g]) is the class of the ideal \((\nu (g_{f}))\) associated with the idele \(\nu (g_{f})\in {\mathcal {K}}_{f}^{\times }\). This ideal is in the same class as \((\det (g_{f}))\), because \(\mu (g_{f})\in \mathbb {Q}_{f}^{\times }\), so \((\mu (g_{f}))\) is principal. But the class of \((\det (g_{f}))\) is the Steinitz class of \(L_{g}\), since the Steinitz class of L is trivial.
 (ii)To find \(\nu _{N}([g])\), we first project the idele \(\mu (g_{f})\) to \(\mathbb {{\hat{Z}}}^{\times }\) using \(\mathbb {Q}_{f}^{\times }=\mathbb {Q}_{+}^{\times }\mathbb {{\hat{Z}}}^{\times }\). But this is just \({\tilde{\mu }}(g_{f})^{1}\mu (g_{f})\). We then take the result modulo N, soNow the definition of the tuple \((A_{g},\lambda _{g},\iota _{g},\alpha _{g})\) is such that if \(u,v\in N^{1}L/L\), then$$\begin{aligned} \nu _{N}([g])={\tilde{\mu }}(g_{f})^{1}\mu (g_{f})\mod N. \end{aligned}$$(1.48)Part (ii) follows if we identify \(\nu _{N,g}\in Isom_{\mathbb {C}}(N^{1}\mathbb {Z}/\mathbb {Z},\mu _{N})\) with \(\nu _{N}([g])\in (\mathbb {Z}/N\mathbb {Z})^{\times }\) using \(\exp (2\pi i(\cdot ))\).$$\begin{aligned} \left\langle \alpha _{g}(u),\alpha _{g}(v)\right\rangle _{\lambda _{g}}= & {} \exp \left( 2\pi iN\left\langle g_{f}u,g_{f}v\right\rangle _{g}\right) \nonumber \\= & {} \exp \left( 2\pi i{\tilde{\mu }}(g_{f})^{1}N\left\langle g_{f}u,g_{f}v\right\rangle \right) \nonumber \\= & {} \exp \left( 2\pi i{\tilde{\mu }}(g_{f})^{1}\mu (g_{f})N\left\langle u,v\right\rangle \right) \nonumber \\= & {} \exp \left( 2\pi i\nu _{N}([g])N\left\langle u,v\right\rangle \right) \end{aligned}$$(1.49)
1.3.5 The complex uniformization
Note that \(\varGamma _{1}=\varGamma \) is the principal levelN congruence subgroup in \(\mathbf {G}_{\mathbb {Z}}(\mathbb {Z})\), the stabilizer of L. Similarly, \(\varGamma _{j}\) is the principal levelN congruence subgroup in the stabilizer of \(L_{g_{j}}\), and is thus a group of the type considered in 1.2.1, except that we have dropped the assumption on the Steinitz class of \(L_{g_{j}}\). As \(N\ge 3\), \(\det (\gamma )=1\) and \(\mu (\gamma )=1\) for all \(\gamma \in \varGamma _{j}\), for every j. Indeed, on the one hand these are in \({\mathcal {K}}^{\times }\) and \(\mathbb {Q}_{+}^{\times }\), respectively. On the other hand, they are local units which are congruent to \(1\mod N\) everywhere. It follows that \(\varGamma _{j}\) are subgroups of \(\mathbf {G}^{\prime }(\mathbb {Q})=\mathbf {SU}(\mathbb {Q})\).
1.4 Smooth compactifications
1.4.1 The smooth compactification of \(X_{\varGamma }\)
We begin by working in the complex analytic category and follow the exposition of [5]. The Baily–Borel compactification \(X_{\varGamma }^{*}\) is singular at the cusps and does not admit a modular interpretation. For general unitary Shimura varieties, the theory of toroidal compactifications provides smooth compactifications that depend, in general, on extra data. It is a unique feature of Picard modular surfaces, stemming from the finiteness of \({\mathcal {O}}_{{\mathcal {K}}}^{\times }\), that this smooth compactification is canonical. As all cusps are equivalent (if we vary the lattice L or \(\varGamma \)), it is enough, as usual, to study the smooth compactification at \(c_{\infty }\). In [5] this is described for an arbitrary L (not even \({\mathcal {O}}_{{\mathcal {K}}}\)free), but for simplicity we write it down only for \(L=L_{0}\).
Lemma 1.6
Let \(N\ge 3\) be even. The matrix \(n(s,r)\in \varGamma _\mathrm{cusp}\) if and only if: (i) (\(d_{{\mathcal {K}}}\equiv 1\mod 4\)) \(s\in N{\mathcal {O}}_{{\mathcal {K}}}\), \(r\in ND_{{\mathcal {K}}}\mathbb {Z}\), (ii) (\(d_{{\mathcal {K}}}\equiv 2,3\mod 4\)) \(s\in N{\mathcal {O}}_{{\mathcal {K}}}\) and \(r\in 2^{1}ND_{{\mathcal {K}}}\mathbb {Z}\) .
Proposition 1.7
Proof
This follows from the discussion so far and the fact that \(\lambda (z,u)>R\) is equivalent to the above condition on \(t=q(z)\) ([5], Prop. 2.1).
To get a smooth compactification \({\bar{X}}_{\varGamma }\) of \(X_{\varGamma }\) (as a complex surface), we glue the disk bundle \(\mathcal {T}_{R}\) to \(X_{\varGamma }\) along \(\mathcal {T}_{R}^{\prime }\). In other words, we complete \(\mathcal {T}_{R}^{\prime }\) by adding the zero section, which is isomorphic to E. The same procedure should be carried out at any other cusp of \(\mathcal {C}_{\varGamma }\).
Note that the geodesic (1.15) connecting \((z,u)\in \mathfrak {X}\) to the cusp \(c_{\infty }\) projects in \({\bar{X}}_{\varGamma }\) to a geodesic which meets E transversally at the point \(u\mod \Lambda \). We caution that this geodesic in \(X_{\varGamma }\) depends on (z, u) and \(c_{\infty }\) and not only on their images modulo \(\varGamma \).
Recall that with any \(x=(z,u)\in \mathfrak {X}\) we associated a complex abelian variety \(A_{x}\), and another model \(A_{x}^{\prime }\) of the same abelian variety (1.27). This allowed us to define sections \(\text {d}\zeta _{1},\text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\) of \(\omega _{{\mathcal {A}}/\mathfrak {X}}\). A simple matrix computation gives the following.
Lemma 1.8
The sections \(\text {d}\zeta _{1}\) and \(\text {d}\zeta _{3}\) are invariant under \(\varGamma _\mathrm{cusp}\). The section \(\text {d}\zeta _{2}\) is invariant modulo the subbundle generated by \(\text {d}\zeta _{1}\).
Thus \(\text {d}\zeta _{1},\text {d}\zeta _{3}\) and \(\text {d}\zeta _{2}\mod \left\langle \text {d}\zeta _{1}\right\rangle \) descend to welldefined sections in the neighborhood \(\mathcal {T}_{R}\simeq \varGamma _\mathrm{cusp}\backslash \varOmega _{R}\cup E\) of E in \({\bar{X}}_{\varGamma }\).
1.4.2 The smooth compactification of S
The arithmetic compactification \({\bar{S}}\) of the Picard surface S (over \(R_{0}\)) is due to Larsen [28, 29] (see also [2, 26]). We summarize the results in the following theorem. We mention first that as \(S_{\mathbb {C}}\) has a canonical model S over \(R_{0}\), its Baily–Borel compactification \(S_{\mathbb {C}}^{*}\) has a similar model \(S^{*}\) over \(R_{0}\), and S embeds in \(S^{*}\) as an open dense subscheme.
Theorem 1.9
 (ii)As a complex manifold, there is an isomorphismextending the isomorphism of \(S_{\mathbb {C}}\) with \(\coprod _{j=1}^{m}X_{\varGamma _{j}}\).$$\begin{aligned} {\bar{S}}_{\mathbb {C}}\simeq \coprod _{j=1}^{m}{\bar{X}}_{\varGamma _{j}}, \end{aligned}$$(1.61)
 (iii)
Let \(C=p^{1}(S^{*}\backslash S)\). Let \(R_{N}\) be the integral closure of \(R_{0}\) in the ray class field \({\mathcal {K}}_{N}\) of conductor N over \({\mathcal {K}}\). Then the connected components of \(C_{R_{N}}\) are geometrically irreducible and are indexed by the cusps of \(S_{R_{N}}^{*}\) over which they sit. Furthermore, each component \(E\subset C_{R_{N}}\) is an elliptic curve with complex multiplication by \({\mathcal {O}}_{{\mathcal {K}}}\).
We call C the cuspidal divisor. If \(c\in S_{\mathbb {C}}^{*}\backslash S_{\mathbb {C}}\) is a cusp, we denote the complex elliptic curve \(p^{1}(c)\) by \(E_{c}\). Although \(E_{c}\) is in principle definable over the Hilbert class field \({\mathcal {K}}_{1}\), no canonical model of it over that field is provided by \({\bar{S}}\). On the other hand, \(E_{c}\) does come with a canonical model over \({\mathcal {K}}_{N}\), and even over \(R_{N}\).
We refer to [2, 28] for a modulitheoretic interpretation of C as a moduli space for semiabelian schemes with a suitable action of \({\mathcal {O}}_{{\mathcal {K}}}\) and a “levelN structure”.
1.4.3 Change of level
Assume that \(N\ge 3\) is even, and \(N^{\prime }=QN\). We then obtain a covering map \(X_{\varGamma (N^{\prime })}\rightarrow X_{\varGamma (N)}\) where by \(\varGamma (N)\) we denote the group previously denoted by \(\varGamma \). Near any of the cusps, the analytic model allows us to analyze this map locally. Let \(E^{\prime }\) be an irreducible cuspidal component of \({\bar{X}}_{\varGamma (N^{\prime })}\) mapping to the irreducible component E of \({\bar{X}}_{\varGamma (N)}\). The following is a consequence of the discussion in the previous sections.
Proposition 1.10
The map \(E^{\prime }\rightarrow E\) is a multiplicationbyQ isogeny, hence étale of degree \(Q^{2}\). When restricted to a neighborhood of \(E^{\prime }\), the covering \({\bar{X}}_{\varGamma (N^{\prime })}\rightarrow {\bar{X}}_{\varGamma (N)}\) is of degree \(Q^{3}\) and has ramification index Q along E, in the normal direction to E.
Corollary 1.11
The pullback to \(E^{\prime }\) of the normal bundle \(\mathcal {T}(N)\) of E is the Qth power of the normal bundle \(\mathcal {T}(N^{\prime })\) of \(E^{\prime }\).
1.5 The universal semiabelian scheme \({\mathcal {A}}\)
1.5.1 The universal semiabelian scheme over \({\bar{S}}\)
As Larsen and Bellaïche explain, the universal abelian scheme \(\pi :{\mathcal {A}}\rightarrow S\) extends canonically to a semiabelian scheme \(\pi :{\mathcal {A}}\rightarrow {\bar{S}}\). The polarization \(\lambda \) extends over the boundary \(C={\bar{S}}\backslash S\) to a principal polarization \(\lambda \) of the abelian part of \({\mathcal {A}}\). The action \(\iota \) of \({\mathcal {O}}_{{\mathcal {K}}}\) extends to an action on the semiabelian variety, which necessarily induces separate actions on the toric part and on the abelian part.
Let E be a connected component of \(C_{R_{N}}\), mapping (over \(\mathbb {C}\) and under the projection p) to the cusp \(c\in S_{\mathbb {C}}^{*}\). Then there exist (1) a principally polarized elliptic curve B defined over \(R_{N}\), with complex multiplication by \({\mathcal {O}}_{{\mathcal {K}}}\) and CM type \(\varSigma \), and (2) an ideal \(\mathfrak {a}\) of \({\mathcal {O}}_{{\mathcal {K}}}\), such that every fiber \({\mathcal {A}}_{x}\) of \({\mathcal {A}}\) over E is an \({\mathcal {O}}_{{\mathcal {K}}}\)group extension of B by the \({\mathcal {O}}_{{\mathcal {K}}}\)torus \(\mathfrak {a}\otimes \mathbb {G}_{m}\). Both B (with its polarization) and the ideal class \([\mathfrak {a}]\in Cl_{{\mathcal {K}}}\) are uniquely determined by the cusp c. Only the extension class in the category of \({\mathcal {O}}_{{\mathcal {K}}}\)groups varies as we move along E. Note that since the Lie algebra of the torus is of type (1, 1), the Lie algebra of such an extension \({\mathcal {A}}_{x}\) is of type (2, 1), as is the case at an interior point \(x\in S\). If we extend scalars to \(\mathbb {C}\), the isomorphism type of B is given by another ideal class \([\mathfrak {b}]\) (i.e. \(B(\mathbb {C})\simeq \mathbb {C}/\mathfrak {b}\)). In this case, we say that the cusp c is of type \((\mathfrak {a},\mathfrak {b})\).
1.5.2 \({\mathcal {O}}_{{\mathcal {K}}}\)semiabelian schemes of type \((\mathfrak {a},\mathfrak {b})\)
Thus over \(\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\), there is a universal semiabelian scheme \(\mathcal {G}(\mathfrak {a},B)\) of type \((\mathfrak {a},B)\), and any \(\mathcal {G}\) as above, over any base \(R^{\prime }/R\), is obtained from \(\mathcal {G}(\mathfrak {a},B)\) by pullback (specialization) with respect to a unique map Spec\((R^{\prime })\rightarrow \delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\).
1.6 Degeneration of \({\mathcal {A}}\) along a geodesic connecting to a cusp
1.6.1 The degeneration to a semiabelian variety
It is instructive to use the “moving lattice model” to compute the degeneration of the universal abelian scheme along a geodesic, as we approach a cusp. To simplify the computations, assume for the rest of this section, as before, that \(N\ge 3\) is even and that the cusp is the standard cusp at infinity \(c=c_{\infty }\). In this case, we have shown that \(E_{c}=\mathbb {C}/\Lambda \), where \(\Lambda =N{\mathcal {O}}_{{\mathcal {K}}}\), and we have given a neighborhood of \(E_{c}\) in \({\bar{X}}_{\varGamma }\) the structure of a disk bundle in a line bundle \(\mathcal {T}\). See Proposition 1.7.
1.6.2 The analytic uniformization of the universal semiabelian variety of type \((\mathfrak {a},\mathfrak {b})\)
We now compare the description that we have found for the degeneration of \({\mathcal {A}}\) along the geodesic connecting (z, u) to \(c_{\infty }\) with the analytic description of the universal semiabelian variety of type \((\mathfrak {a},\mathfrak {b})\).
Proposition 1.12
Proof
Corollary 1.13
Proof
The extra data carried by \(u\in E_{c}\), which are forgotten by the map of the corollary, come from the level N structure. As mentioned before, according to [28] and [2] the cuspidal divisor C has a modular interpretation as the moduli space for semiabelian schemes of the type considered above, together with levelN structure (\(\mathcal {M}_{\infty ,N} \) structures in the language of [2]). A levelN structure on a semiabelian variety \(\mathcal {G}\) of type \((\mathfrak {a},\mathfrak {b})\) consists of (i) a levelN structures \(\alpha :N^{1}{\mathcal {O}}_{{\mathcal {K}}}/{\mathcal {O}}_{{\mathcal {K}}}\simeq \mathfrak {a}\otimes \mu _{N}\) on the toric part (ii) a levelN structure \(\beta :N^{1}{\mathcal {O}}_{{\mathcal {K}}}/{\mathcal {O}}_{{\mathcal {K}}}\simeq N^{1}\mathfrak {b}/\mathfrak {b}=B[N]\) on the abelian part (iii) an \({\mathcal {O}}_{{\mathcal {K}}}\)splitting \(\gamma \) of the map \(\mathcal {G}[N]\rightarrow B[N]\).
1.7 The basic automorphic vector bundles
1.7.1 Definition and first properties
1.7.2 The factors of automorphy corresponding to \(\mathcal {L}\) and \(\mathcal {P}\)
The formulae below can be deduced also from the matrix calculations in the first few pages of [36]. Let \(\varGamma =\varGamma _{j}\) be one of the groups used in the complex uniformization of \(S_{\mathbb {C}}\), cf Sect. 1.3.5. Via the analytic isomorphism \(X_{\varGamma }\simeq S_{\varGamma }\) with the jth connected component, the vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) are pulled back to \(X_{\varGamma }\) and then to the symmetric space \(\mathfrak {X}\), where they can be trivialized, hence described by means of factors of automorphy. Let us denote by \(\mathcal {P}_{an}\) and \(\mathcal {L}_{an}\) the two vector bundles on \(X_{\varGamma }\), in the complex analytic category, or their pullbacks to \(\mathfrak {X}\).
Lemma 1.14
Proof
Let \(\mathcal {V}=Lie({\mathcal {A}}/\mathfrak {X})=\omega _{{\mathcal {A}}/\mathfrak {X}}^{\vee }\) and \(\mathcal {W}=\mathcal {V}({\bar{\varSigma }})=\mathcal {L}_{an}^{\vee }\) (a line bundle). At a point \(x=(z,u)\in \mathfrak {X}\) the fiber \(\mathcal {V}_{x}\) is identified canonically with \((V_{\mathbb {R}},J_{x})\) and then \(\mathcal {W}_{x}=W_{x}=\mathbb {C} \cdot \, ^{t}(z,u,1)\).
Proposition 1.15
Proof
Consider next the plane bundle \(\mathcal {P}_{an}\). As we will only be interested in scalarvalued modular forms, we do not compute its matrixvalued factor of automorphy (but see [36]). It is important to know, however, that the line bundle \(\det \mathcal {P}_{an}\) gives nothing new.
Proposition 1.16
Proof
1.7.3 The relation \(\det \mathcal {P}\simeq \mathcal {L}\) over \({\bar{S}}_{{\mathcal {K}}}\)
The isomorphism between \(\det \mathcal {P}\) and \(\mathcal {L}\) is in fact algebraic and even extends to the generic fiber \({\bar{S}}_{{\mathcal {K}}}\) of the smooth compactification.
Proposition 1.17
One has \(\det \mathcal {P}\simeq \mathcal {L}\) over \({\bar{S}}_{{\mathcal {K}}}\).
Proof
An alternative proof is to use Theorem 4.8 of [14]. In our case, it gives a functor \(\mathcal {V}\mapsto [\mathcal {V]}\) from the category of \(\mathbf {G}(\mathbb {C})\)equivariant vector bundles on the compact dual \(\mathbb {P}_{\mathbb {C}}^{2}\) of \({\text {Sh}}_{K}\) to the category of vector bundles with \(\mathbf {G}(\mathbb {A}_{f})\)action on the inverse system of Shimura varieties \({\text {Sh}}_{K}\). Here \(\mathbb {P}_{\mathbb {C}}^{2}=\mathbf {G}(\mathbb {C})/\mathbf {H}(\mathbb {C})\), where \(\mathbf {H}(\mathbb {C})\) is the parabolic group stabilizing the line \(\mathbb {C\cdot }\) \(^{t}(\delta /2,0,1)\) in \(\mathbf {G}(\mathbb {C})=GL_{3}(\mathbb {C})\times \mathbb {C}^{\times }\), and the irreducible \(\mathcal {V}\) are associated with highest weight representations of the Levi factor \(\mathbf {L}(\mathbb {C})\) of \(\mathbf {H}(\mathbb {C})\). It is straightforward to check that \(\det \mathcal {P}\) and \(\mathcal {L}\) are associated with the same character of \(\mathbf {L}(\mathbb {C})\), up to a twist by a character of \(\mathbf {G}(\mathbb {C})\), which affects the \(\mathbf {G}(\mathbb {A}_{f})\)action (hence the normalization of Hecke operators), but not the structure of the line bundles themselves. The functoriality of Harris’ construction implies that \(\det \mathcal {P}\) and \(\mathcal {L}\) are isomorphic also algebraically.
We de not know whether \(\det \mathcal {P}\) and \(\mathcal {L}\) are isomorphic as algebraic line bundles over S. This would be equivalent, by (1.84), to the statement that for every PEL structure \((A,\lambda ,\iota ,\alpha )\in \mathcal {M}(R)\), for any \(R_{0}\)algebra R, \(\det (H_{dR}^{1}(A/R)(\varSigma ))\) is the trivial line bundle on Spec(R). To our regret, we have not been able to establish this, although a similar statement in the “Siegel case”, namely that for any principally polarized abelian scheme \((A,\lambda )\) over R, \(\det H_{dR}^{1}(A/R)\) is trivial, follows at once from the Hodge filtration (1.81). Our result, however, suffices to guarantee the following corollary, which is all that we will be using in the sequel.
Corollary 1.18
For any characteristic p geometric point \(\text {Spec}(k)\rightarrow \text {Spec}(R_{0})\), we have \(\det \mathcal {P}\simeq \mathcal {L}\) on \({\bar{S}}_{k}\). A similar statement holds for morphisms \(\text {Spec}W(k)\rightarrow \text {Spec}(R_{0})\).
Proof
Since \({\bar{S}}\) is a regular scheme, \(\det \mathcal {P}\otimes \mathcal {L}^{1}\simeq {\mathcal {O}}(D)\) for a Weil divisor D supported on vertical fibers over \(R_{0}\). Since any connected component Z of \({\bar{S}}_{k}\) is irreducible, we can modify D so that D and Z are disjoint, showing that \(\det \mathcal {P}\otimes \mathcal {L}^{1}_{Z}\) is trivial. The second claim is proved similarly.
1.7.4 Modular forms

\(f(\underline{A},\lambda \omega )=\lambda ^{k}f(\underline{A},\omega )\) for every \(\lambda \in H^{0}(T,{\mathcal {O}}_{T})^{\times }\)

The “rule” f is compatible with base change \(T^{\prime }/T\).
Let \(R\rightarrow R^{\prime }\) be a homomorphism of \(R_{0}\)algebras. Then Bellaïche proved the following theorem ([2], 1.1.5).
Theorem 1.19
Bellaïche considers only weights divisible by 3, but his proofs generalize to all k (cf remark on the bottom of p. 43 in [2]).
1.8 The Kodaira–Spencer isomorphism
Lemma 1.20
Proof
The first claim follows from the fact that the Gauss–Manin connection commutes with the endomorphisms, hence preserves CM types. The second claim is a consequence of the symmetry of the polarization, see [11], Prop. 9.1 on p.81 (in the Siegel modular case).
Observe that \(\omega _{A}(\varSigma )\otimes _{{\mathcal {O}}_{S}}\omega _{A^{t}}(\varSigma )\), as well as \(\omega _{A}({\bar{\varSigma }})\otimes _{{\mathcal {O}}_{S}}\omega _{A^{t}}({\bar{\varSigma }})\), are vector bundles of rank 2.
Lemma 1.21
Proof
This is well known and follows from deformation theory. For a selfcontained proof, see [2], Prop. II.2.1.5.
Proposition 1.22
Proof
We need only use \(\lambda ^{*}\) to identify \(\omega _{{\mathcal {A}}^{t}}(\varSigma )\) with \(\omega _{{\mathcal {A}}}({\bar{\varSigma }})\).
We refer to Corollary 1.29 for an extension of this result to \({\bar{S}}\).
Corollary 1.23
There is an isomorphism of line bundles \(\mathcal {L}^{3}\simeq \varOmega _{S}^{2}\).
Proof
Take determinants and use \(\det \mathcal {P}\simeq \mathcal {L}\). We emphasize that while \(\text {KS}(\varSigma )\) is canonical, the identification of \(\det \mathcal {P}\) with \(\mathcal {L}\) depends on a choice, which we shall fix later on once and for all.
The last corollary should be compared to the case of the open modular curve Y(N), where the square of the Hodge bundle \(\omega _{\mathcal {E}}\) of the universal elliptic curve becomes isomorphic to \(\varOmega _{Y(N)}^{1}\). Over \(\mathbb {C}\), as the isomorphism between \(\mathcal {L}^{3}\) and \(\varOmega _{S}^{2}\) takes \(\text {d}\zeta _{3}^{\otimes 3}\) to a constant multiple of \(\text {d}z\wedge \text {d}u\) (see Corollary 1.31), the differential form corresponding to a modular form ( \(f_{j})_{1\le j\le m}\) of weight 3, is (up to a constant) \((f_{j}(z,u)\text {d}z\wedge \text {d}u)_{1\le j\le m}\).
1.9 Extensions to the boundary of S
1.9.1 The vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) over C
Let \(E\subset C_{R_{N}}\) be a connected component of the cuspidal divisor (over the integral closure \(R_{N}\) of \(R_{0}\) in the ray class field \({\mathcal {K}}_{N})\). As we have seen, E is an elliptic curve with CM by \({\mathcal {O}}_{{\mathcal {K}}}\). If the cusp at which E sits is of type \((\mathfrak {a},B)\) (\(\mathfrak {a}\) an ideal of \({\mathcal {O}}_{{\mathcal {K}}}\), B an elliptic curve with CM by \({\mathcal {O}}_{{\mathcal {K}}}\) defined over \(R_{N})\), then E maps via an isogeny to \(\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}=\text {Ext}_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(B,\mathfrak {a}\otimes \mathbb {G}_{m})\). In particular, E and B are isogenous over \({\mathcal {K}}_{N}\).
Consider \(\mathcal {G}\), the universal semiabelian \({\mathcal {O}}_{{\mathcal {K}}} \)threefold of type \((\mathfrak {a},B)\), over \(\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\). The semiabelian scheme \({\mathcal {A}}\) over E is the pullback of this \(\mathcal {G}\). Clearly, \(\omega _{{\mathcal {A}}/E}=\mathcal {P}\oplus \mathcal {L}\) and \(\mathcal {P}=\omega _{{\mathcal {A}}/E}(\varSigma )\) admits over E a canonical rank 1 subbundle \(\mathcal {P}_{0}=\omega _{B}\). Since the toric part and the abelian part of \(\mathcal {G}\) are constant, \(\mathcal {L},\mathcal {P}_{0}\) and \(\mathcal {P}_{\mu }=\mathcal {P}/\mathcal {P}_{0}\) are all trivial line bundles when restricted to E. It can be shown that \(\mathcal {P}\) itself is not trivial over E.
1.9.2 More identities over \({\bar{S}}\)
We have seen that \(\varOmega _{S}^{2}\simeq \mathcal {L}^{3}\). For the following proposition, compare [2], Lemme II.2.1.7.
Proposition 1.24
Proof
1.10 Fourier–Jacobi expansions
1.10.1 The infinitesimal retraction
1.10.2 Arithmetic Fourier–Jacobi expansions
To understand the structure of \(H^{0}(C,{\mathcal {O}}_{{\widehat{S}}})\) let \(\mathcal {I}\subset {\mathcal {O}}_{{\bar{S}}}\) be the sheaf of ideals defining C, so that \(C^{(n)}\) is defined by \(\mathcal {I}^{n}\). The conormal sheaf \(\mathcal {N}=\mathcal {I}/\mathcal {I}^{2}\) is the restriction \(i^{*}{\mathcal {O}}_{{\bar{S}}}(C)\) of \(\mathcal {I}={\mathcal {O}}_{{\bar{S}}}(C)\) to C. It is an ample invertible sheaf on C, since (over \(R_{N}\)) its degree on each component \(E_{j}\) is \(E_{j}^{2}>0\).
1.10.3 Fourier–Jacobi expansions over \(\mathbb {C}\)
Lemma 1.25
The infinitesimal retraction \(r:E^{(n)}\rightarrow E\) coincides with the map induced by the geodesic retraction (1.15).
Proof
To prove the lemma, note that the infinitesimal retraction \(r:E^{(n)}\rightarrow E\) is uniquely characterized by the fact that the \({\mathcal {O}}_{{\mathcal {K}}}\)semiabelian variety \({\mathcal {A}}_{x}=x^{*}{\mathcal {A}}\) at any point \(x:\text {Spec}(R)\rightarrow E^{(n)}\) is equal to \({\mathcal {A}}_{r\circ x}\) (an equality respecting the PEL structures). See [2], II.2.4.2. The computations of Sect. 1.6 show that the same is true for the infinitesimal retraction obtained from the geodesic retraction. We conclude that the two retractions agree on the level of “truncated Taylor expansions”.
Consider now a modular form of weight k and level N over \(\mathbb {C}\), \(f\in M_{k}(N,\mathbb {C})\). Using the trivialization of \(\mathcal {L}_{an}\) over the symmetric space \(\mathfrak {X}\) given by \(2\pi i\cdot \text {d}\zeta _{3}\) as discussed in Sect. 1.7.2, we identify f with a collection of functions \(f_{j}\) on \(\mathfrak {X}\), transforming under \(\varGamma _{j}\) according to the automorphy factor \(j(\gamma ;z,u)^{k}\). As usual, we look at \(\varGamma =\varGamma _{1}\) only, and at the expansion of \(f=f_{1} \) at the standard cusp \(c_{\infty }\), the other cusps being in principle similar. On the arithmetic FJ expansion side, this means that we concentrate on one connected component E of C, which lies on the connected component of \(S_{\mathbb {C}}\) corresponding to \(g_{1}=1\). It also means that as the section s used to trivialize \(\mathcal {L}\) along E, we must use a section that, analytically, coincides with \(2\pi i\cdot \text {d}\zeta _{3}\).
1.11 The Gauss–Manin connection in a neighborhood of a cusp
1.11.1 A computation of \(\nabla \) in the complex model
We shall now compute the Gauss–Manin connection in the complex model near the standard cusp \(c_{\infty }\). Recall that we use the coordinates \((z,u,\zeta _{1},\zeta _{2},\zeta _{3})\) as in Sect. 1.2.4. Here \(\text {d}\zeta _{1}\) and \(\text {d}\zeta _{2}\) form a basis for \(\mathcal {P}\) and \(\text {d}\zeta _{3}\) for \(\mathcal {L}\). The same coordinates served to define also the semiabelian variety \(\mathcal {G}_{u}\) (denoted also \({\mathcal {A}}_{u}\)) over the cuspidal component E at \(c_{\infty }\), cf Sect. 1.6. As explained there (1.69), the projection to the abelian part is given by the coordinate \(\zeta _{1}\) (modulo \({\mathcal {O}}_{{\mathcal {K}}}\)), so \(\text {d}\zeta _{1}\) is a basis for the sublinebundle of \(\omega _{{\mathcal {A}}/E}\) coming from the abelian part, which was denoted \(\mathcal {P}_{0}\). In Sect. 1.10.1, it is explained how to extend the filtration \(\mathcal {P}_{0}\subset \mathcal {P}\) canonically to the formal neighborhood \({\widehat{S}}\) of E using the retraction r, by pulling back from the boundary. It was also noted that complex analytically, the retraction r is the germ of the geodesic retraction introduced earlier. From the analytic description of the degeneration of \({\mathcal {A}}_{(z,u)}\) along a geodesic, it becomes clear that \(\mathcal {P}_{0}=r^{*}(\mathcal {P}_{0}_{E})\) is just the line bundle \({\mathcal {O}}_{{\widehat{S}}}\cdot \text {d}\zeta _{1}\subset \omega _{{\mathcal {A}}/{\widehat{S}}}\). It follows that \(\mathcal {P}_{\mu }={\mathcal {O}}_{{\widehat{S}}}\cdot \text {d}\zeta _{2}\mod \mathcal {P}_{0}\).
1.11.2 A computation of KS in the complex model
To perform the computation, we need two lemmas.
Lemma 1.26
Proof
This is an easy computation using the transition map T between L and \(L_{x}^{\prime }\) and the fact that on L the Riemann form is the alternating form \(\left\langle ,\right\rangle =\text {Im}_{\delta }(,)\).
For the formulation of the next lemma recall that if A is a complex abelian variety, a polarization \(\lambda :A\rightarrow A^{t}\) induces an alternating form \(\left\langle ,\right\rangle _{\lambda }\) on \(H_{dR}^{1}(A)\) as well as a Riemann form on the integral homology \(H_{1}(A,\mathbb {Z})\). We compare the two.
Lemma 1.27
Let \((A,\lambda )\) be a principally polarized complex abelian variety. If \(\alpha _{1},\dots ,\alpha _{2g}\) is a symplectic basis for \(H_{1}(A,\mathbb {Z})\) in which the associated Riemann form is given by a matrix J, and \(\beta _{1},\dots ,\beta _{2g}\) is the dual basis of \(H_{dR}^{1}(A)\), then the matrix of the bilinear form \(\left\langle ,\right\rangle _{\lambda }\) on \(H_{dR}^{1}(A)\) in the basis \(\beta _{1},\dots ,\beta _{2g}\) is \((2\pi i)^{1}J\).
Proof
Proposition 1.28
Corollary 1.29
The Kodaira–Spencer isomorphism \(\mathcal {P}\otimes \mathcal {L}\simeq \varOmega _{S}^{1}\) extends meromorphically over \({\bar{S}}\). Moreover, in a formal neighborhood \({\widehat{S}}\) of C, its restriction to the line subbundle \(\mathcal {P}_{0}\otimes \mathcal {L}\) is holomorphic, and on any direct complement of \(\mathcal {P}_{0}\otimes \mathcal {L}\) in \(\mathcal {P}\otimes \mathcal {L}\), it has a simple pole along C.
Proof
As we have seen, \(\text {d}\zeta _{1}\otimes \text {d}\zeta _{3}\) and \(\text {d}\zeta _{2}\otimes \text {d}\zeta _{3}\) define a basis of \(\mathcal {P}\otimes \mathcal {L}\) at the boundary, with \(\text {d}\zeta _{1}\otimes \text {d}\zeta _{3}\) spanning the line subbundle \(\mathcal {P}_{0}\otimes \mathcal {L}\). On the other hand \(\text {d}u\) is holomorphic there, while \(\text {d}z\) has a simple pole along the boundary.
Corollary 1.30
Proof
As we have seen, d\(\zeta _{1}\) is a basis for \(\mathcal {P}_{0}\).
Corollary 1.31
The isomorphism \(\mathcal {L}^{3}\simeq \varOmega _{S}^{2}\) maps \(\mathrm{d}\zeta _{3}^{\otimes 3}\) to a constant multiple of \(\mathrm{d}z\wedge \mathrm{d}u\).
Proof
The isomorphism \(\det \mathcal {P}\simeq \mathcal {L}\) carries \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) to a constant multiple of \(\text {d}\zeta _{3}\), so the corollary follows from (1.141).
1.11.3 Transferring the results to the algebraic category
The computations leading to (1.141) of course descend (still in the analytic category) to \(S_{\mathbb {C}}\), because they are local in nature. They then hold a fortiori in the formal completion \({\widehat{S}}_{\mathbb {C}}\) along the cuspidal component E. Recall that the sections \(\text {d}\zeta _{1},\text {d}\zeta _{3}\) and \(\text {d}\zeta _{2} \mod \left\langle \text {d}\zeta _{1}\right\rangle \) (respectively, \(\text {d}u\) and \(\text {d}z\mod \left\langle \text {d}u\right\rangle \)) are well defined in \({\widehat{S}}_{\mathbb {C}}\), because as global sections defined over \(\mathfrak {X}\) they are invariant under \(\varGamma _\mathrm{cusp}\) (see Lemma 1.8). But the Gauss–Manin and Kodaira–Spencer maps are defined algebraically on S, and both \(\varOmega _{{\widehat{S}}}^{1}\) and \(\omega _{{\mathcal {A}}/{\widehat{S}}}\) are flat over \(R_{0}\), so from the validity of the formulae over \(\mathbb {C}\) we deduce their validity in \({\widehat{S}}\) over \(R_{0}\), provided we identify the differential forms figuring in them (suitably normalized) with elements of \(\varOmega _{{\widehat{S}}}^{1}\) and \(\omega _{{\mathcal {A}}/{\widehat{S}}}\) defined over \(R_{0}\). In particular, they hold in the characteristic p fiber as well.
Finally, although we have done all the computations at one specific cusp, it is clear that similar computations hold at any other cusp.
1.12 Fields of rationality
1.12.1 Rationality of local sections of \(\mathcal {P}\) and \(\mathcal {L}\)
We have compared the arithmetic surface S with the complex analytic surfaces \(\varGamma _{j}\backslash \mathfrak {X}\) (\(1\le j\le m\)), and the compactifications of these two models. We have also compared the universal semiabelian scheme \({\mathcal {A}}\) and the automorphic vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) in both models. In this section, we want to compare the local parameters obtained from the two presentations, and settle the question of rationality. For simplicity, we shall work rationally and not integrally, which is all we need. In order to work integrally, one would have to study degeneration and periods of abelian varieties integrally, which is more delicate, see [28], Ch.I, Sections 3,4.
We shall need to look at local parameters at the cusps, and as the cusps are defined only over \({\mathcal {K}}_{N}\), we shall work with \(S_{{\mathcal {K}}_{N}}\) instead of \(S_{{\mathcal {K}}}\). With a little more care, working with Galois orbits of cusps, we could probably prove rationality over \({\mathcal {K}}\), but for our purpose \({\mathcal {K}}_{N}\) is good enough.
If \(\xi \) and \(\eta \) belong to a \({\mathcal {K}}_{N}\)module, we write \(\xi \sim \eta \) to mean that \(\eta =c\xi \) for some \(c\in {\mathcal {K}}_{N}^{\times }\).
Proposition 1.32
 (i)
\(2\pi i\cdot \mathrm{d}\zeta _{3}\in H^{0}({\widehat{S}}_{{\mathcal {K}}_{N}},\mathcal {L})\). In other words, this section is \({\mathcal {K}}_{N}\)rational.
 (ii)
Similarly \(2\pi i\cdot \mathrm{d}\zeta _{2}\) projects (modulo \(\mathcal {P}_{0}\)) to a \({\mathcal {K}}_{N}\)rational section of \(\mathcal {P}_{\mu }.\)
 (iii)
Let B be the elliptic curve over \({\mathcal {K}}_{N}\) associated with the cusp \(c_{\infty }\) as in Sect. 1.5.1. Let \(\varOmega _{B}\in \mathbb {C}^{\times }\) be a period of a basis \(\omega \) of \(\omega _{B}=H^{0}(B,\varOmega _{B/{\mathcal {K}}_{N}}^{1})\) (i.e. the lattice of periods of \(\omega \) is \(\varOmega _{B}\cdot {\mathcal {O}}_{{\mathcal {K}}}\)). This \(\varOmega _{B}\) is well defined up to an element of \({\mathcal {K}}_{N}^{\times }.\) Then \(\varOmega _{B}\cdot \text {d}\zeta _{1}\in H^{0}({\widehat{S}}_{{\mathcal {K}}_{N}},\mathcal {P}_{0})\) is \({\mathcal {K}}_{N}\)rational.
Proof
Let E be the component of \(C_{{\mathcal {K}}_{N}}\) which over \(\mathbb {C}\) becomes \(E_{c}.\) Let \(\mathcal {G}\) be the universal semiabelian scheme over E. Then \(\mathcal {G}\) is a semiabelian scheme which is an extension of \(B\times _{{\mathcal {K}}_{N}}E\) by the torus \(({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m,{\mathcal {K}}_{N}})\times _{{\mathcal {K}}_{N}}E.\) At any point \(u\in E(\mathbb {C})\), we have the analytic model \(\mathcal {G}_{u}\) (1.69) for the fiber of \(\mathcal {G}\) at u, but the abelian part and the toric part are constant. Over E, the line bundle \(\mathcal {P}_{0}\) is (by definition) \(\omega _{B\times E/E}.\) As the lattice of periods of a suitable \({\mathcal {K}}_{N}\)rational differential is \(\varOmega _{B}\cdot {\mathcal {O}}_{{\mathcal {K}}}\), while the lattice of periods of \(\text {d}\zeta _{1}\) is \({\mathcal {O}}_{{\mathcal {K}}}\), part (iii) follows. For parts (i) and (ii), observe that the toric part of \(\mathcal {G}\) is in fact defined over \({\mathcal {K}}\) and that \(e_{{\mathcal {O}}_{{\mathcal {K}}}}^{*}\) maps the cotangent space of \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m,{\mathcal {K}}}\) isomorphically to the \({\mathcal {K}}\)span of \(2\pi id\zeta _{2}\) and \(2\pi id\zeta _{3}\).
Corollary 1.33
\(\varOmega _{B}\cdot \sigma _{an}\) is a nowhere vanishing global section of \(\det \mathcal {P}\otimes \mathcal {L}^{1}\) over \(S_{\varGamma }\), rational over \({\mathcal {K}}_{N}.\)
Proof
Recall that we denote by \(S_{\varGamma }\) the connected component of \(S_{{\mathcal {K}}_{N}}\) whose associated analytic space is the complex manifold \(X_{\varGamma }.\) We have seen that as an analytic section \(\varOmega _{B}\cdot \sigma _{an}\) descends to \(X_{\varGamma }\) and extends to the smooth compactification \({\bar{X}}_{\varGamma }.\) By GAGA, it is algebraic. Since \({\bar{X}}_{\varGamma }\) is connected, to check its field of definition, it is enough to consider it at one of the cusps. By the Proposition, its restriction to the formal neighborhood of \(E_{c}\) (\(c=c_{\infty }\)) is defined over \({\mathcal {K}}_{N}.\)
The complex periods \(\varOmega _{B}\) (and their powers) appear as the transcendental parts of special values of Lfunctions associated with Grossencharacters of \({\mathcal {K}}\). They are therefore instrumental in the construction of padic Lfunctions on \({\mathcal {K}}.\) We expect them to appear in the padic interpolation of holomorphic Eisenstein series on the group \(\mathbf {G}\), much as powers of \(2\pi i\) (values of \(\zeta (2k)\)) appear in the padic interpolation of Eisenstein series on \(GL_{2}(\mathbb {Q}).\)
1.12.2 Rationality of local parameters at the cusps
Proposition 1.34
(i) The section \(2\pi i\cdot \mathrm{d}z\) mod \(\left\langle \mathrm{d}u\right\rangle \) is \({\mathcal {K}}_{N}\)rational, i.e. it is the analytification of a section of \(r^{*}\mathcal {N}\). (ii) The section \(\varOmega _{B}\cdot \mathrm{d}u\) is \({\mathcal {K}}_{N}\)rational, i.e. belongs to \(H^{0}(E,\varOmega _{E/{\mathcal {K}}_{N}}^{1}).\)
Proof
Remark 1.1
The parameter q is not a welldefined parameter at x and depends not only on x, but also on the point u used to uniformize it. If we change u to \(u+s\) (\(s\in \Lambda \)), then q is multiplied by the factor \(e^{2\pi i\delta {\bar{s}}(u+s/2)/M}\), so although \({\mathcal {O}}_{{\bar{S}}_{\mathbb {C}},x}^{hol}\subset {\widehat{{\mathcal {O}}}}_{{\bar{S}}_{\mathbb {C}},x}\) and analytic parameters may be considered as formal parameters, the question whether q itself is \({\mathcal {K}}_{N}\)rational is not well defined (in sharp contrast to the case of modular curves!).
1.12.3 Normalizing the isomorphism \(\det \mathcal {P}\simeq \mathcal {L}\)
2 Picard modular schemes modulo an inert prime
2.1 The stratification
2.1.1 The three strata
Recall that an abelian variety over an algebraically closed field of characteristic p is called supersingular if the Newton polygon of its pdivisible group has a constant slope 1 / 2. It is called superspecial if it is isomorphic to a product of supersingular elliptic curves. The following theorem combines various results proved in [4, 39, 40]. See also [8], Theorem 2.1.
Theorem 2.1
 (i)
There exists a closed reduced 1dimensional subscheme \(S_\mathrm{ss}\subset {\bar{S}}\), disjoint from the cuspidal divisor (i.e. contained in S), which is uniquely characterized by the fact that for any geometric point x of S, the abelian variety \({\mathcal {A}}_{x}\) is supersingular if and only if x lies on \(S_\mathrm{ss}\). The scheme \(S_\mathrm{ss}\) is defined over \(\kappa _{0}.\)
 (ii)Let \(S_\mathrm{ssp}\) be the singular locus of \(S_\mathrm{ss}.\) Then x lies in \(S_\mathrm{ssp}\) if and only if \({\mathcal {A}}_{x}\) is superspecial. If \(x\in S_\mathrm{ssp}\), then$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{S_\mathrm{ss},x}\simeq \kappa [[u,v]]/\left( u^{p+1}+v^{p+1}\right) . \end{aligned}$$(2.2)
 (iii)Assume that N is large enough (depending on p). Then the irreducible components of \(S_\mathrm{ss}\) are nonsingular and in fact are all isomorphic to the Fermat curve \(\mathcal {C}_{p}\) given by the equationThere are \(p^{3}+1\) points of \(S_\mathrm{ssp}\) on each irreducible component and through each such point pass \(p+1\) irreducible components. Any two irreducible components are either disjoint or intersect transversally at a unique point.$$\begin{aligned} x^{p+1}+y^{p+1}+z^{p+1}=0. \end{aligned}$$(2.3)
 (iv)
Without the assumption of N being large (but under \(N\ge 3\) as usual), the irreducible components of \(S_\mathrm{ss}\) may have multiple intersections with each other, including selfintersections. Their normalizations are nevertheless still isomorphic to \(\mathcal {C}_{p}.\)
We call \({\bar{S}}_{\mu }={\bar{S}}\backslash S_\mathrm{ss}\) (or \(S_{\mu }={\bar{S}}_{\mu }\cap S\)) the \(\mu \) ordinary or generic locus, \(S_\mathrm{gss}=S_\mathrm{ss}\backslash S_\mathrm{ssp}\) the general supersingular locus, and \(S_\mathrm{ssp}\) the superspecial locus. Then \({\bar{S}}={\bar{S}}_{\mu }\cup S_\mathrm{gss}\cup S_\mathrm{ssp}\) is a stratification.
2.1.2 The pdivisible group
Let \(x:\text {Spec}(k)\rightarrow S\) (k an algebraically closed field) be a geometric point of S, \({\mathcal {A}}_{x}\) the corresponding fiber of \({\mathcal {A}}\), and \({\mathcal {A}}_{x}(p)\) its pdivisible group. Let \(\mathfrak {G} \) be the pdivisible group of a supersingular elliptic curve over k (the group denoted by \(G_{1,1}\) in the ManinDieudonné classification). The following theorem can be deduced from [4, 39].
Theorem 2.2
 (i)If \(x\in S_{\mu }\), then$$\begin{aligned} {\mathcal {A}}_{x}(p)\simeq ({\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{\infty }})\times \mathfrak {G}\times ({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {Q}_{p}/\mathbb {Z}_{p}). \end{aligned}$$(2.4)
 (ii)
If \(x\in S_\mathrm{ss}\), then \({\mathcal {A}}_{x}(p)\) is isogenous to \(\mathfrak {G}^{3}\), and \(x\in S_\mathrm{ssp}\) if and only if the two groups are isomorphic.
2.2 New relations between \(\mathcal {P}\) and \(\mathcal {L}\) in characteristic p
For proofs and more details on this subsection, see [8], Section 2.2.
2.2.1 The line bundles \(\mathcal {P}_{0}\) and \(\mathcal {P}_{\mu }\) over \({\bar{S}}_{\mu }\)
2.2.2 Frobenius and Verschiebung
Let \({\mathcal {A}}^{(p)}={\mathcal {A}}\times _{{\bar{S}},\varPhi }{\bar{S}}\) be the base change of \({\mathcal {A}}\) with respect to the absolute Frobenius morphism \(\varPhi \) of degree p of \({\bar{S}}.\) The relative Frobenius is an \({\mathcal {O}}_{{\bar{S}}}\)linear isogeny \(Frob_{{\mathcal {A}}}:{\mathcal {A}}\rightarrow {\mathcal {A}}^{(p)}\), characterized by the fact that \(pr_{1}\circ Frob_{{\mathcal {A}}}\) is the absolute Frobenius morphism of \({\mathcal {A}}.\) Over S (but not over the boundary C), we have the dual abelian scheme \({\mathcal {A}}^{t}\), and the Verschiebung \(Ver_{{\mathcal {A}}}:{\mathcal {A}}^{(p)}\rightarrow {\mathcal {A}}\) is the \({\mathcal {O}}_{S}\)linear isogeny which is dual to \(Frob_{{\mathcal {A}}^{t}}:{\mathcal {A}}^{t}\rightarrow ({\mathcal {A}}^{t})^{(p)}.\)
Proposition 2.3
Recall that over any base scheme in characteristic p, and for any line bundle \(\mathcal {M}\), its base change \(\mathcal {M}^{(p)}\) under the absolute Frobenius is canonically isomorphic to its pth power \(\mathcal {M}^{p}.\)
Corollary 2.4
Corollary 2.5
Over \({\bar{S}}_{\mu }\), \(\mathcal {L}^{p^{2}1},\mathcal {P}_{\mu }^{p^{2}1}\) and \(\mathcal {P}_{0}^{p+1}\) are trivial line bundles.
2.2.3 Extending the filtration on \(\mathcal {P}\) over \(S_\mathrm{gss}\)
In order to determine to what extent the filtration on \(\mathcal {P}\) and the relation between \(\mathcal {L}\) and the two graded pieces of the filtration extend into the supersingular locus, we have to employ Dieudonné theory. The following is proved in [8].
Proposition 2.6
 (i)
Let \(\mathcal {P}_{0}=\ker V_{\mathcal {P}}\) (this agrees with what was denoted by \(\mathcal {P}_{0}\) over \({\bar{S}}_{\mu }\)). Then outside \(S_\mathrm{ssp}\), \(V(\mathcal {P})=\mathcal {L}^{(p)}\) and \(\mathcal {P}_{0}\) is a rank 1 submodule.
 (ii)
Let \(\mathcal {P}_{\mu }=\mathcal {P}/\mathcal {P}_{0}.\) Then outside \(S_\mathrm{ssp}\) we have \(\mathcal {P}_{\mu }\simeq \mathcal {L}^{p}\) and \(\mathcal {P}_{0}\simeq \mathcal {L}^{1p}\).
For \(V_{\mathcal {L}}\), we similarly get the following.
Proposition 2.7
Outside \(S_\mathrm{ssp}\), \(V_{\mathcal {L}}\) maps \(\mathcal {L}\) injectively onto a sublinebundle of \(\mathcal {P}^{(p)}.\)
At a superspecial point, both \(V_{\mathcal {P}}\) and \(V_{\mathcal {L}}\) vanish.
2.2.4 The Hasse invariant
As we have just seen, the fact that \(V_{\mathcal {P}}\) and \(V_{\mathcal {L}}\) are both of rank 1 “extends” across the general supersingular locus \(S_\mathrm{gss}.\) However, while \(\text {Im}(V_{\mathcal {L}})\) and \(\ker (V_{\mathcal {P}}^{(p)})=\mathcal {P}_{0}^{(p)}\) made up a frame of \(\mathcal {P}\) over \({\bar{S}}_{\mu }\), over \(S_\mathrm{gss}\) these two line bundles coincide. To state a more precise result, we make the following definition.
Definition 2.8
Theorem 2.9
The divisor of \(h_{{\bar{\varSigma }}}\) is \(S_\mathrm{ss}\) (with its reduced subscheme structure).
2.3 The open Igusa surfaces
2.3.1 The Igusa scheme

\(\mathcal {M}_{Ig(p^{n})}(R)\) is the set of isomorphism classes of pairs \((\underline{A},\varepsilon )\) where \(\underline{A}\in \mathcal {M}(R)\) andis a closed immersion of \({\mathcal {O}}_{{\mathcal {K}}}\)group schemes over R.$$\begin{aligned} \varepsilon :\delta _{{\mathcal {K}}}^{1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\hookrightarrow A[p^{n}] \end{aligned}$$(2.16)
Proposition 2.10
The morphism \(\tau :Ig_{\mu }(p^{n})\rightarrow S_{\mu }\) is finite and étale, with the Galois group \(\varDelta (p^{n})=({\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}})^{\times }\) acting as a group of deck transformations.
Proof
2.3.2 A compactification over the cusps
The proof of the following proposition mimics the construction of \({\bar{S}}.\) We omit it.
Proposition 2.11
Let \(\overline{Ig}_{\mu }(p^{n})\) be the normalization of \({\bar{S}}_{\mu }={\bar{S}}\backslash S_\mathrm{ss}\) in \(Ig_{\mu }(p^{n}).\) Then, \(\overline{Ig}_{\mu }(p^{n})\rightarrow {\bar{S}}_{\mu }\) is finite étale and the action of \(\varDelta (p^{n})\) extends to it. The boundary \(\overline{Ig}_{\mu }(p^{n})\backslash Ig_{\mu }(p^{n})\) is noncanonically identified with \(\varDelta (p^{n})\times C.\)
We define similarly \(Ig_{\mu }^{*}\), and note that it is finite étale over \(S_{\mu }^{*}.\)
Proposition 2.12
2.3.3 A trivialization of \(\mathcal {L}\) over the Igusa surface
Proposition 2.13
The line bundles \(\mathcal {L}\), \(\mathcal {P}_{0}\) and \(\mathcal {P}_{\mu }\) are trivial over \(\overline{Ig}_{\mu }.\)
Proof
Use \(\varepsilon ^{*}\) as an isomorphism between vector bundles and note that \(\mathcal {L}=\omega _{{\mathcal {A}}}^{\mu }({\bar{\varSigma }})\) and \(\mathcal {P}_{\mu }=\omega _{{\mathcal {A}}}^{\mu }(\varSigma ).\) The relation \(\mathcal {P}_{0}\otimes \mathcal {P}_{\mu }=\det \mathcal {P}\simeq \mathcal {L}\) implies the triviality of \(\mathcal {P}_{0}\) as well.
Note that the trivialization of \(\mathcal {L}\) and \(\mathcal {P}_{\mu }\) is canonical, because it uses only the tautological map \(\varepsilon \) which exists over the Igusa scheme. The trivialization of \(\mathcal {P}_{0}\) on the other hand depends on how we realize the isomorphism \(\det \mathcal {P}\simeq \mathcal {L}.\)
2.4 Compactification of the Igusa surface along the supersingular locus
2.4.1 Extracting a \(p^{2}1\) root from \(h_{{\bar{\varSigma }}}\) over \(\overline{Ig}_{\mu }\)
Proposition 2.14
 (i)Let \(\gamma \in \varDelta (p)=\mathcal {(O}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}})^{\times }.\) Then \(\varDelta (p)\) acts on \(H^{0}(\overline{Ig}_{\mu },\mathcal {L)}\) and$$\begin{aligned} \gamma ^{*}a={\bar{\varSigma }}(\gamma )^{1}\cdot a. \end{aligned}$$(2.25)
 (ii)The section a is a \(p^{2}1\) root of the Hasse invariant over \(\overline{Ig}_{\mu }\), i.e.$$\begin{aligned} a\left( p^{2}1\right) =h_{{\bar{\varSigma }}}. \end{aligned}$$(2.26)
Proof
 (i)This part is a restatement of the action of \(\varDelta (p).\) At two points of \(Ig_{\mu }(R)\) lying over the same point of \(S_{\mu }(R)\) and differing by the action of \(\gamma \in \varDelta (p)\), the canonical embeddingsdiffer by \(\iota (\gamma )\) (2.17). The induced trivializations of \(Lie(A)({\bar{\varSigma }})\) differ by \({\bar{\varSigma }}(\gamma )\) and by duality we get (i).$$\begin{aligned} \delta _{{\mathcal {K}}}^{1}\otimes \mu _{p}\hookrightarrow A[p] \end{aligned}$$(2.27)
 (ii)Since over any \(\mathbb {F}_{p}\)base, \(Ver_{\mathbb {G}_{m}}=1\), we have a commutative diagramUsing the isomorphism \(Lie({\mathcal {A}})({\bar{\varSigma }})^{(p^{2})}\simeq Lie({\mathcal {A}})({\bar{\varSigma }})^{p^{2}}\), we get the commutative diagram$$\begin{aligned} \begin{array}{lll} Lie({\mathcal {A}})({\bar{\varSigma }})^{(p^{2})} &{} \overset{V_{*}^{2}}{\rightarrow } &{} Lie({\mathcal {A}})({\bar{\varSigma }}) \\ \downarrow a^{(p^{2})} &{} &{} \downarrow a \\ \delta _{{\mathcal {K}}}^{1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) &{} = &{} \delta _{{\mathcal {K}}}^{1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) \end{array} . \end{aligned}$$(2.28)from which we deduce that \(h_{{\bar{\varSigma }}}=a(p^{2}1)\).$$\begin{aligned} \begin{array}{lll} Lie({\mathcal {A}})({\bar{\varSigma }})^{p^{2}} &{} \overset{h_{{\bar{\varSigma }}}}{\rightarrow } &{} Lie({\mathcal {A}})({\bar{\varSigma }}) \\ \downarrow a(p^{2}) &{} &{} \downarrow a \\ \delta _{{\mathcal {K}}}^{1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) &{} = &{} \delta _{{\mathcal {K}}}^{1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) \end{array} , \end{aligned}$$(2.29)
2.4.2 The compactification \(\overline{Ig}\) of \(\overline{Ig}_{\mu }\)
The map \(L\rightarrow L^{n}\) is finite flat of degree n and if n is invertible on the base, finite étale is away from the zero section. Indeed, locally on X it is the map \(\mathbb {A}^{1}\times X\rightarrow \mathbb {A}^{1}\times X\) which is just raising to nth power in the first coordinate. By base change, it follows that the same is true for the map \(p:Y\rightarrow X:\) this map is finite flat of degree n and étale away from the vanishing locus of the section s (assuming n is invertible). We remark that if L is the trivial line bundle, we recover usual Kummer theory.
Theorem 2.15
 (i)
It is finite flat of degree \(p^{2}1\), étale over \({\bar{S}}_{\mu }\), totally ramified over \(S_\mathrm{ss}\).
 (ii)
\(\varDelta (p)\) acts on \(\overline{Ig}\) as a group of deck transformations and the quotient is \({\bar{S}}.\)
 (iii)
Let \(s_{0}\in S_\mathrm{gss}(\mathbb {{\bar{F}}}_{p}).\) Then there exist local parameters u, v at \(s_{0}\) such that \({\widehat{{\mathcal {O}}}}_{S,s_{0}}=\mathbb {{\bar{F}}}_{p}[[u,v]]\), \(S_\mathrm{gss}\subset S\) is formally defined by \(u=0\), and if \({\tilde{s}}_{0}\in Ig\) maps to \(s_{0}\) under \(\tau \), then \({\widehat{{\mathcal {O}}}}_{Ig,{\tilde{s}}_{0}}=\mathbb {{\bar{F}}}_{p}[[w,v]]\) where \(w^{p^{2}1}=u.\) In particular, Ig is regular in codimension 1.
 (iv)Let \(s_{0}\in S_\mathrm{ssp}(\mathbb {{\bar{F}}}_{p}).\) Then there exist local parameters u, v at \(s_{0}\) such that \({\widehat{{\mathcal {O}}}}_{S,s_{0}}=\mathbb {{\bar{F}}}_{p}[[u,v]]\), \(S_\mathrm{ss}\subset S\) is formally defined at \(s_{0}\) by \(u^{p+1}+v^{p+1}=0\), and if \({\tilde{s}}_{0}\in Ig\) maps to \(s_{0}\) under \(\tau \), thenIn particular, \({\tilde{s}}_{0}\) is a normal singularity of Ig.$$\begin{aligned} {\widehat{{\mathcal {O}}}}_{Ig,{\tilde{s}}_{0}}=\mathbb {{\bar{F}}}_{p}[[w,u,v]]/\left( w^{p^{2}1}u^{p+1}v^{p+1}\right) \end{aligned}$$(2.36)
2.4.3 Irreducibility of Ig

Let \(q=p^{2}.\) For any r sufficiently large and for any \(\gamma \in ({\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}})^{\times }\) there exists a \(\mu \)ordinary abelian variety with PEL structure \(\underline{A}\in S_{\mu }(\mathbb {F}_{q^{r}})\) such that the image of \(Gal(\mathbb {{\bar{F}}}_{q}/\mathbb {F}_{q^{r}})\) incontains \(\gamma .\)$$\begin{aligned} Aut\left( Isom_{\mathbb {{\bar{F}}}_{q}}(\delta _{{\mathcal {K}}}^{1}\otimes \mu _{p^{n}},A[p^{n}]^{\mu })\right) =({\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}})^{\times } \end{aligned}$$(2.37)
Proposition 2.16
The morphism \(\tau :\overline{Ig}\rightarrow {\bar{S}}\) induces a bijection on irreducible components.
Proof
Since \(\overline{Ig}\) is a normal surface, connected components and irreducible components are the same. Let T be a connected component of \({\bar{S}}\) and \(T_\mathrm{ss}=T\cap S_\mathrm{ss}\). Let \(\tau ^{1}(T)=\coprod Y_{i}\) be the decomposition into connected components. As \(\tau \) is finite and flat, each \(\tau (Y_{i})=T.\) Since \(\tau \) is totally ramified over \(T_\mathrm{ss}\), there is only one \(Y_{i}.\)
3 Modular forms modulo p and the theta operator
3.1 Modular forms mod p as functions on Ig
3.1.1 Representing modular forms by functions on Ig
Lemma 3.1
Proof
Take \(f\in H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)}\), so that \(f\cdot a(k)\in H^{0}(Ig_{\mu },\mathcal {L}^{k})^{(0)}\) and hence descends to \(g\in H^{0}(S_{\mu },\mathcal {L}^{k}).\) This g may have poles along \(S_\mathrm{ss}\), but some \(h_{{\bar{\varSigma }}}^{n}g\) will extend holomorphically to S and hence represents a modular form of weight \(k+n(p^{2}1)\), which will map to f because \(a(k+n(p^{2}1))=h_{{\bar{\varSigma }}}^{n}a(k).\)
Proposition 3.2
Proof
3.1.2 Fourier–Jacobi expansions modulo p
Proposition 3.3
The Fourier–Jacobi expansion \({\widetilde{FJ}}(h_{{\bar{\varSigma }}})\) of the Hasse invariant is 1. Moreover, for \(f_{1}\) and \(f_{2}\) in the graded ring \(M_{*}(N,R)\), \(r(f_{1})=r(f_{2})\) if and only if \({\widetilde{FJ}}(f_{1})={\widetilde{FJ}}(f_{2}).\)
Proof
The first statement is tautologically true. For the second, note that for \(f\in M_{k}(N,R)\), \({\widetilde{FJ}}(f)\) is the (expansion of the) image of f / a(k) in \(H^{0}({\tilde{C}},{\mathcal {O}}_{{\widehat{Ig}}})\) where \({\widehat{Ig}}\) is the formal completion of Ig along \({\tilde{C}}\), while r(f) is the image of f / a(k) in \(H^{0}(\overline{Ig}_{\mu },{\mathcal {O}}).\) The proposition follows from the fact that by Proposition 2.16 the irreducible components of \(\overline{Ig}_{\mu }\) are in bijection with the connected components of \({\bar{S}}\), so every irreducible component of \(\overline{Ig}_{\mu }\) contains at least one cuspidal component (“qexpansion principle”). A function on \(\overline{Ig}_{\mu }\) that vanishes in the formal neighborhood of any cuspidal component must therefore vanish on any irreducible component, so is identically 0.
3.1.3 The filtration of a modular form modulo p
Let \(f\in M_{k}(N,R)\), where R is a \(\kappa _{0}\)algebra as before. Define the filtration \(\omega (f)\) to be the minimal \(j\ge 0\) such that \(r(f)=r(f^{\prime })\) (equivalently \(FJ(f)=FJ(f^{\prime })\)) for some \(f^{\prime }\in M_{j}(N,R).\) The following proposition follows immediately from previous results.
Proposition 3.4
3.2 The theta operator
3.2.1 Definition of \(\varTheta (f)\)
We work over \(\kappa =\mathbb {{\bar{F}}}_{p}\). Let S be the (open) Picard surface over \(\kappa \) and \(Ig=Ig(p)\) the Igusa surface of level p (completed along the supersingular locus as explained above). To simplify the notation, we denote by \(Z=S_\mathrm{ss}=S\backslash S_{\mu }\) the supersingular locus of S, by \({\tilde{Z}}=Ig_\mathrm{ss}=Ig\backslash Ig_{\mu }\) its preimage under the covering map \(\tau :Ig\rightarrow S\), by \(Z^{\prime }=S_\mathrm{gss}=S_\mathrm{ss}\backslash S_\mathrm{ssp}\) the smooth part of Z, and by \({\tilde{Z}}^{\prime }=Ig_\mathrm{gss}=Ig_\mathrm{ss}\backslash Ig_\mathrm{ssp}\) the preimage of \(Z^{\prime } \) under \(\tau .\)
3.2.2 The main theorem
Theorem 3.5
 (i)
The operator \(\varTheta \) maps \(H^{0}(S,\mathcal {L}^{k}) \) to \(H^{0}(S,\mathcal {L}^{k+p+1}).\)
 (ii)The effect of \(\varTheta \) on Fourier–Jacobi expansions is a “Tate twist”. More precisely, letbe the canonical Fourier–Jacobi expansion of f along \({\tilde{E}}\) (thus \(c_{m}(f)\in H^{0}({\tilde{E}},\mathcal {N}^{m})\)). Then$$\begin{aligned} {\widetilde{FJ}}(f)=\sum _{m=0}^{\infty }c_{m}(f) \end{aligned}$$(3.20)Here M (equal to \(ND_{{\mathcal {K}}}\) or \(2^{1}ND_{{\mathcal {K}}}\)) is the width of the cusp.$$\begin{aligned} {\widetilde{FJ}}(\varTheta (f))=M^{1}\sum _{m=0}^{\infty }mc_{m}(f). \end{aligned}$$(3.21)
 (iii)If \(f\in H^{0}(S,\mathcal {L}^{k})\) and \(g\in H^{0}(S,\mathcal {L}^{l})\), then$$\begin{aligned} \varTheta (fg)=f\varTheta (g)+\varTheta (f)g. \end{aligned}$$(3.22)
 (iv)
\(\varTheta (h_{{\bar{\varSigma }}}f)=h_{{\bar{\varSigma }}}\varTheta (f)\) (equivalently, \(\varTheta (h_{{\bar{\varSigma }}})=0\)).
Corollary 3.6
The operator \(\varTheta \) extends to a derivation of the graded ring of modular forms \(\mod p\), and for any f, \(\varTheta (f)\) is a cusp form.
Parts (iii) and (iv) of the theorem are clear from the construction. The proof of (i), that \(\varTheta (f)\) is in fact \(holomorphic \) along \(S_\mathrm{ss}\), will be given in 3.4. We shall now study its effect on Fourier–Jacobi expansions, i.e. part (ii). That a factor like \(M^{1}\) is necessary in (ii) becomes evident if we consider what happens to FJ expansions under level change. If N is replaced by \(N^{\prime }=NQ\), then the conormal bundle becomes the Qth power of the conormal bundle of level \(N^{\prime }\), i.e. \(\mathcal {N}=\mathcal {N}^{\prime Q}\) (see Sect. 1.4.3). It follows that what was the mth FJ coefficient at level N becomes the Qmth coefficient at level \(N^{\prime }.\) The operator \(\varTheta \) commutes with level change, but the factor \(M^{1}\), which changes to \((QM)^{1}\), takes care of this.
3.3 The effect of \(\varTheta \) on FJ expansions
Let E be the standard cuspidal component of \({\bar{S}}\) (over the ring \(R_{N} \)). We have earlier trivialized the line bundle \(\mathcal {L}\) along E in two seemingly different ways, that we must now compare. On the one hand, after reducing modulo \(\mathfrak {P}\) (the prime of \(R_{N}\) above p fixed above) and pulling \(\mathcal {L}\) back to the Igusa surface, we got a canonical nowhere vanishing section a trivializing \(\mathcal {L}\) over \(\overline{Ig}_{\mu }\), and in particular along any of the \(p^{2}1\) cuspidal components lying over E in \(\overline{Ig}_{\mu }.\) Using \({\tilde{E}}\) as a reference, there is a unique section of \(\mathcal {L}\) along E which pulls back to \(a_{{\tilde{E}}}.\) On the other hand, extending scalars from \(R_{N}\) to \(\mathbb {C}\), shifting to the analytic category, restricting to the connected component \({\bar{X}}_{\varGamma }\) on which E lies, and then pulling back to a neighborhood of the cusp \(c_{\infty }\) in the unit ball \(\mathfrak {X}\), we have trivialized \(\mathcal {L}_{E}\) by means of the section \(2\pi id\zeta _{3}\), which we showed to be \({\mathcal {K}}_{N}\)rational.
Lemma 3.7
The sections \(a_{{\tilde{E}}}\) and \(2\pi id\zeta _{3}\) “coincide” in the sense that they come from the same section in \(H^{0}(E,\mathcal {L}).\)
Proof
Lemma 3.8
The sections \(a_{{\tilde{E}}}^{p+1}\) and \(2\pi id\zeta _{2}\otimes 2\pi id\zeta _{3}\mod \mathcal {P}_{0}\otimes \mathcal {L}\) “coincide” in the sense that they come from the same section in \(H^{0}(E,\mathcal {P}_{\mu }\otimes \mathcal {L}).\)
Proof
3.4 A study of the theta operator along the supersingular locus
3.4.1 De Rham cohomology in characteristic p
There is a similar connection on \(D^{(p)}.\) The isogenies Frob and Ver, like any isogeny, take horizontal sections with respect to the Gauss–Manin connection to horizontal sections, e.g. if \(x\in D\) and \(\nabla x=0\), then \(Vx\in D^{(p)}\) satisfies \(\nabla (Vx)=0.\)
Remark 3.1
In the theory of Dieudonné modules, one works over a perfect base. It is then customary to identify D with \(D^{(p)}\) via \(x\leftrightarrow 1\otimes x.\) This identification is only \(\sigma \)linear where \(\sigma =\phi \), now viewed as an automorphism of R. The operator F becomes \(\sigma \)linear, V becomes \(\sigma ^{1}\)linear, and (3.35) reads \(\left\langle Fx,y\right\rangle =\left\langle x,Vy\right\rangle ^{\sigma }.\) With this convention, F and V switch types, rather than preserve them.
3.4.2 The Dieudonné module at a gss point
Assume from now on that \(s_{0}\in Z^{\prime }=S_\mathrm{gss}\) is a closed point of the general supersingular locus. We write \(D_{0}\) for \(H_{dR}^{1}(A_{s_{0}}/\kappa ).\)
Lemma 3.9
 (i)
\({\mathcal {O}}_{{\mathcal {K}}}\) acts on the \(e_{i}\) via \(\varSigma \) and on the \(f_{i}\) via \({\bar{\varSigma }}\) (hence it acts on the \(e_{i}^{(p)}\) via \({\bar{\varSigma }}\) and on the \(f_{i}^{(p)}\) via \(\varSigma ).\)
 (ii)The symplectic pairing satisfies$$\begin{aligned} \left\langle e_{i},f_{j}\right\rangle =\left\langle f_{j},e_{i}\right\rangle =\delta _{ij},\,\,\,\left\langle e_{i},e_{j}\right\rangle =\left\langle f_{i},f_{j}\right\rangle =0. \end{aligned}$$(3.41)
 (iii)
The vectors \(e_{1},e_{2},f_{3}\) form a basis for the cotangent space \(\omega _{A_{0}/\kappa }.\) Hence \(e_{1}\) and \(e_{2}\) span \(\mathcal {P}\) and \(f_{3}\) spans \(\mathcal {L}.\)
 (iv)
\(\ker (V)\) is spanned by \(e_{1},f_{2},e_{3}.\) Hence \(\mathcal {P}_{0}=\mathcal {P}\cap \ker (V)\) is spanned by \(e_{1}.\)
 (v)
\(Ve_{2}=f_{3}^{(p)},\,Vf_{3}=e_{1}^{(p)},\,Vf_{1}=e_{2}^{(p)}.\)
 (vi)
\(Ff_{1}^{(p)}=e_{3},\,Ff_{2}^{(p)}=e_{1},Fe_{3}^{(p)}=f_{2}.\)
Proof
Up to a slight change of notation, this is the unitary Dieudonné module which Bültel and Wedhorn call a “braid of length 3” and denote by \(\bar{B}(3)\), cf [4] (3.2). The classification in loc. cit. Proposition 3.6 shows that the Dieudonné module of a \(\mu \)ordinary abelian variety is isomorphic to \(\bar{B}(2)\oplus {\bar{S}}\), that of a gss abelian variety is isomorphic to \(\bar{B}(3)\) and in the superspecial case we get \(\bar{B}(1)\oplus {\bar{S}}^{2}.\)
3.4.3 Infinitesimal deformations
Let \({\mathcal {O}}_{S,s_{0}}\) be the local ring of S at \(s_{0}\), \(\mathfrak {m}\) its maximal ideal, and \(R={\mathcal {O}}_{S,s_{0}}/\mathfrak {m}^{2}.\) This R is a truncated polynomial ring in two variables, isomorphic to \(\kappa [u,v]/(u^{2},uv,v^{2}).\)
As remarked above, the de Rham cohomology \(D=H_{dR}^{1}(A/R)\) has a basis of horizontal sections and we may identify \(D^{\nabla }\) with \(D_{0}\) and D with \(R\otimes _{\kappa }D_{0}.\)
Since V and F preserve horizontality, \(e_{1},f_{2},e_{3}\) span \(\ker (V)\) over R in D, and the relations in (v) and (vi) of Lemma 3.9 continue to hold. Indeed, the matrix of V in the basis at \(s_{0}\) prescribed by that lemma, continues to represent V over \(\text {Spec}(R)\) by “horizontal continuation”. The matrix of F is then derived from the relation (3.35).
Lemma 3.10
Let \(s_{0}\in S_\mathrm{gss}\) and the notation be as above. Then the closed subscheme \(S_\mathrm{gss}\cap \mathrm{Spec}(R)\) is given by the equation \(u=0.\)
3.4.4 The Kodaira–Spencer isomorphism along the general supersingular locus
Proposition 3.11
Let \(s_{0}\in Z^{\prime }=S_\mathrm{gss}\). Choose local parameters u and v at \(s_{0}\) so that in \({\mathcal {O}}_{S,s_{0}}\) the local equation of \(Z^{\prime }\) becomes \(u=0.\) Then at \(s_{0}\), \(\psi (\mathrm{d}u)\) has a zero along \(Z^{\prime }.\)
Proof
3.4.5 A computation of poles along the supersingular locus
We are now ready to prove the following.
Proposition 3.12
Let \(k\ge 0\), and let \(f\in H^{0}(S,\mathcal {L}^{k})\) be a modular form of weight k in characteristic p. Then \(\varTheta (f)\in H^{0}(S,\mathcal {L}^{k+p+1})\).
Proof
It is amusing to compare the reasons for the increase by \(p+1\) in the weight of \(\varTheta (f)\) for modular curves and for Picard modular surfaces. In the case of modular curves the Kodaira–Spencer isomorphism is responsible for a shift by 2 in the weight, but the section acquires simple poles at the supersingular points. One has to multiply it by the Hasse invariant, which has weight \(p1\), to make the section holomorphic and hence a total increase by \(p+1=2+(p1)\) in the weight. In our case, the map \(\psi \) is responsible for a shift by \(p+1\) (the p coming from \(\mathcal {P}_{\mu }\simeq \mathcal {L}^{p}\)), but the section turns out to be holomorphic along the supersingular locus. See Section 4.2.
4 Further results on \(\varTheta \)
4.1 Relation to the filtration and theta cycles
Proposition 4.1
 (1)
If \(k\le p^{2}1\), then \(\omega (f)=k.\)
 (2)
If \(k<p+1\), then \(\omega (\varTheta ^{i}(f))=k+i(p+1)\) for \(0\le i\le p2, \) so the drop occurs at the last step of the theta cycle, i.e. at weight \(k+(p2)(p+1)\), which is congruent to \(k2\) modulo p.
 (3)
If \(k<p+1\) but \(r(f)\notin \text {Im}(\varTheta )\), then starting with \(\varTheta (f)\) instead of f, one sees that the drop in the theta cycle of \(\varTheta (f)\) occurs either in passing from \(\varTheta ^{p2}(f)\) to \(\varTheta ^{p1}(f)\), or in passing from \(\varTheta ^{p1}(f)\) to \(\varTheta ^{p}(f).\)
4.2 Compatibility between theta operators for elliptic and Picard modular forms
4.2.1 The theta operator for elliptic modular forms
The theta operator for elliptic modular forms modulo p was introduced by Serre and SwinnertonDyer in terms of qexpansions, but its geometric construction was given by Katz [20, 21]. Katz relied on a canonical splitting of the Hodge filtration over the ordinary locus, but Gross gave in [13], Proposition 5.8, the construction after which we modeled our \(\varTheta \).
4.2.2 An embedding of a modular curve in \({\bar{S}}\)
To illustrate our idea, and to simplify the computations, we assume that \(N=1 \) and \(d_{{\mathcal {K}}}\equiv 1\mod 4\), so that \(D=D_{{\mathcal {K}}}=d_{{\mathcal {K}}}.\) This conflicts of course with our running hypothesis \(N\ge 3\), but for the current section does not matter much. We shall treat only one special embedding of the modular curve \({\bar{X}}=X_{0}(D)\) into \({\bar{S}}\) (there are many more).
It is now clear that over any \(R_{0}\)algebra R we have the same modulitheoretic construction, sending a pair (E, M) where M is a cyclic subgroup of degree D to A(E, M), with \({\mathcal {O}}_{{\mathcal {K}}}\) structure and polarization given by the same formulae. This gives a modular embedding \(j:X\rightarrow S\) which is generically injective. To make this precise at the level of schemes (rather than stacks), one would have to add a level N structure and replace the base ring \(R_{0}\) by \(R_{N}.\)
4.2.3 Comparison of the two theta operators
Lemma 4.2
The line bundle \(j^{*}\mathcal {P}_{0}\) is constant, and \(\mathcal {P}_{\mu }\), originally a quotient bundle of \(\mathcal {P}\), becomes a direct summand when restricted to \({\bar{X}}\).
Proof
This is straightforward from the construction of j, and the fact that \(E_{0}\) is supersingular, while E is ordinary over \({\bar{X}}_\mathrm{ord}.\) Note that \({\mathcal {O}}_{{\mathcal {K}}}\otimes E/\delta _{{\mathcal {K}}}\otimes M\) and \({\mathcal {O}}_{{\mathcal {K}}}\otimes E\) have the same cotangent space.
Proposition 4.3
Proof
Remark 4.1
The proposition follows, of course, also from the effect of \(\theta \) and \(\varTheta \) on qexpansions, once we compare FJ expansions on \({\bar{S}}\) to qexpansions on the embedded \({\bar{X}}.\) The geometric proof given here has the advantage that it explains the precise way in which \(V_{\mathcal {P}}\otimes 1\) replaces “multiplication by \(h''.\)
5 The Igusa tower and padic modular forms
We shall be very brief, since from now on the development follows closely the classical case of padic modular forms on GL(2), with minor modifications. A general reference for this section is Hida’s book [16], although, strictly speaking, our case (p inert) is excluded there.
5.1 Geometry modulo \(p^{m}\)
5.1.1 The Picard surface modulo \(p^{m}\)
5.1.2 pAdic modular forms of integral weight k
Note that if R is not of topologically finite type over \({\mathcal {K}}_{p}\) our definition of “overconvergent” is a priori stronger than asking f to extend to a strict neighborhood of \({\bar{S}}_{\mu }^{rig}{\widehat{\otimes }}_{{\mathcal {K}}_{p}}R\) in \({\bar{S}}^{rig}{\widehat{\otimes }}_{{\mathcal {K}}_{p}}R.\)
5.1.3 qExpansion principle
Whether we are dealing with an \(f\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k})\) or an \(f\in M_{k}^{p}(N;{\mathcal {K}}_{p})\) the same procedure as in Sect. 1.10.2 allows us to associate with f a Fourier–Jacobi expansion FJ(f) (1.125). Recall, however, that FJ(f) depends on the section \(s\in H^{0}(C,\mathcal {L})\) used to trivialize \(\mathcal {L}_{C}.\) Note that if \(f\in M_{k}^{p}(N;{\mathcal {K}}_{p})\), the coefficients of FJ(f) are theta functions with bounded denominators, since a suitable \({\mathcal {K}}_{p} \)multiple of f lies in \(M_{k}^{p}(N;{\mathcal {O}}_{p}).\)
As with classical modular forms, we have the qexpansion principle, stemming from the fact that C meets every component of \({\bar{S}}_{\mu }^{rig}.\)
Lemma 5.1
If \(FJ(f)=0\), then \(f=0.\)
Corollary 5.2
If \(f\in M_{k}^{p}(N;{\mathcal {O}}_{p})\) and FJ(f) is divisible by p (in the sense that every \(c_{j}(f)\in H^{0}(C,\mathcal {N}^{j})\) is divisible by p with respect to the integral structure on \({\bar{S}}\)), then \(f\in pM_{k}^{p}(N;{\mathcal {O}}_{p}).\)
5.2 The Igusa scheme of level \(p^{n}\)
5.2.1 \(\mu \)Ordinary abelian schemes over \(R_{m}\)algebras
Let \(m\ge 1\) and let R be an \(R_{m}\)algebra. If \(\underline{A}\in S_{\mu }^{(m)}(R)\subset \mathcal {M}(R)\), then A is fiberbyfiber \(\mu \)ordinary, hence \(A[p^{n}]^{\mu }\), the largest Rsubgroup scheme of \(A[p^{n}]\) of multiplicative type (dual to the étale quotient \(A[p^{n}]^{et}\)), is a finite flat \({\mathcal {O}}_{{\mathcal {K}}}\)subgroup scheme of rank \(p^{2n}.\) Locally in the étale topology it is isomorphic to \(\delta _{{\mathcal {K}}}^{1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}.\)
5.2.2 Igusa level structure of level \(p^{n}\)
5.2.3 The trivialization of \(\mathcal {L}\) when \(m\le n\)
The same holds of course for \(\mu _{p^{n}}\) and \(\mathbb {G}_{m}.\) The reasoning used for \(m=n=1\) applies and shows that \(\varepsilon \) induces a canonical isomorphism between \(\mathcal {L}_{\overline{Ig}(p^{n})_{\mu }^{(m)}}\) and \({\mathcal {O}}_{\overline{Ig}(p^{n})_{\mu }^{(m)}}.\) We denote by \(a=a_{n}^{(m)} \) the section which corresponds to \(1\in {\mathcal {O}}_{\overline{Ig}(p^{n})_{\mu }^{(m)}}\), i.e. the trivializing section.
5.2.4 Congruences between FJ expansions force congruences between the weights
Let \(k_{1}\le k_{2}\) be two integers. The following lemma follows formally from the definitions.
Lemma 5.3
Let \(f_{i}\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k_{i}})\) and assume that \(f_{1}\) is not divisible by p. Suppose \({\widetilde{FJ}}(f_{1})={\widetilde{FJ}}(f_{2}).\) Then \(k_{1}\equiv k_{2}\mod (p^{2}1)p^{m1}.\)
Proof
In practice, one would like to deduce the same result from congruences between FJ expansions along C, not along \({\tilde{C}}.\) This is deeper and depends on Igusa’s irreducibility theorem.
Theorem 5.4
Consider \(\tau =\tau _{n}^{(1)}:\overline{Ig}(p^{n})_{\mu }^{(1)}\rightarrow {\bar{S}}_{\mu }^{(1)}={\bar{S}}_{\mu ,\kappa _{0}}\) and extend scalars from \(\kappa _{0}\) to \(\kappa .\) Let T be an irreducible component of \({\bar{S}}_{\mu ,\kappa }.\) Then \(\tau ^{1}(T)\) is irreducible in \(\overline{Ig}(p^{n})_{\mu ,\kappa }.\)
Proof
The theorem can be proved by the same method used by Hida [16, 8.4], [17], or by the method of Ribet to which we alluded in 2.4.3. In that section, we proved the theorem for \(n=1\) by a third method, due to Igusa, studying the image of inertia around \(S_\mathrm{ss}\). See also the discussion of the big Igusa tower BigIg below, which turns out to be reducible.
Theorem 5.5
Let \(f_{1}\) and \(f_{2}\) be \(\mod p^{m}\) modular forms as above and assume that \(f_{1}\) is not divisible by p. Trivialize \(\mathcal {L}_{C}\) by choosing a lift of C to \({\tilde{C}}\) (i.e. a section of the map \(\tau _{{\tilde{C}}}:{\tilde{C}}\rightarrow C\)) and using the trivialization of \(\mathcal {L}\) along this lift which is supplied by the section a. Then if \(FJ(f_{1})=FJ(f_{2})\), \(k_{1}\equiv k_{2}\mod (p^{2}1)p^{m1}.\)
Here \(FJ(f)=\sum _{j=0}^{\infty }c_{j}(f)\) and \(c_{j}(f)\in H^{0}(C,\mathcal {N}^{j})\). The lift of C to \({\tilde{C}}\) exists since \({\tilde{C}}\simeq \varDelta (p^{m})\times C\) (noncanonically). If we change the lift (locally on the base) by \(\gamma \in \varDelta (p^{m})\), then \(FJ(f_{i})\) changes by the factor \({\bar{\varSigma }}(\gamma )^{k_{i}}.\)
Proof
By Igusa’s irreducibility theorem, it is enough to know that \(FJ(f_{i})\) (\(i=1,2\)) agree on the given lift of C, to conclude that \(\tau ^{*}f_{1}/a^{k_{1}}=\tau ^{*}f_{2}/a^{k_{2}}\) on the whole of \(\overline{Ig}(p^{m})_{\mu }^{(m)}\), hence the result follows by the Lemma. Note that the underlying topological spaces of \(\overline{Ig}(p^{m})_{\mu }^{(m)}\) and \(\overline{Ig}(p^{m})_{\mu }^{(1)}\) are the same, and hence for the irreducibility theorem, it is enough to deal with the special fiber.
Corollary 5.6
5.2.5 Irreducibility of the Igusa tower and the big Igusa tower
5.3 padic modular forms of padic weights
5.3.1 The space of padic weights
5.3.2 padic modular forms à la Serre
We work with \({\bar{S}}\) (hence also the cuspidal divisor C) over the base \({\mathcal {O}}_{p}\), the padic completion of \(R_{0}.\) Little is lost by extending the base further to \({\mathcal {O}}_{N,\mathfrak {P}}\), the completion of the ring of integers of the ray class field \({\mathcal {K}}_{N}\) at a prime \(\mathfrak {P}\) above p. After such a base extension the irreducible components of C become absolutely irreducible. The reader may assume that this is the case.
Consider the pdivisible group of the toric part of the universal semiabelian variety \({\mathcal {A}}_{C}\). Once and for all fix an \({\mathcal {O}}_{{\mathcal {K}}}\)isomorphism of it with \(\delta _{{\mathcal {K}}}^{1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{\infty }}\), and use this isomorphism to identify \(\mathcal {L}_{C}=\omega _{{\mathcal {A}}/C}({\bar{\varSigma }})\) with \({\mathcal {O}}_{C}.\) This choice is unique up to multiplication by \({\mathcal {O}}_{p}^{\times }\) on each irreducible component of C. It determines a FJ expansion FJ(f) for every \(f\in M_{k}^{p}(N;{\mathcal {K}}_{p})\) as in (1.125), and is equivalent to splitting the projection \(\tau _{{\tilde{C}}}:{\tilde{C}}\rightarrow C\) from the boundary of the Igusa tower \(\left( \overline{Ig}_{\mu }(p^{n})\right) _{n=1}^{\infty }\) to the boundary of the Picard modular surface.
Proposition 5.7
 (i)
If \(k\in \mathbb {Z}\), then \(M_{k}^\mathrm{Serre}(N;{\mathcal {K}}_{p})=M_{k}^{p}(N;{\mathcal {K}}_{p}).\) In other words, we do not get any new padic modular forms by allowing limits of padic modular forms of varying weights, if the weights converge to an integral k.
 (ii)
In the definition of \(M_{k}^\mathrm{Serre}(N;{\mathcal {K}}_{p})\) we can require \(f_{\nu }\in M_{k_{\nu }}(N;{\mathcal {K}}_{p})\) (classical modular forms of integral weight \(k_{\nu }\)) and still get the same space.
 (iii)
\(M_{k}^\mathrm{Serre}(N;{\mathcal {K}}_{p})\) is a closed subspace of \(\mathcal {FJ}_{p}.\) The product of two \(f_{i}\in M_{k_{i}}^\mathrm{Serre}\) is in \(M_{k_{1}+k_{2}}^\mathrm{Serre}\).
 (iv)
If \(f\in M_{k}^\mathrm{Serre}(N;{\mathcal {O}}_{p})\), then its reduction modulo p appears in \(M_{k^{\prime }}(N;\kappa _{0})\) for some positive integer \(k^{\prime }\) sufficiently close to k in \(\mathfrak {X}_{p}.\)
Proof
Let \(H_{{\bar{\varSigma }}}\in M_{p^{2}1}(N;{\mathcal {O}}_{p})\) be a lift of the Hasse invariant \(h_{{\bar{\varSigma }}}\) to characteristic 0. Such a lift exists by general principles, whenever p is large enough. For the few exceptional primes p we may replace \(h_{\mathbf {\varSigma }}\) by a high enough power of it, which is liftable, and use the same argument. This lift satisfies \(FJ(H_{{\bar{\varSigma }}})\equiv 1\mod p\), so \(H_{{\bar{\varSigma }}}^{1}\in M_{1p^{2}}^{p}(N;{\mathcal {O}}_{p})\) is a padic modular form defined over \({\mathcal {O}}_{p}.\) Indeed, \(H_{{\bar{\varSigma }}}\mod p^{m}\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{p^{2}1})\) is nowhere vanishing over \({\bar{S}}_{\mu }^{(m)}\), and taking the limit of its inverse over m we get \(H_{{\bar{\varSigma }}}^{1}.\) Suppose, as in (i), that \(k,k_{\nu }\in \mathbb {Z}\), \(k_{\nu }\rightarrow k\) in \(\mathfrak {X}_{p}\), and \(f_{\nu }\in M_{k_{\nu }}^{p}(N;{\mathcal {K}}_{p})\) are such that \(FJ(f_{\nu })\) converge in \(\mathcal {FJ}_{p}\) to f. Replacing \(f_{\nu }\) by \(f_{\nu }H_{{\bar{\varSigma }}}^{p^{e_{\nu }}}\) for suitable \(e_{\nu }\), we may assume that the \(k_{\nu }\) are increasing and are all in the same congruence class modulo \(p^{2}1.\) But then \(f_{\nu }H_{{\bar{\varSigma }}}^{(kk_{\nu })/(p^{2}1)}\) are in \(M_{k}^{p}(N;{\mathcal {K}}_{p})\) and their FJ expansions converge to f in \(\mathcal {FJ}_{p}.\) This proves (i). For (ii) note that if \(f\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k})\), then for all sufficiently large e, \(fH_{{\bar{\varSigma }}}^{p^{e}}\) extends to an element of \(M_{k+(p^{2}1)p^{e}}(N;R_{m})\) and has the same FJ expansion as f. Thus every padic modular form of integral weight is the padic limit of classical forms of varying weights, and the same is therefore true for Serre modular forms of padic weight. Points (iii) and (iv) are obvious.
5.3.3 pAdic modular forms à la Katz
Proposition 5.8
Proof
From now on, it is therefore legitimate to denote these spaces by the common notation \(M_{k}^{p}(N;{\mathcal {O}}_{p})\) and refer to them simply as p adic modular forms of p adic weight k.
5.4 pAdic modular forms of padic biweights
5.4.1 The space of biweights
A new feature of padic modular forms on Picard modular surfaces, that does not show up in the classical theory of \(GL_{2}(\mathbb {Q})\), is that even if we restrict attention to scalarvalued padic modular forms, we sometimes need to consider classical vectorvalued forms to approach them. This phenomenon, as we shall explain below, does not show up in the \(\mod p\) theory, but is essential to the padic theory.
5.4.2 The line bundle \(\mathcal {L}^{(k_{1},k_{2})}\) over \({\bar{S}}_{\mu }^{rig}\) and padic modular forms of integral biweights
5.4.3 The trivialization of \(\mathcal {L}^{(k_{1},k_{2})}\) over the Igusa tower
Let \(a^{k_{1},k_{2}}=a^{k_{1}}{\bar{a}}^{k_{2}}.\) Then we may trivialize \(\tau ^{*}\mathcal {L}^{(k_{1},k_{2})}\) by \(s\mapsto s/a^{k_{1},k_{2}}\) to get a function on \(\overline{Ig}(p^{n})_{\mu }^{(m)}\). This allows us to define, as usual, canonical Fourier–Jacobi expansion \({\widetilde{FJ}}(f)\) (along \({\tilde{C}}\)), and if we make a choice of a splitting of \(\tau :{\tilde{C}}\rightarrow C\), a Fourier–Jacobi expansion FJ(f) (along C) for every \(f\in M_{k_{1},k_{2}}^{p}(N;{\mathcal {K}}_{p}).\)
5.4.4 pAdic modular forms of padic biweights
5.5 The theta operator for padic modular forms
Theorem 5.9
 (i)When one reduces \(M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p})\) modulo p, and uses the isomorphism \(\mathcal {P}_{\mu }\simeq \mathcal {L}^{p}\), \(\varTheta \) reduces to the operatoron \(\mod p\) modular forms.$$\begin{aligned} \varTheta :M_{k}(N;\kappa )\rightarrow M_{k+p+1}(N;\kappa ) \end{aligned}$$(5.41)
 (ii)
The effect of \(\varTheta \) on the canonical FJ expansion \({\widetilde{FJ}}(f) \) is given by “\(q\frac{d}{dq}\)”, i.e. by the formula (3.21).
We omit the proof of (ii), which goes along the same lines as in the \(\mod p\) theory.
The Steinitz class of a finite projective \({\mathcal {O}}_{{\mathcal {K}}}\)module is the class of its top exterior power as an invertible module.
No confusion should arise from the use of the letter N to denote both the level and the unipotent radical of P.
Notes
Declarations
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