Open Access

A theta operator on Picard modular forms modulo an inert prime

Research in the Mathematical Sciences20163:28

DOI: 10.1186/s40687-016-0075-8

Received: 5 July 2015

Accepted: 27 May 2016

Published: 1 September 2016

Swinnerton-Dyer [38] and Serre [34] introduced a certain differential operator \(\theta \) on (elliptic) modular forms over \(\mathbb {{\bar{F}}}_{p}\). In terms of the q-expansion
$$\begin{aligned} f=\sum _{n=0}^{\infty }a_{n}q^{n} \end{aligned}$$
(0.1)
(\(a_{n}\in \mathbb {{\bar{F}}}_{p}\)) of such a form, \(\theta \) is given by qd / dq. It lifts, by the same formula, to the space of p-adic modular forms. This suggests a relation with the Tate twist of the mod p Galois representation attached to f, if the latter is a Hecke eigenform.

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 p-adic operator has been studied in relation to mod p Galois representations, congruences between modular forms, p-adic families of modular forms and p-adic L-functions. 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 p-adic 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 p-adic 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 unit-root 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 non-empty 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 (nn)]. Böcherer and Nagaoka [3] defined theta operators on Siegel modular forms by studying their q-expansions.

The assumption that the ordinary locus is non-empty 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(nm) 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(nm) admit Fourier–Jacobi (FJ) expansions at the cusps, which are q-expansions with theta functions as coefficients.

One of the main goals of this paper is to define the theta operator for Picard modular surfaces at a good inert prime and study its properties. To explain how we overcome the need to consider the unit-root splitting of the cohomology of the universal abelian variety, let us re-examine the case of the modular curve X of full level \(N\ge 3\) over \(\mathbb {Z}_{p}\) (\((p,N)=1\)). We follow an approach of Gross [13], see also [1], who extended it to Hilbert modular varieties. Let \(\kappa \) be a fixed algebraic closure of \(\mathbb {F}_{p}\), and consider the geometric characteristic p fiber \(X_{\kappa }\). Let \({\mathcal {A}}\) be the universal elliptic curve over \(Y=X\backslash C\) (the complement of the cusps) and let \(\mathcal {L}=\omega _{{\mathcal {A}}/Y}\) be its cotangent bundle at the origin. Then \({\mathcal {A}}\) extends to a semi-abelian variety over X, and so does \(\mathcal {L}=\omega _{{\mathcal {A}}/X}\). By definition, a weight k, level N modular form over \(\kappa \), is a global section of \(\mathcal {L}^{k}\) over \(X_{\kappa }\), i.e.
$$\begin{aligned} M_{k}(N;\kappa )=H^{0}\left( X_{\kappa },\mathcal {L}^{k}\right) . \end{aligned}$$
(0.2)
Let \(X_{\kappa }^\mathrm{ord}\) be the ordinary locus in \(X_{\kappa }\). Let \(\tau :I\rightarrow X_{\kappa }^\mathrm{ord}\) be the Igusa curve of level p, classifying (besides the elliptic curve A and level structure classified by \(X_{\kappa }\)) embeddings of finite flat group schemes \(\iota :\mu _{p}\hookrightarrow A[p]\). Let
$$\begin{aligned} h\in H^{0}\left( X_{\kappa },\mathcal {L}^{p-1}\right) \end{aligned}$$
(0.3)
be the Hasse invariant. As the universal \(\iota :\mu _{p}\hookrightarrow {\mathcal {A}}[p]\) over I induces an isomorphism
$$\begin{aligned} \tau ^{*}\mathcal {L}=\omega _{{\mathcal {A}}/I}=\omega _{{\mathcal {A}}[p]/I}\overset{\iota ^{*}}{\simeq }\omega _{\mu _{p}/I}={\mathcal {O}}_{I}, \end{aligned}$$
(0.4)
the line bundle \(\tau ^{*}\mathcal {L}\) is trivialized over I by a canonical section a. In fact, \(a^{p-1}=\tau ^{*}h\).
Now, given a \(\kappa \)-valued modular form \(f\in H^{0}(X_{\kappa },\mathcal {L}^{k})\), we consider its pull-back \(\tau ^{*}f\) to I, divide by \(a^{k}\) to get a function on I, and take its differential
$$\begin{aligned} \eta _{f}={\hbox {d}}\left( \tau ^{*}f/a^{k}\right) \in \varOmega _{I}^{1}. \end{aligned}$$
(0.5)
The Gauss–Manin connection induces the Kodaira–Spencer isomorphism
$$\begin{aligned} \text {KS}:\mathcal {L}^{2}\otimes {\mathcal {O}}(C)^{\vee }\simeq \varOmega _{X}^{1}. \end{aligned}$$
(0.6)
As \(\tau \) is étale, \(\varOmega _{I}^{1}=\tau ^{*}\varOmega _{X_{\kappa }^\mathrm{ord}}^{1}\) and we may pull KS back to a similar isomorphism over I. We can therefore look at
$$\begin{aligned} a^{k}\cdot \text {KS}^{-1}(\eta _{f}). \end{aligned}$$
(0.7)
This is a section of \(\tau ^{*}(\mathcal {L}^{k+2}\otimes {\mathcal {O}}(C)^{\vee })\) over I. Since we divided and multiplied by the same power of a, it descends to \(X_{\kappa }^\mathrm{ord}\). A calculation shows that it has at most simple poles at the supersingular points \(X_{\kappa }^\mathrm{ss}\), so
$$\begin{aligned} \theta (f)=h\cdot a^{k}\cdot \text {KS}^{-1}(\eta _{f}) \end{aligned}$$
(0.8)
extends to a global section of \(\mathcal {L}^{k+p+1}\otimes {\mathcal {O}}(C)^{\vee }\) over \(X_{\kappa }\), i.e. to a cusp form of weight \(k+p+1\) and level N over \(\kappa \). Note that \(\theta (f)\) and \(a^{k}\cdot \text {KS}^{-1}(\eta _{f})\) have the same q-expansions, since the q-expansion of h is 1. It can be checked that \(\theta \) coincides with the operator denoted by \(A\theta \) in [21].

The absence of the unit-root splitting from the above-mentioned 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 one-dimensional piece.

The resulting operator \(\varTheta \) enjoys all the desired properties. It has the right effect on Fourier–Jacobi expansions, extends holomorphically across the 1-dimensional 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).

Let us now review the contents of the paper in more detail. We denote by \({\mathcal {K}}\) a quadratic imaginary field and by \({\bar{S}}\) a compactified integral model of the Picard modular surface of full level \(N\ge 3\), associated with \({\mathcal {K}}\). The surface \({\bar{S}}\) is defined over \(R_{0}={\mathcal {O}}_{{\mathcal {K}}}[1/2D_{{\mathcal {K}}}N]\) and we may consider its reduction modulo the prime p, which is assumed to be relatively prime to 2N and inert in \({\mathcal {K}}\). For simplicity, fix an algebraic closure \(\kappa \) of \(R_{0}/pR_{0}\) and consider the geometric fiber \({\bar{S}}_{\kappa }={\bar{S}}\times _{\mathrm{Spec}(R_{0})}{\text {Spec}}(\kappa )\). Let \({\mathcal {A}}\) be the universal semi-abelian variety over \({\bar{S}}\). It is relatively 3-dimensional, has complex multiplication by \({\mathcal {O}}_{{\mathcal {K}}}\), and the cotangent bundle at the origin, \(\omega _{{\mathcal {A}}/{\bar{S}}}\), is of type (2, 1). This means that if \(\varSigma :{\mathcal {O}}_{{\mathcal {K}}}\hookrightarrow R_{0}\) is the canonical embedding and \({\bar{\varSigma }}\) its complex conjugate, then
$$\begin{aligned} \omega _{{\mathcal {A}}/{\bar{S}}}=\mathcal {P}\oplus \mathcal {L}, \end{aligned}$$
(0.9)
where \(\mathcal {P}=\omega _{{\mathcal {A}}/{\bar{S}}}(\varSigma )\) is a plane bundle on which \({\mathcal {O}}_{{\mathcal {K}}}\) acts via \(\varSigma \), and \(\mathcal {L}=\omega _{{\mathcal {A}}/{\bar{S}}}({\bar{\varSigma }})\) is a line bundle on which it acts via \({\bar{\varSigma }}\). Scalar modular forms of weight \(k\ge 0\) defined over an \(R_{0}\)-algebra R are by definition elements of
$$\begin{aligned} M_{k}(N;R):=H^{0}\left( {\bar{S}}_{R},\mathcal {L}^{k}\right) . \end{aligned}$$
(0.10)
Our main interest is in \(R=\kappa \). In this case, there are homomorphisms of vector bundles \(V_{\mathcal {P}}:\mathcal {P}\rightarrow \mathcal {L}^{(p)}\) and \(V_{\mathcal {L}}:\mathcal {L}\rightarrow \mathcal {P}^{(p)}\) deduced from the Verschiebung homomorphism. Here, for any vector bundle \(\mathcal {V}\) over \({\bar{S}}_{\kappa }\), \(\mathcal {V}^{(p)}\) stands for its base change under the absolute Frobenius morphism of degree \(p,\,\varPhi :{\bar{S}}_{\kappa }\rightarrow {\bar{S}}_{\kappa }\). The Hasse invariant is the map
$$\begin{aligned} h_{{\bar{\varSigma }}}=V_{\mathcal {P}}^{(p)}\circ V_{\mathcal {L}}:\mathcal {L}\rightarrow \mathcal {L}^{(p^{2})}. \end{aligned}$$
(0.11)
Since \(\mathcal {L}\) is a line bundle, \(\mathcal {L}^{(p)}\simeq \mathcal {L}^{p}\), so \(h_{{\bar{\varSigma }}}\in H^{0}({\bar{S}}_{\kappa },\mathcal {L}^{p^{2}-1}) \) is a modular form of weight \(p^{2}-1\) over \(\kappa \). The divisor of \(h_{{\bar{\varSigma }}}\) is precisely the supersingular locus \(S_\mathrm{ss}\subset {\bar{S}}_{\kappa }\). This is a reduced 1-dimensional closed subscheme whose geometric points x are characterized by the fact that \({\mathcal {A}}_{x}\) is supersingular (the Newton polygon of its p-divisible group has constant slope 1 / 2). The structure of \(S_\mathrm{ss}\) has been determined by Vollaard [39], following work of Bültel and Wedhorn [4]. Its irreducible components are curves whose normalizations are all isomorphic to the Fermat curve of degree \(p+1\). (If N is large enough, depending on p, these components are even non-singular.) They intersect transversally at finitely many points, which form the singular locus of \(S_\mathrm{ss}\). This singular locus is also the superspecial locus \(S_\mathrm{ssp}\) in \({\bar{S}}_{\kappa }\), characterized by the fact that \(x\in S_\mathrm{ssp}\) if and only if \({\mathcal {A}}_{x}\) is isomorphic to a product of three supersingular elliptic curves. At \(x\in S_\mathrm{ssp}\), the maps \(V_{\mathcal {P}}\) and \(V_{\mathcal {L}}\) vanish, but over the general supersingular locus \(S_\mathrm{gss}=S_\mathrm{ss}\backslash S_\mathrm{ssp}\), they are both of rank 1. The complement of \(S_\mathrm{ss}\) in \({\bar{S}}_{\kappa }\) is the dense, open \(\mu \)-ordinary locus \({\bar{S}}_{\mu }\). Over a \(\mu \)-ordinary point which does not belong to a cuspidal component, the p-divisible group of \({\mathcal {A}}_{x}\) is a product of a height 2 group of multiplicative type, a height 2 group of local–local type, and a height 2 étale group (all stable under \({\mathcal {O}}_{{\mathcal {K}}}\)). See [8] and Sect. 2.1.2.

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].

In Sect. 2 we review the geometry of \({\bar{S}}\) and the automorphic vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) modulo an inert prime p. Here we follow [4, 39], and the exposition in [8]. We construct the Igusa surface of level p. It is a finite étale Galois cover
$$\begin{aligned} \tau :{\bar{I}}g_{\mu }\rightarrow {\bar{S}}_{\mu } \end{aligned}$$
(0.12)
of the \(\mu \)-ordinary part in \({\bar{S}}_{\kappa }\), with Galois group \(\varDelta (p)=({\mathcal {O}}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}})^{\times }\). We prove that it is relatively irreducible, and compactify it over the supersingular locus to get a normal surface \({\bar{I}}g\), finite and flat over \({\bar{S}}_{\kappa }\), which is totally ramified over \(S_\mathrm{ss}\). The Hasse invariant has a tautological \(p^{2}-1\) root a over the whole of \({\bar{I}}g\). Thus \(a\in H^{0}({\bar{I}}g,\tau ^{*}\mathcal {L})\) and \(a^{p^{2}-1}=\tau ^{*}h_{{\bar{\varSigma }}}. \)
In Sect. 3 we construct the theta operator. We pull back \(f\in H^{0}({\bar{S}}_{\kappa },\mathcal {L}^{k})\) to \({\bar{I}}g_{\mu }\), divide by the non-vanishing section \(a^{k}\) to get a function, and let
$$\begin{aligned} \eta _{f}=\text {d}\left( \tau ^{*}f/a^{k}\right) \in H^{0}\left( {\bar{I}}g_{\mu },\varOmega ^{1}\right) . \end{aligned}$$
(0.13)
The Kodaira–Spencer isomorphism over S is an isomorphism of rank two vector bundles
$$\begin{aligned} \text {KS}:\mathcal {P}\otimes \mathcal {L}\simeq \varOmega _{S}^{1}. \end{aligned}$$
(0.14)
When we try to extend it to \({\bar{S}}\), we find out that it has a pole along the cuspidal divisor \(C={\bar{S}}\backslash S\). Nevertheless, in the characteristic p fiber, the map
$$\begin{aligned} (V_{\mathcal {P}}\otimes 1)\circ \text {KS}^{-1}:\varOmega _{S_{\kappa }}^{1}\rightarrow \mathcal {L}^{(p)}\otimes \mathcal {L}=\mathcal {L}^{p+1} \end{aligned}$$
(0.15)
extends holomorphically across C, and even acquires a simple zero there. We pull it back from \({\bar{S}}_{\mu }\) to \({\bar{I}}g_{\mu }\) under the étale map \(\tau \), and define
$$\begin{aligned} \varTheta (f)=a^{k}\cdot (V_{\mathcal {P}}\otimes 1)\circ \text {KS}^{-1}(\eta _{f})\in H^{0}\left( {\bar{I}}g_{\mu },\tau ^{*}\mathcal {L}^{k+p+1}\right) . \end{aligned}$$
(0.16)
Thanks to the fact that we have multiplied by \(a^{k}\), this section descends to \({\bar{S}}_{\mu }\). A pleasant computation reveals that \(\varTheta (f)\) has no poles along \(S_\mathrm{ss}\). We end up with
$$\begin{aligned} \varTheta (f)\in H^{0}\left( {\bar{S}}_{\kappa },\mathcal {L}^{k+p+1}\right) , \end{aligned}$$
(0.17)
a weight \(k+p+1\), level N modular form over \(\kappa \).

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 p-adic and complex formalisms. It may be possible to perform it entirely on the “Mumford-Tate 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 p-adic 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 \(U(n-1,1)\) 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 lay-out 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

Let \({\mathcal {K}}\) be an imaginary quadratic field, contained in \(\mathbb {C}\). We denote by \(\varSigma :{\mathcal {K}}\hookrightarrow \mathbb {C}\) the inclusion and by \({\bar{\varSigma }}:{\mathcal {K}}\hookrightarrow \mathbb {C}\) its complex conjugate. We use the following notation:
  • \(d_{{\mathcal {K}}}\)—the square-free 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 \(a\in {\mathcal {K}}\).

We fix an integer \(N\ge 3\) (the “tame level”) and let \(R_{0}={\mathcal {O}}_{{\mathcal {K}}}[1/(2d_{{\mathcal {K}}}N)]\). This is our base ring. If R is any \(R_{0}\)-algebra and M is any R-module with \({\mathcal {O}}_{{\mathcal {K}}}\)-action, then M becomes an \({\mathcal {O}}_{{\mathcal {K}}}\otimes R\)-module and we have a canonical type decomposition
$$\begin{aligned} M=M(\varSigma )\oplus M({\bar{\varSigma }}) \end{aligned}$$
(1.1)
where \(M(\varSigma )=e_{\varSigma }M\) and \(M({\bar{\varSigma }})=e_{{\bar{\varSigma }}}M\), and where the idempotents \(e_{\varSigma }\) and \(e_{{\bar{\varSigma }}}\) are defined by
$$\begin{aligned} e_{\varSigma }=\frac{1\otimes 1}{2}+\frac{\delta \otimes \delta ^{-1}}{2},\quad e_{{\bar{\varSigma }}}=\frac{1\otimes 1}{2}-\frac{\delta \otimes \delta ^{-1}}{2}. \end{aligned}$$
(1.2)
Then \(M(\varSigma )\) (resp. \(M({\bar{\varSigma }})\)) is the part of M on which \({\mathcal {O}}_{{\mathcal {K}}}\) acts via \(\varSigma \) (resp. \({\bar{\varSigma }}\)). The same notation will be used for sheaves of modules on R-schemes, endowed with an \({\mathcal {O}}_{{\mathcal {K}}}\) action. If M is locally free, we say that it has type (pq) if \(M(\varSigma )\) is of rank p and \(M({\bar{\varSigma }})\) is of rank q.
We denote by
$$\begin{aligned} \mathbf {T}={\text {res}}_{\mathbb {Q}}^{{\mathcal {K}}}\mathbb {G}_{m} \end{aligned}$$
(1.3)
the non-split torus whose \(\mathbb {Q}\)-points are \({\mathcal {K}}^{\times }\), and by \(\rho \) the non-trivial automorphism of \(\mathbf {T}\), which on \(\mathbb {Q}\)-points induces \(\rho (a)={\bar{a}}\). The group \(\mathbb {G}_{m}\) embeds in \(\mathbf {T}\) and the homomorphism \(a\mapsto a\cdot \rho (a)\) from \(\mathbf {T}\) to itself factors through a homomorphism
$$\begin{aligned} N:\mathbf {T}\rightarrow \mathbb {G}_{m}, \end{aligned}$$
(1.4)
the norm homomorphism. Its kernel \(\ker (N)\) is denoted \(\mathbf {T}^{1}\).

1.1.2 The unitary group

Let \(V={\mathcal {K}}^{3}\) and endow it with the hermitian pairing
$$\begin{aligned} (u,v)=\,^{t}{\bar{u}}\left( \begin{array}{ccc} &{} &{}\quad \delta ^{-1} \\ &{}\quad 1 &{} \\ -\delta ^{-1} &{} &{} \end{array} \right) v. \end{aligned}$$
(1.5)
We identify \(V_{\mathbb {R}}\) with \(\mathbb {C}^{3}\) (\({\mathcal {K}}\) acting via the natural inclusion \(\varSigma \)). It then becomes a hermitian space of signature (2, 1). Conversely, any 3-dimensional hermitian space over \({\mathcal {K}}\) whose signature at the infinite place is (2, 1) is isomorphic to V after rescaling the hermitian form by a positive rational number.
Let
$$\begin{aligned} \mathbf {G=GU}(V,(,)) \end{aligned}$$
(1.6)
be the general unitary group of V, regarded as an algebraic group over \(\mathbb {Q}\). For any \(\mathbb {Q}\)-algebra A, we have
$$\begin{aligned} \mathbf {G}(A)=\left\{ (g,\mu )\in GL_{3}(A\otimes {\mathcal {K}})\otimes A^{\times }|\,\,(gu,gv)=\mu \cdot (u,v)\,\,\forall u,v\in V_{A}\right\} . \end{aligned}$$
(1.7)
We write \(G=\mathbf {G}(\mathbb {Q})\), \(G_{\infty }=\mathbf {G}(\mathbb {R})\) and \(G_{p}=\mathbf {G}(\mathbb {Q}_{p})\). A similar notational convention will apply to any algebraic group over \(\mathbb {Q}\) without further ado. If p splits in \({\mathcal {K}}\), \(\mathbb {Q}_{p}\otimes {\mathcal {K}}\simeq \mathbb {Q}_{p}^{2}\) and \(G_{p}\) becomes isomorphic to \(GL_{3}(\mathbb {Q}_{p})\times \mathbb {Q}_{p}^{\times }\). The isomorphism depends on the embedding of \({\mathcal {K}}\) in \(\mathbb {Q}_{p}\), i.e. on the choice of a prime above p in \({\mathcal {K}}\). For a non-split prime p, the group \(G_{p}\), like \(G_{\infty }\), is of (semisimple) rank 1.
As \(\mu \) is determined by g, we often abuse notation and write g for the pair \((g,\mu )\) and \(\mu (g)\) for the similitude factor (multiplier) \(\mu \). It is a character of algebraic groups over \(\mathbb {Q},\,\mu :\mathbf {G\rightarrow }\mathbb {G}_{m}\). Another character is \(\det :\mathbf {G\rightarrow T}\), defined by \(\det (g,\mu )=\det (g)\). If we let
$$\begin{aligned} \nu =\mu ^{-1}\cdot \det :\mathbf {G}\rightarrow \mathbf {T} \end{aligned}$$
(1.8)
then both \(\mu \) and \(\det \) are expressible in terms of \(\nu \), namely \(\mu =\nu \cdot (\rho \circ \nu )\) and \(\det =\nu ^{2}\cdot (\rho \circ \nu )\).
The groups
$$\begin{aligned} \mathbf {U}=\ker \mu ,\quad \mathbf {SU}=\ker \nu =\ker \mu \cap \ker (\det ) \end{aligned}$$
(1.9)
are the unitary and the special unitary group, respectively.
We also introduce an alternating \(\mathbb {Q}\)-linear pairing \(\left\langle ,\right\rangle :V\times V\rightarrow \mathbb {Q}\) (the polarization form) defined by \(\left\langle u,v\right\rangle =\text {Im}_{\delta }(u,v)\). We then have the formulae
$$\begin{aligned} \left\langle au,v\right\rangle =\left\langle u,{\bar{a}}v\right\rangle ,\quad 2(u,v)=\left\langle u,\delta v\right\rangle +\delta \left\langle u,v\right\rangle . \end{aligned}$$
(1.10)

1.1.3 The hermitian symmetric domain

The group \(G_{\infty }=\mathbf {G}(\mathbb {R})\) acts on \(\mathbb {P}_{\mathbb {C}}^{2}=\mathbb {P}(V_{\mathbb {R}})\) by projective linear transformations and preserves the open subdomain \(\mathfrak {X}\) of negative definite lines (in the metric (, )), which is biholomorphic to the open unit ball in \(\mathbb {C}^{2}\). Every negative definite line is represented by a unique vector \(^{t}(z,u,1)\), and such a vector represents a negative definite line if and only if
$$\begin{aligned} \lambda (z,u)\overset{\text {def}}{=}\,\mathrm {Im}_{\delta }(z)-u{\bar{u}}>0. \end{aligned}$$
(1.11)
One refers to the realization of \(\mathfrak {X}\) as the set of points \((z,u)\in \mathbb {C}^{2}\) satisfying this inequality as a Siegel domain of the second kind. It is convenient to think of the point \(x_{0}=\) \((\delta /2,0)\) as the “center” of \(\mathfrak {X.}\)

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 \(\mathfrak {X}\)

The boundary \(\partial \mathfrak {X}\) of \(\mathfrak {X}\) is the set of points (zu) where \(\text {Im}_{\delta }(z)=u{\bar{u}}\), together with a unique point “at infinity” \(c_{\infty }\) represented by the line \(^{t}(1:0:0)\). The lines represented by \(\partial \mathfrak {X}\) are the isotropic lines in \(V_{\mathbb {R}}\). The set of cusps \(\mathcal {C}\mathfrak {X}\) is the set of \({\mathcal {K}}\)-rational isotropic lines. If \(s\in {\mathcal {K}}\) and \(r\in \mathbb {Q}\), we write
$$\begin{aligned} c_{s}^{r}=(r+\delta s{\bar{s}}/2,s). \end{aligned}$$
(1.12)
Then \(\mathcal {C}\mathfrak {X}=\left\{ c_{s}^{r}|r\in \mathbb {Q},s\in {\mathcal {K}}\right\} \cup \{c_{\infty }\}\). The group \(G=\mathbf {G}(\mathbb {Q})\) acts transitively on the cusps.
The stabilizer of a cusp is a Borel subgroup in \(G_{\infty }\). Since G acts transitively on the cusps, we may assume that our cusp is \(c_{\infty }\). It is then easy to check that its stabilizer \(P_{\infty }\) has the form \(P_{\infty }=M_{\infty }N_{\infty }\), where
$$\begin{aligned} M_{\infty }= & {} \left\{ tm(\alpha ,\beta )=t\left( \begin{array}{ccc} \alpha &{} &{} \\ &{}\quad \beta &{} \\ &{} &{}\quad {\bar{\alpha }}^{-1} \end{array} \right) |\,t\in \mathbb {R}_{+}^{\times },\,\alpha \in \mathbb {C}^{\times },\beta \in \mathbb {C}^{1}\right\} , \end{aligned}$$
(1.13)
$$\begin{aligned} N_{\infty }= & {} \left\{ n(u,r)=\left( \begin{array}{ccc} 1 &{}\quad \delta {\bar{u}} &{}\quad r+\delta u{\bar{u}}/2 \\ &{}\quad 1 &{}\quad u \\ &{} &{}\quad 1 \end{array} \right) |\,u\in \mathbb {C},r\in \mathbb {R}\right\} . \end{aligned}$$
(1.14)
The matrix \(tm(\alpha ,\beta )\) belongs to \(U_{\infty }\) if and only if \(t=1, \) and to \(SU_{\infty }\) if furthermore \(\beta ={\bar{\alpha }}/\alpha \). The group \(N_{\infty }\) is contained in \(SU_{\infty }\). Since \(N=N_{\infty }\cap G\) still acts transitively on the set of finite cusps \(c_{s}^{r}\), we conclude that G acts doubly transitively on \(\mathcal {C}\mathfrak {X}. \)
Of particular interest to us will be the geodesics connecting an interior point (zu) to a cusp \(c\in \mathcal {C}\mathfrak {X}\). If \((z,u)=n(u,r)m(d,1)x_{0}\) (recall \(x_{0}=\,^{t}(\delta /2:0:1)\)) where d is real and positive (i.e. \(r=\mathfrak {R}z\) and \(d=\sqrt{\lambda (z,u)}\)), then the geodesic connecting (zu) to \(c_{\infty }\) can be described by the formula
$$\begin{aligned} \gamma _{u}^{r}(t)= & {} n(u,r)m(t,1)x_{0} \nonumber \\= & {} (r+\delta (u{\bar{u}}+t^{2})/2,u)\text { (}d\le t<\infty ). \end{aligned}$$
(1.15)
The same geodesic extends in the opposite direction for \(0<t\le d\), and if u and r lie in \({\mathcal {K}}\), it ends there in the cusp \(c_{u}^{r}\). We shall call \(\gamma _{u}^{r}(t)\) the geodesic retraction of \(\mathfrak {X}\) to the cusp \(c_{\infty }\). As \(0<t<\infty \), these parallel geodesics exhaust \(\mathfrak {X}\).

1.2 Picard modular surfaces over \(\mathbb {C}\)

1.2.1 Lattices and their arithmetic groups

Fix an \({\mathcal {O}}_{{\mathcal {K}}}\)-invariant lattice \(L\subset V\) which is self-dual in the sense that
$$\begin{aligned} L=\left\{ u\in V|\,\left\langle u,v\right\rangle \in \mathbb {Z\,\,\forall }v\in L\right\} . \end{aligned}$$
(1.16)
Equivalently, L is its own \({\mathcal {O}}_{{\mathcal {K}}}\)-dual with respect to the hermitian pairing (, ). We assume also that the Steinitz class1 of L as an \({\mathcal {O}}_{{\mathcal {K}}}\)-module is \([{\mathcal {O}}_{{\mathcal {K}}}]\), or, what amounts to the same, that L is a free \({\mathcal {O}}_{{\mathcal {K}}}\)-module. When we introduce the Shimura variety later on, we shall relax this last assumption, but the resulting scheme will be disconnected (over \(\mathbb {C} \)).
Fix an integer \(N\ge 1\) and let
$$\begin{aligned} \varGamma =\varGamma _{L}(N)=\left\{ g\in G|\,gL=L\,\text {and }g(u)\equiv u \mod NL\,\,\forall u\in L\right\} , \end{aligned}$$
(1.17)
a discrete subgroup of \(G_{\infty }\). It is easy to see that if \(N\ge 3\), then \(\varGamma \) is torsion free, acts freely and faithfully on \(\mathfrak {X}\), and is contained in \(SU_{\infty }\). From now on, we assume that this is the case.

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 \(\varGamma \)).

It is important to know the classification of lattices L as above (self-dual and \({\mathcal {O}}_{{\mathcal {K}}}\)-free). Let \(e_{1},e_{2},e_{3}\) be the standard basis of \({\mathcal {K}}^{3}\). Let
$$\begin{aligned} L_{0}=\text {Span}_{{\mathcal {O}}_{{\mathcal {K}}}}\{\delta e_{1},e_{2},e_{3}\} \end{aligned}$$
(1.18)
and
$$\begin{aligned} L_{1}={\text {Span}}_{{\mathcal {O}}_{{\mathcal {K}}}}\left\{ \frac{\delta }{2}e_{1}+e_{3},e_{2},\frac{\delta }{2}e_{1}-e_{3}\right\} . \end{aligned}$$
(1.19)
These two lattices are self-dual and, of course, \({\mathcal {O}}_{{\mathcal {K}}}\)-free. The following theorem is based on the local–global principle and a classification of lattices over \(\mathbb {Q}_{p}\) by Shimura [35].

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

We denote by \(X_{\varGamma }\) the complex surface \(\varGamma \backslash \mathfrak {X.}\) Since the action of \(\varGamma \) is free, \(X_{\varGamma }\) is smooth. We describe a topological compactification of \(X_{\varGamma }\). A standard neighborhood of the cusp \(c_{\infty }\) in \(\mathfrak {X}\) is an open set of the form
$$\begin{aligned} \varOmega _{R}=\left\{ (z,u)|\lambda (z,u)>R\right\} . \end{aligned}$$
(1.20)
The set \(\mathcal {C}_{\varGamma }=\varGamma \backslash \mathcal {C}\mathfrak {X}\) is finite, and we write \(c_{\varGamma }=\varGamma c\). We let \(X_{\varGamma }^{*}\) be the disjoint union of \(X_{\varGamma }\) and \(\mathcal {C}_{\varGamma }\). Let \(\varGamma _\mathrm{cusp}\) be the stabilizer of \(c_{\infty }\) in \(\varGamma \). We topologize \(X_{\varGamma }^{*}\) by taking \(\varGamma _\mathrm{cusp}\backslash \varOmega _{R}\cup \{c_{\infty ,\varGamma }\}\) as a basis of neighborhoods at \(c_{\infty ,\varGamma }. \) If \(c=g(c_{\infty })\) where \(g\in U\), we take \(g(g^{-1}\varGamma _\mathrm{cusp}g\backslash \varOmega _{R})\cup \{c_{\varGamma }\}\) instead. The following theorem is well known.

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 \(X_{\varGamma }\)

With \(x\in \mathfrak {X}\) and with our choice of L, we shall now associate a PEL structure \(\underline{A}_{x}=(A_{x},\lambda _{x},\iota _{x},\alpha _{x})\) where
  1. (1)

    \(A_{x}\) is a 3-dimensional complex abelian variety,

     
  2. (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. (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. (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)
     
Let \(W_{x}\) be the negative definite complex line in \(V_{\mathbb {R}}=\mathbb {C}^{3} \) defined by x, and \(W_{x}^{\perp }\) its orthogonal complement, a positive definite plane. Let \(J_{x}\) be the complex structure which is multiplication by i on \(W_{x}^{\perp }\) and by \(-i\) on \(W_{x}\). Let \(A_{x}=(V_{\mathbb {R}},J_{x})/L\). Then the polarization form \(\langle ,\rangle \) is a Riemann form on L, which determines a principal polarization on \(A_{x} \) as usual. The action of \({\mathcal {O}}_{{\mathcal {K}}}\) is derived from the underlying \({\mathcal {K}}\) structure of V. As we have reversed the complex structure on \(W_{x}\), the CM type is now (2,1). Finally the level N structure \(\alpha _{x}\) is the identity map.
If \(\gamma \in \varGamma \), then \(\gamma \) induces an isomorphism between \(\underline{A}_{x}\) and \(\underline{A}_{\gamma (x)}\). Conversely, if \(\underline{A}_{x}\) and \(\underline{A}_{x^{\prime }}\) are isomorphic structures, it is easy to see that \(x^{\prime }\) and x must belong to the same \(\varGamma \)-orbit. It follows that points of \(X_{\varGamma }\) are in a bijection with PEL structures of the above type for which the triple
$$\begin{aligned} (H_{1}(A_{x},\mathbb {Z}),\iota _{x},\left\langle ,\right\rangle _{\lambda _{x}}) \end{aligned}$$
(1.22)
is isomorphic to \((L,\iota ,\left\langle ,\right\rangle )\) (here \(\iota \) refers to the \({\mathcal {O}}_{{\mathcal {K}}}\) action on L), with the further condition that \(\alpha _{x}\) is compatible with the isomorphism between L and \(H_{1}(A_{x},\mathbb {Z})\) in the sense that we have a commutative diagram
$$\begin{aligned} \begin{array}{lllllllll} 0 &{} \rightarrow &{} L &{} \rightarrow &{} N^{-1}L &{} \rightarrow &{} N^{-1}L/L &{} \rightarrow &{} 0 \\ &{} &{} \downarrow &{} &{} \downarrow &{} &{} \downarrow \alpha _{x} &{} &{} \\ 0 &{} \rightarrow &{} H_{1}(A_{x},\mathbb {Z}) &{} \rightarrow &{} N^{-1}H_{1}(A_{x},\mathbb {Z}) &{} \rightarrow &{} A_{x}[N] &{} \rightarrow &{} 0 \end{array} . \end{aligned}$$
(1.23)

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 5-dimensional 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.

Let \(\mathbb {C}^{3}\) be given the usual complex structure, and let \(a\in {\mathcal {O}}_{{\mathcal {K}}}\) act on it via the matrix
$$\begin{aligned} \iota ^{\prime }(a)=\left( \begin{array}{lll} a &{} &{} \\ &{}\quad a &{} \\ &{} &{}\quad {\bar{a}} \end{array} \right) . \end{aligned}$$
(1.24)
Given \(x=(z,u)\in \mathfrak {X}\), consider the lattice
$$\begin{aligned} L_{x}^{\prime }={\text {Span}}_{\iota ^{\prime }({\mathcal {O}}_{{\mathcal {K}}})}\left\{ \left( \begin{array}{c} 0 \\ 1 \\ 1 \end{array} \right) ,\left( \begin{array}{c} -1 \\ 0 \\ -u \end{array} \right) ,\left( \begin{array}{c} u \\ -z/\delta \\ z/\delta \end{array} \right) \right\} \subset \mathbb {C}^{3}. \end{aligned}$$
(1.25)
The map \(T_{x}:\mathbb {C}^{3}\rightarrow \mathbb {C}^{3}\) which sends \(\zeta =\,^{t}(\zeta _{1},\zeta _{2},\zeta _{3})\) to
$$\begin{aligned} T(\zeta )=\lambda (z,u)^{-1}\left\{ -\zeta _{1}\left( \begin{array}{c} {\bar{u}}z \\ (z-{\bar{z}})/\delta \\ {\bar{u}} \end{array} \right) -\zeta _{2}\left( \begin{array}{c} {\bar{z}}+\delta u{\bar{u}} \\ u \\ 1 \end{array} \right) +{\bar{\zeta }}_{3}\left( \begin{array}{c} z \\ u \\ 1 \end{array} \right) \right\} \end{aligned}$$
(1.26)
is a complex linear isomorphism between \(\mathbb {C}^{3}\) and \((V_{\mathbb {R}},J_{x})\). In fact, it sends \(\mathbb {C}e_{1}+\mathbb {C}e_{2}\) linearly to \(W_{x}^{\perp }\) and \(\mathbb {C}e_{3}\) conjugate-linearly to \(W_{x}\). It intertwines the \(\iota ^{\prime }\) action of \({\mathcal {O}}_{{\mathcal {K}}}\) on \(\mathbb {C}^{3}\) with its \(\iota \)-action on \((V_{\mathbb {R}},J_{x})\). It furthermore sends \(L_{x}^{\prime }\) to L. In fact, an easy computation shows that it sends the three generating vectors of \(L_{x}^{\prime }\) to \(\delta e_{1},e_{2}\) and \(e_{3}\), respectively. We conclude that \(T_{x}\) induces an isomorphism
$$\begin{aligned} T_{x}:A_{x}^{\prime }=\mathbb {C}^{3}/L_{x}^{\prime }\simeq A_{x}. \end{aligned}$$
(1.27)
Consider the differential forms \(\text {d}\zeta _{1},\text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\). As their periods along any \(l\in L_{x}^{\prime }\) vary holomorphically in z and u, the five coordinates \(\zeta _{1},\zeta _{2},\zeta _{3},z,u\) form a local system of coordinates on the family \(A^{\prime }\rightarrow \mathfrak {X}\). Identifying \(A^{\prime }\) with A allows us to put the desired complex structure on the family A. Alternatively, we may define \(A^{\prime }\) as the quotient of \(\mathbb {C}^{3}\times \mathfrak {X}\) by \(\zeta \mapsto \zeta +l(z,u)\) where l(zu) varies over the holomorphic lattice-sections.

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

Fix \(N\ge 3\) and \(L=L_{0}\subset V\) as before. Let R be an \(R_{0}\)-algebra. Let \(\mathcal {M}(R)\) be the collection of (isomorphism classes of) PEL structures \((A,\lambda ,\iota ,\alpha )\) where
  1. (1)

    A / R is an abelian scheme of relative dimension 3

     
  2. (2)

    \(\lambda :A\simeq A^{t}\) is a principal polarization

     
  3. (3)

    \(\iota :\) \({\mathcal {O}}_{{\mathcal {K}}}\rightarrow End(A/R)\) is a homomorphism such that (1) \(\iota \) makes Lie(A / R) a locally free R-module 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. (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 that
    $$\begin{aligned} \left\langle \alpha \left( \frac{l}{N}\right) ,\alpha \left( \frac{l^{\prime }}{N}\right) \right\rangle _{\lambda }=\nu _{N}\left( \left\langle l,l^{\prime }\right\rangle \mod N\right) . \end{aligned}$$
    (1.28)
    In 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).
     
In view of Lemma 1.1, the last condition of symplectic liftability is void if \(D_{{\mathcal {K}}}\) is odd, while if \(D_{{\mathcal {K}}}\) is even it is equivalent to the following condition ([2], I.3.1):
  • 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 \(\mathbb {Z}_{2}\).

The choice of \(L_{0}\) was arbitrary. If we took \(L_{1}\) as our basic lattice, we would get a similar moduli problem.

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 quasi-projective 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 R-point 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 \(\text {Sh}_{K}\)

We briefly recall the interpretation of the Picard modular surface as a canonical model of a Shimura variety. The symmetric domain \(\mathfrak {X}\) can be interpreted as a \(G_{\infty }\)-conjugacy class of homomorphisms
$$\begin{aligned} h:\mathbb {S}=\text {res}_{\mathbb {R}}^{\mathbb {C}}\mathbb {G}_{m}\rightarrow \mathbf {G}_{\mathbb {R}} \end{aligned}$$
(1.29)
turning \((\mathbf {G},\mathfrak {X})\) into a Shimura datum in the sense of Deligne [7]. In fact \(h_{x}(i)=J_{x}\). The reflex field associated with this datum turns out to be \({\mathcal {K}}\). Let \(K_{\infty }\) be the stabilizer of \(x_{0}\) in \(G_{\infty }\) and \(K_{f}^{0}\subset \mathbf {G}(\mathbb {A}_{f})\) the subgroup stabilizing \({\widehat{L}}=L\otimes {\widehat{\mathbb {Z}}}\). Let \(K_{f}\) be the subgroup of \(K_{f}^{0}\) inducing the identity on L / NL. Let \(K=K_{\infty }K_{f}\subset \mathbf {G}(\mathbb {A})\). Then the Shimura variety \({\text {Sh}}_{K}\) is a complex quasi-projective variety whose complex points are isomorphic, as a complex manifold, to the double coset space
$$\begin{aligned} {\text {Sh}}_{K}(\mathbb {C})\simeq & {} \mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K\nonumber \\= & {} \mathbf {G}(\mathbb {Q})\backslash (\mathfrak {X}\times \mathbf {G}(\mathbb {A}_{f})/K_{f}). \end{aligned}$$
(1.30)
The theory of Shimura varieties provides a canonical model for \({\text {Sh}}_{K} \) over \({\mathcal {K}}\). The following important theorem complements the one on the representability of the functor \(\mathcal {M}\).

Theorem 1.4

The canonical model of \({\text {Sh}}_{K}\) is the generic fiber \(S_{{\mathcal {K}}}\) of S.

Let us explain only how to associate with a point of \({\text {Sh}}_{K}(\mathbb {C})\) a point in \(S(\mathbb {C})\). For that we have to associate an element of \(\mathcal {M}(\mathbb {C})\) with \(g\in \mathbf {G}(\mathbb {A})\), and show that the structures associated with g and to \(\gamma gk\) (\(\gamma \in G,k\in K\)) are isomorphic. Let \(x=x_{g}=g_{\infty }(x_{0})\in \mathfrak {X}\). Let \(L_{g}=g_{f}({\widehat{L}})\cap V\) (the intersection taking place in \(V_{\mathbb {A}}={\widehat{L}}\otimes \mathbb {Q}\)) and
$$\begin{aligned} A_{g}=(V_{\mathbb {R}},J_{x})/L_{g}. \end{aligned}$$
(1.31)
Note that \(J_{x}\) depends only on \(g_{\infty }K_{\infty }\) and \(L_{g}\) only on \(g_{f}K_{f}^{0}\), so \(A_{g}\) depends only on \(gK^{0}\).
Let \({\tilde{\mu }}(g)\) be the unique positive rational number such that for every prime p,
$$\begin{aligned} {\text {ord}}_{p}{\tilde{\mu }}(g)={\text {ord}}_{p}\mu (g_{p}). \end{aligned}$$
(1.32)
Such a rational number exists since \(\mu (g_{p})\) is a p-adic unit for almost all p and \(\mathbb {Q}\) has class number 1. We claim that
$$\begin{aligned} \left\langle ,\right\rangle _{g}={\tilde{\mu }}(g)^{-1}\left\langle ,\right\rangle :L_{g}\times L_{g}\rightarrow \mathbb {Q} \end{aligned}$$
induces a principal polarization \(\lambda _{g}\) on \(A_{g}\). That this is a (rational) Riemann form follows from the fact that \((u,v)_{J_{x}}=\left\langle u,J_{x}v\right\rangle +i\left\langle u,v\right\rangle \) is hermitian positive definite. That \(\left\langle ,\right\rangle _{g}\) is indeed \(\mathbb {Z} \)-valued and \(L_{g}\) is self-dual follows from the choice of \({\tilde{\mu }}(g)\) since locally at p the dual of \(g_{p}L_{p}\) under \(\left\langle ,\right\rangle :V_{p}\times V_{p}\rightarrow \mathbb {Q}_{p}\) is \(\mu (g_{p})^{-1}g_{p}L_{p}\). We conclude that there exists a unique polarization \(\lambda _{g}:A_{g}\rightarrow A_{g}^{t}\) such that
$$\begin{aligned} \left\langle u,v\right\rangle _{\lambda _{g}}=\exp (2\pi il\left\langle u,v\right\rangle _{g}) \end{aligned}$$
(1.33)
for every \(u,v\in A_{g}[l]=l^{-1}L_{g}/L_{g}\) and every \(l\ge 1\). This polarization is principal.

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.

Finally \(\alpha _{g}\) is derived from
$$\begin{aligned} N^{-1}L/L=N^{-1}{\widehat{L}}/{\widehat{L}}\overset{g_{f}}{\rightarrow }N^{-1}{\widehat{L}}_{g}/{\widehat{L}}_{g}=N^{-1}L_{g}/L_{g}=A_{g}[N]. \end{aligned}$$
(1.34)
We note that \(\alpha _{g}\) depends only on gK because \(K_{f}\subset K_{f}^{0}\) is the principal level-N subgroup, and that it lifts to level \(N^{\prime }\) structure for any multiple \(N^{\prime }\) of N, by the same formula. The isomorphism \(\nu _{N,g}\) between \(\mathbb {Z}/N\mathbb {Z}\) and \(\mu _{N}(\mathbb {C})\) that makes (1.28) work is self-evident [see (1.49)]. Let \(\underline{A}_{g}\in \mathcal {M}(\mathbb {C})\) be the structure just constructed.

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 \({\text {Sh}}_{K}\)

Recall that \(\mathbf {G}^{\prime }=\mathbf {SU}=\ker (\nu :\mathbf {G}\rightarrow \mathbf {T})\). Since \(\mathbf {G}^{\prime }\) is simple and simply connected, strong approximation holds and
$$\begin{aligned} \mathbf {G}^{\prime }(\mathbb {A})=\mathbf {G}^{\prime }(\mathbb {Q})G_{\infty }^{\prime }K_{f}^{\prime }. \end{aligned}$$
(1.35)
Here \(K^{\prime }=K\cap \mathbf {G}^{\prime }(\mathbb {A})\), \(K_{f}^{\prime }=K\cap \mathbf {G}^{\prime }(\mathbb {A}_{f})\). From the connectedness of \(G_{\infty }^{\prime }\), we deduce that
$$\begin{aligned} \mathbf {G}^{\prime }(\mathbb {Q})\backslash \mathbf {G}^{\prime }(\mathbb {A})/K^{\prime } \end{aligned}$$
(1.36)
is connected.
As \(N\ge 3\), \(\nu (K)\cap {\mathcal {K}}^{\times }=\{1\}\). Here \({\mathcal {K}}^{\times }=\nu (\mathbf {G}(\mathbb {Q}))\), and it follows that
$$\begin{aligned} \mathbf {G}^{\prime }(\mathbb {Q})\backslash \mathbf {G}^{\prime }(\mathbb {A})/K^{\prime }\hookrightarrow \mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K \end{aligned}$$
(1.37)
is injective. We now claim (see also Theorem 2.4 and 2.5 of [7]) that
$$\begin{aligned} \nu :\pi _{0}(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K)\simeq \pi _{0}(\mathbf {T}(\mathbb {Q})\backslash \mathbf {T}(\mathbb {A})/\nu (K)) \end{aligned}$$
(1.38)
is a bijection. For \(\nu \) is surjective ([7] (0.2)) and continuous (on double coset spaces) so clearly induces a surjective map between the sets of connected components. On the other hand, if \([g_{1}]\) and \([g_{2}]\) (double cosets of \(g_{i}\in \mathbf {G}(\mathbb {A})\)) are mapped by \(\nu \) to the same connected component in \(\mathbf {T}(\mathbb {Q})\backslash \mathbf {T}(\mathbb {A})/\nu (K)\), then since \(G_{\infty }\) is mapped onto the connected component of the identity in \(\mathbf {T}(\mathbb {Q})\backslash \mathbf {T}(\mathbb {A})/\nu (K)\), modifying \(g_{1}\) by an element of \(G_{\infty }\) we may assume that
$$\begin{aligned} \nu ([g_{1}])=\nu ([g_{2}])\in \mathbf {T}(\mathbb {Q})\backslash \mathbf {T}(\mathbb {A})/\nu (K), \end{aligned}$$
(1.39)
without changing the connected component in which \([g_{1}]\) lies. Once this has been established, for appropriate representatives \(g_{i}\) of the double cosets, \(g_{1}^{-1}g_{2}\in \mathbf {G}^{\prime }(\mathbb {A})\), so by the connectedness of \(\mathbf {G}^{\prime }(\mathbb {Q})\backslash \mathbf {G}^{\prime }(\mathbb {A})/K^{\prime }\), \([g_{1}]\) and \([g_{2}]\) lie in the same connected component of \(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K\).
The group \(\pi _{0}(\mathbf {T}(\mathbb {Q})\backslash \mathbf {T}(\mathbb {A})/\nu (K))\) is the group
$$\begin{aligned} {\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{\mathbb {A}}^{\times }/\mathbb {C}^{\times }\nu (K_{f})={\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{f}^{\times }/\nu (K_{f}). \end{aligned}$$
(1.40)
It sits in a short exact sequence
$$\begin{aligned} 0\rightarrow \mu _{{\mathcal {K}}}\backslash U_{{\mathcal {K}}}/\nu (K_{f})\rightarrow {\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{f}^{\times }/\nu (K_{f})\overset{cl}{\rightarrow }Cl_{{\mathcal {K}}}\rightarrow 0, \end{aligned}$$
(1.41)
where \(U_{{\mathcal {K}}}\) is the product of local units at all the finite primes and \(Cl_{{\mathcal {K}}}\) is the class group.

1.3.4 The cl and \(\nu _{N}\) invariants of a connected component

The norm \(N:{\mathcal {K}}^{\times }\rightarrow \mathbb {Q}^{\times }\) satisfies \(N\circ \nu =\nu \nu ^{\rho }=\mu \) and hence induces a map
$$\begin{aligned} {\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{f}^{\times }/\nu (K_{f})\rightarrow \mathbb {Q}_{+}^{\times }\backslash \mathbb {Q}_{f}^{\times }/\mu (K_{f}). \end{aligned}$$
(1.42)
Using the lattice L as an integral structure in V, we see that \(\mathbf {G}\) comes from a group scheme \(\mathbf {G}_{\mathbb {Z}}\) over \(\mathbb {Z}\), whose points in any ring A are
$$\begin{aligned} \mathbf {G}_{\mathbb {Z}}(A)=\left\{ (g,\mu )\in GL_{{\mathcal {O}}_{{\mathcal {K}}}\otimes A}(L_{A})\times A^{\times }|\,\left\langle gu,gv\right\rangle =\mu \left\langle u,v\right\rangle \right\} . \end{aligned}$$
(1.43)
We likewise get that \(\mu \) is a homomorphism from \(\mathbf {G}_{\mathbb {Z}}\) to \(\mathbb {G}_{m}\). The diagram
$$\begin{aligned} \begin{array}{lll} \mathbf {G}_{\mathbb {Z}}(\mathbb {Z}_{p}) &{} \overset{\mu }{\rightarrow } &{} \mathbb {Z}_{p}^{\times } \\ \downarrow &{} &{} \downarrow \\ \mathbf {G}_{\mathbb {Z}}(\mathbb {Z}_{p}/N\mathbb {Z}_{p}) &{} \overset{\mu }{\rightarrow } &{} (\mathbb {Z}_{p}/N\mathbb {Z}_{p})^{\times } \end{array} \end{aligned}$$
(1.44)
commutes, \(\mathbf {G}_{\mathbb {Z}}(\mathbb {Z}_{p})=K_{p}^{0}\) and the kernel of \(\mathbf {G}_{\mathbb {Z}}(\mathbb {Z}_{p})\rightarrow \mathbf {G}_{\mathbb {Z}}(\mathbb {Z}_{p}/N\mathbb {Z}_{p})\) is \(K_{p}\). This shows that \(\mu (K_{f})\subset \mathbb {{\hat{Z}}}^{\times }(N)\), the product of local units congruent to \(1\mod N\). But
$$\begin{aligned} \mathbb {Q}_{+}^{\times }\backslash \mathbb {Q}_{f}^{\times }/\mathbb {{\hat{Z}}}^{\times }(N)=(\mathbb {Z}/N\mathbb {Z})^{\times }. \end{aligned}$$
(1.45)
To conclude, we have shown the existence of two maps from the set of connected components:
$$\begin{aligned}&cl:\pi _{0}(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K)\rightarrow Cl_{{\mathcal {K}}} \end{aligned}$$
(1.46)
$$\begin{aligned}&\nu _{N}:\pi _{0}(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K)\rightarrow (\mathbb {Z}/N\mathbb {Z})^{\times }. \end{aligned}$$
(1.47)
These two maps are independent: together they map \(\pi _{0}(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K)\) onto \(Cl_{{\mathcal {K}}}\times (\mathbb {Z}/N\mathbb {Z})^{\times }\). On the other hand, they have a non-trivial common kernel, which grows with N, as is evident from the interpretation of \({\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{f}^{\times }/\nu (K_{f})\) as the Galois group of a certain class field extension of \({\mathcal {K}}\). The map cl gives the restriction to the Hilbert class field, while the map \(\nu _{N}\) gives the restriction to the cyclotomic field \(\mathbb {Q}(\mu _{N})\). We have singled out cl and \(\nu _{N}\), because when \(N\ge 3\), they have an interpretation in terms of the complex points of \({\text {Sh}}_{K}\).

Proposition 1.5

Let \([g]\in \mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K={\text {Sh}}_{K}(\mathbb {C})\). Then
  1. (i)

    cl([g]) is the Steinitz class of the lattice \(L_{g}=g_{f}({\hat{L}})\cap V\) in \(Cl_{{\mathcal {K}}}\).

     
  2. (ii)

    \(\nu _{N}([g])\) is (essentially) the \(\nu _{N,g}\) that appears in the definition of \(\alpha _{g}\) (see 1.3.1).

     

Proof

  1. (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.

     
  2. (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, so
    $$\begin{aligned} \nu _{N}([g])={\tilde{\mu }}(g_{f})^{-1}\mu (g_{f})\mod N. \end{aligned}$$
    (1.48)
    Now 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
    $$\begin{aligned} \left\langle \alpha _{g}(u),\alpha _{g}(v)\right\rangle _{\lambda _{g}}= & {} \exp \left( 2\pi iN\left\langle g_{f}u,g_{f}v\right\rangle _{g}\right) \nonumber \\= & {} \exp \left( 2\pi i{\tilde{\mu }}(g_{f})^{-1}N\left\langle g_{f}u,g_{f}v\right\rangle \right) \nonumber \\= & {} \exp \left( 2\pi i{\tilde{\mu }}(g_{f})^{-1}\mu (g_{f})N\left\langle u,v\right\rangle \right) \nonumber \\= & {} \exp \left( 2\pi i\nu _{N}([g])N\left\langle u,v\right\rangle \right) \end{aligned}$$
    (1.49)
    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 \(\exp (2\pi i(\cdot ))\).
     

1.3.5 The complex uniformization

Recall that \(\mathfrak {X}=G_{\infty }/K_{\infty }\) and that it was equipped with a base point \(x_{0}\) (corresponding to \((z,u)=(\delta _{{\mathcal {K}}}/2,0)\) in the Siegel domain of the second kind). Let \(1=g_{1},\dots ,g_{m}\in \mathbf {G}(\mathbb {A}_{f})\) \((m=\#({\mathcal {K}}^{\times }\backslash {\mathcal {K}}_{f}^{\times }/\nu (K_{f})))\) be representatives of the connected components of \(\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K\), and define congruence groups
$$\begin{aligned} \varGamma _{j}=\mathbf {G}(\mathbb {Q})\cap g_{j}K_{f}g_{j}^{-1}. \end{aligned}$$
(1.50)
We write \([x,g_{j}]\) for \(\mathbf {G}(\mathbb {Q})(x,g_{j}K_{f})\in \mathbf {G}(\mathbb {Q})\backslash (\mathfrak {X}\times \mathbf {G}(\mathbb {A}_{f})/K_{f})=\mathbf {G}(\mathbb {Q})\backslash \mathbf {G}(\mathbb {A})/K\). Then \([x^{\prime },g_{j}]=[x,g_{j}]\) if and only if \(x^{\prime }=\gamma x\) for \(\gamma \in \varGamma _{j}\). The map
$$\begin{aligned} \coprod _{j=1}^{m}X_{\varGamma _{j}}=\coprod _{j=1}^{m}\varGamma _{j}\backslash \mathfrak {X}\simeq {\text {Sh}}_{K}(\mathbb {C}) \end{aligned}$$
(1.51)
sending \(\varGamma _{j}x\) to \([x,g_{j}]\) is an isomorphism.

Note that \(\varGamma _{1}=\varGamma \) is the principal level-N congruence subgroup in \(\mathbf {G}_{\mathbb {Z}}(\mathbb {Z})\), the stabilizer of L. Similarly, \(\varGamma _{j}\) is the principal level-N 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 \(\mathbf {G}^{\prime }(\mathbb {Q})=\mathbf {SU}(\mathbb {Q})\).

We get a similar decomposition to connected components (as an algebraic surface)
$$\begin{aligned} S_{\mathbb {C}}=\coprod _{j=1}^{m}S_{\varGamma _{j}} \end{aligned}$$
(1.52)
and we write \(S_{\mathbb {C}}^{*}=\coprod _{j=1}^{m}S_{\varGamma _{j}}^{*}\) for the Baily–Borel compactification.

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 \(L=L_{0}\).

As \(N\ge 3\), elements of \(\varGamma \) stabilizing \(c_{\infty }\) lie in \(N_{\infty }\).2 The computations, which we omit, are somewhat simpler if N is even, an assumption made for the rest of this section. Let
$$\begin{aligned} \varGamma _\mathrm{cusp}=\varGamma \cap N_{\infty }. \end{aligned}$$
(1.53)

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 \(r\in 2^{-1}ND_{{\mathcal {K}}}\mathbb {Z}\) .

Let \(M=N|D_{{\mathcal {K}}}|\) in case (i) and \(M=2^{-1}N|D_{{\mathcal {K}}}| \) in case (ii). This is the width of the cusp \(c_{\infty }\). Let
$$\begin{aligned} q=q(z)=e^{2\pi iz/M}. \end{aligned}$$
(1.54)
For \(R>0\), the domain \(\varOmega _{R}=\left\{ (z,u)\in \mathfrak {X}|\,\lambda (z,u)>R\right\} \) is invariant under \(\varGamma _\mathrm{cusp}\) and if R is large enough, two points of it are \(\varGamma \)-equivalent if and only if they are \(\varGamma _\mathrm{cusp}\)-equivalent. A sufficiently small punctured neighborhood of \(c_{\infty }\) in \(X_{\varGamma }^{*}\) therefore looks like \(\varGamma _\mathrm{cusp}\backslash \varOmega _{R}\). As
$$\begin{aligned} n(s,r)(z,u)=(z+\delta {\bar{s}}(u+s/2)+r,u+s) \end{aligned}$$
(1.55)
we obtain the following description of \(\varGamma _\mathrm{cusp}\backslash \varOmega _{R}. \) Let \(\Lambda =N{\mathcal {O}}_{{\mathcal {K}}}\) and \(E=\mathbb {C}/\Lambda \), an elliptic curve with complex multiplication by \({\mathcal {O}}_{{\mathcal {K}}}\). Let \(\mathcal {T}\) be the quotient
$$\begin{aligned} \mathcal {T}=(\mathbb {C}\times \mathbb {C})/\Lambda \end{aligned}$$
(1.56)
where the action of \(s\in \Lambda \) is via
$$\begin{aligned}{}[s]:(t,u)\mapsto \left( e^{2\pi i\delta {\bar{s}}(u+s/2)/M}t,u+s\right) . \end{aligned}$$
(1.57)
It is a line bundle over E via the second projection. We denote the class of (tu) modulo the action of \(\Lambda \) by [tu].

Proposition 1.7

Let \(\mathcal {T}_{R}\subset \mathcal {T}\) be the disk bundle consisting of all the points [tu] where
$$\begin{aligned} |t|<e^{-\pi |\delta |(R+u{\bar{u}})/M}. \end{aligned}$$
(1.58)
(This condition is invariant under the action of \(\Lambda \).) Let \(\mathcal {T}_{R}^{\prime }\) be the punctured disk bundle obtained by removing the zero section from \(\mathcal {T}_{R}\). Then the map \((z,u)\mapsto (q(z),u)\) induces an analytic isomorphism between \(\varGamma _\mathrm{cusp}\backslash \varOmega _{R} \) and \(\mathcal {T}_{R}^{\prime }\).

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 (zu) and \(c_{\infty }\) and not only on their images modulo \(\varGamma \).

The line bundle \(\mathcal {T}\) is the inverse of an ample line bundle on E. In fact, \(\mathcal {T}^{\vee }\) is the N-th (resp. 2N-th) power of one of the four basic theta line bundles if \(d_{{\mathcal {K}}}\equiv 1\mod 4\) (resp. \(d_{{\mathcal {K}}}\equiv 2,3\mod 4)\). A basic theta function of the lattice \(\Lambda \) satisfies, for \(u\in \mathbb {C}\) and \(s\in \Lambda \),
$$\begin{aligned} \theta (u+s)=\alpha (s)e^{2\pi {\bar{s}}(u+s/2)/|\delta |N^{2}}\theta (u) \end{aligned}$$
(1.59)
where \(\alpha :\Lambda \rightarrow \pm 1\) is a quasi-character (see [31], p. 25). Recalling the relation between M and N, and the assumption that N was even, we easily get the relation between \(\mathcal {T}\) and the theta line bundles.

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 sub-bundle 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 well-defined 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

(i) There exists a projective scheme \({\bar{S}}\), smooth over \(R_{0}\), of relative dimension 2, together with an open dense immersion of S in \({\bar{S}}\), and a proper morphism \(p:{\bar{S}}\rightarrow S^{*}\), making the following diagram commutative
$$\begin{aligned} \begin{array}{lll} S &{} \rightarrow &{} {\bar{S}} \\ \downarrow &{} \overset{p}{\swarrow } &{} \\ S^{*} &{} &{} \end{array}. \end{aligned}$$
(1.60)
  1. (ii)
    As a complex manifold, there is an isomorphism
    $$\begin{aligned} {\bar{S}}_{\mathbb {C}}\simeq \coprod _{j=1}^{m}{\bar{X}}_{\varGamma _{j}}, \end{aligned}$$
    (1.61)
    extending the isomorphism of \(S_{\mathbb {C}}\) with \(\coprod _{j=1}^{m}X_{\varGamma _{j}}\).
     
  2. (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 moduli-theoretic interpretation of C as a moduli space for semi-abelian schemes with a suitable action of \({\mathcal {O}}_{{\mathcal {K}}}\) and a “level-N 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 multiplication-by-Q 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 pull-back 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 semi-abelian scheme \({\mathcal {A}}\)

1.5.1 The universal semi-abelian scheme over \({\bar{S}}\)

As Larsen and Bellaïche explain, the universal abelian scheme \(\pi :{\mathcal {A}}\rightarrow S\) extends canonically to a semi-abelian 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 semi-abelian 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 \((\mathfrak {a},\mathfrak {b})\).

The above discussion defines a homomorphism (of fppf sheaves over Spec\((R_{N})\))
$$\begin{aligned} E\rightarrow Ext_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(B,\mathfrak {a}\otimes \mathbb {G}_{m}). \end{aligned}$$
(1.62)
As we shall see soon, the Ext group is represented by an elliptic curve with CM by \({\mathcal {O}}_{{\mathcal {K}}}\), defined over \(R_{N}\), and this map is an isogeny.

1.5.2 \({\mathcal {O}}_{{\mathcal {K}}}\)-semi-abelian schemes of type \((\mathfrak {a},\mathfrak {b})\)

We digress to discuss the moduli space for semi-abelian schemes of the type found above points of E. Let R be an \(R_{0}\)-algebra, B an elliptic curve over R with complex multiplication by \({\mathcal {O}}_{{\mathcal {K}}}\) and CM type \(\varSigma \), and \(\mathfrak {a}\) an ideal of \({\mathcal {O}}_{{\mathcal {K}}}. \) Consider a semi-abelian scheme \(\mathcal {G}\) over R, endowed with an \({\mathcal {O}}_{{\mathcal {K}}}\) action \(\iota :{\mathcal {O}}_{{\mathcal {K}}}\rightarrow End(\mathcal {G})\), and a short exact sequence
$$\begin{aligned} 0\rightarrow \mathfrak {a}\otimes \mathbb {G}_{m}\rightarrow \mathcal {G}\rightarrow B\rightarrow 0 \end{aligned}$$
(1.63)
of \({\mathcal {O}}_{{\mathcal {K}}}\)-group schemes over R. We call all this data a semi-abelian scheme of type \((\mathfrak {a},B)\) (over R). The group classifying such structures is \(Ext_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(B,\mathfrak {a}\otimes \mathbb {G}_{m})\). Any \(\chi \in \mathfrak {a}^{*}=\text {Hom}(\mathfrak {a},\mathbb {Z})\) defines, by push-out, an extension \(\mathcal {G}_{\chi }\) of B by \(\mathbb {G}_{m}\), hence a point of \(B^{t}=Ext^{1}(B,\mathbb {G}_{m})\). We therefore get a homomorphism from \(Ext_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(B,\mathfrak {a}\otimes \mathbb {G}_{m})\) to \(\text {Hom}(\mathfrak {a}^{*},B^{t})\). A simple check shows that its image is in \(\text {Hom}_{{\mathcal {O}}_{{\mathcal {K}}}}(\mathfrak {a}^{*},B^{t})=\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\) and that this construction yields an isomorphism
$$\begin{aligned} \text {Ext}_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(B,\mathfrak {a}\otimes \mathbb {G}_{m})\simeq \delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}. \end{aligned}$$
(1.64)
Here we have used the canonical identification \(\mathfrak {a}^{*}=\delta _{{\mathcal {K}}}^{-1}\mathfrak {a}^{-1}\) (via the trace pairing). Although \((\delta _{{\mathcal {K}}})\) is a principal ideal, so can be ignored, it is better to keep track of its presence. We emphasize that the CM type of \(B^{t}\), with the natural action of \({\mathcal {O}}_{{\mathcal {K}}}\) derived from its action on B, is \({\bar{\varSigma }}\) rather than \(\varSigma \).

Thus over \(\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\), there is a universal semi-abelian 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 pull-back (specialization) with respect to a unique map Spec\((R^{\prime })\rightarrow \delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\).

When \(R=\mathbb {C}\), \(B\simeq \mathbb {C}/\mathfrak {b}\) for a unique ideal class \([\mathfrak {b}]\) (with \({\mathcal {O}}_{{\mathcal {K}}}\) acting via \(\varSigma \)). Then, canonically, \(B^{t}=\mathbb {C}/\delta _{{\mathcal {K}}}^{-1}\overline{\mathfrak {b}}^{-1}\) (with \({\mathcal {O}}_{{\mathcal {K}}}\) acting via \({\bar{\varSigma }}\)). The pairing between the lattices, \(\mathfrak {b}\times \delta _{{\mathcal {K}}}^{-1}\overline{\mathfrak {b}}^{-1}\rightarrow \mathbb {Z}\) is \((x,y)\mapsto Tr_{{\mathcal {K}}/\mathbb {Q}}(x{\bar{y}})\). Since the \({\mathcal {O}}_{{\mathcal {K}}}\) action on \(B^{t}\) is via \({\bar{\varSigma }}\),
$$\begin{aligned} \text {Ext}_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(\mathbb {C}/\mathfrak {b},\mathfrak {a}\otimes \mathbb {G}_{m})\simeq \delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}\mathbb {C}/\delta _{{\mathcal {K}}}^{-1}\overline{\mathfrak {b}}^{-1}=\mathbb {C}/\overline{\mathfrak {a}}\overline{\mathfrak {b}}^{-1}. \end{aligned}$$
(1.65)
The universal semi-abelian variety \(\mathcal {G}(\mathfrak {a},B)\) will now be denoted \(\mathcal {G}(\mathfrak {a},\mathfrak {b})\). In 1.6.2, we give a complex analytic model of this \(\mathcal {G}(\mathfrak {a},\mathfrak {b})\).

1.6 Degeneration of \({\mathcal {A}}\) along a geodesic connecting to a cusp

1.6.1 The degeneration to a semi-abelian 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.

Consider the geodesic (1.15) connecting (zu) to \(c_{\infty }\). Consider the universal abelian scheme in the moving lattice model [cf (1.27)]. Of the three vectors used to span \(L_{x}^{\prime }\) over \({\mathcal {O}}_{{\mathcal {K}}}\) in (1.25), the first two do not depend on z. As u is fixed along the geodesic, they are not changed. The third vector represents a cycle that vanishes at the cusp (together with all its \({\mathcal {O}}_{{\mathcal {K}}}\)-multiples). We conclude that \(A_{x}^{\prime }\) degenerates to
$$\begin{aligned} \mathbb {C}^{3}/\text {Span}_{\iota ^{\prime }({\mathcal {O}}_{{\mathcal {K}}})}\left\{ \left( \begin{array}{l} 0 \\ 1 \\ 1 \end{array} \right) ,\left( \begin{array}{l} 1 \\ 0 \\ u \end{array} \right) \right\} . \end{aligned}$$
(1.66)
Making the change of variables \((\zeta _{1}^{\prime },\zeta _{2}^{\prime },\zeta _{3}^{\prime })=(\zeta _{1},\zeta _{2}+{\bar{u}}\zeta _{1},\zeta _{3})\) does not alter the \({\mathcal {O}}_{{\mathcal {K}}}\) action and gives the more symmetric model
$$\begin{aligned} \mathcal {G}_{u}=\mathbb {C}^{3}/\text {Span}_{\iota ^{\prime }({\mathcal {O}}_{{\mathcal {K}}})}\left\{ \left( \begin{array}{l} 0 \\ 1 \\ 1 \end{array} \right) ,\left( \begin{array}{l} 1 \\ {\bar{u}} \\ u \end{array} \right) \right\} \end{aligned}$$
(1.67)
(but note that \(\zeta _{2}^{\prime }\), unlike \(\zeta _{2}\), does not vary holomorphically in the family \(\{\mathcal {G}_{u}\}\), only in each fiber individually).
Let \(e(x)=e^{2\pi ix}:\mathbb {C}\rightarrow \mathbb {C}^{\times }\) be the exponential map, with kernel \(\mathbb {Z}\). For any ideal \(\mathfrak {a}\) of \({\mathcal {O}}_{{\mathcal {K}}}\), it induces a map
$$\begin{aligned} e_{\mathfrak {a}}:\mathfrak {a}\otimes \mathbb {C}\rightarrow \mathfrak {a}\otimes \mathbb {C}^{\times } \end{aligned}$$
(1.68)
with kernel \(\mathfrak {a}\otimes 1\). As usual we identify \(\mathfrak {a}\otimes \mathbb {C}\) with \(\mathbb {C}(\varSigma )\oplus \mathbb {C}({\bar{\varSigma }})\), sending \(a\otimes \zeta \mapsto (a\zeta ,{\bar{a}}\zeta )\). We now note that if we use this identification to identify \(\mathbb {C}^{3}\) with \(\mathbb {C} \oplus ({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C})\) (an identification which is compatible with the \({\mathcal {O}}_{{\mathcal {K}}}\) action), then the \(\iota ^{\prime }({\mathcal {O}}_{{\mathcal {K}}})\)-span of the vector \(^{t}(0,1,1)\) is just the kernel of \(e_{{\mathcal {O}}_{{\mathcal {K}}}}\). We conclude that
$$\begin{aligned} \mathcal {G}_{u}\simeq \{\mathbb {C}\oplus ({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C}^{\times })\}/L_{u} \end{aligned}$$
(1.69)
where \(L_{u}\) is the sub-\({\mathcal {O}}_{{\mathcal {K}}}\)-module
$$\begin{aligned} L_{u}=\left\{ (s,e_{{\mathcal {O}}_{{\mathcal {K}}}}(s{\bar{u}},{\bar{s}}u))|\,s\in {\mathcal {O}}_{{\mathcal {K}}}\right\} . \end{aligned}$$
(1.70)
This clearly gives \(\mathcal {G}_{u}\) the structure of an \({\mathcal {O}}_{{\mathcal {K}}}\)-semi-abelian variety of type \(({\mathcal {O}}_{{\mathcal {K}}},{\mathcal {O}}_{{\mathcal {K}}})\), i.e. an extension
$$\begin{aligned} 0\rightarrow {\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C}^{\times }\rightarrow \mathcal {G}_{u}\rightarrow \mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}}\rightarrow 0. \end{aligned}$$
(1.71)

1.6.2 The analytic uniformization of the universal semi-abelian 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 (zu) to \(c_{\infty }\) with the analytic description of the universal semi-abelian variety of type \((\mathfrak {a},\mathfrak {b})\).

Proposition 1.12

Let \(\mathfrak {a}\) and \(\mathfrak {b}\) be two ideals of \({\mathcal {O}}_{{\mathcal {K}}}\). For \(u\in \mathbb {C}\) consider
$$\begin{aligned} \mathcal {G}_{u}\simeq \{\mathbb {C}\oplus (\mathfrak {a}\otimes \mathbb {C}^{\times })\}/L_{u} \end{aligned}$$
(1.72)
where
$$\begin{aligned} L_{u}=\left\{ (s,e_{\mathfrak {a}}(s{\bar{u}},{\bar{s}}u))|\,s\in \mathfrak {b}\right\} . \end{aligned}$$
(1.73)
Then \(\mathcal {G}_{u}\) is a semi-abelian variety of type \((\mathfrak {a},\mathfrak {b})\), any complex semi-abelian variety of this type is a \(\mathcal {G}_{u}\), and \(\mathcal {G}_{u}\simeq \mathcal {G}_{v}\) if and only if \(u-v\in \overline{\mathfrak {a}}\overline{\mathfrak {b}}^{-1}\).

Proof

That \(\mathcal {G}_{u}\) is a semi-abelian variety of type \((\mathfrak {a},\mathfrak {b}) \) is obvious. That any abelian variety of this type is a \(\mathcal {G}_{u}\) follows by passing to the universal cover \(\mathbb {C}^{2}(\varSigma )\oplus \mathbb {C} ({\bar{\varSigma }})\), and noting that by a change of variables in the \(\varSigma \)- and \({\bar{\varSigma }}\)-isotypical parts, we may assume that the lattice by which we divide is of the form
$$\begin{aligned} \mathfrak {a}\left( \begin{array}{l} 0 \\ 1 \\ 1 \end{array} \right) \oplus \mathfrak {b}\left( \begin{array}{l} 1 \\ {\bar{u}} \\ u \end{array} \right) . \end{aligned}$$
(1.74)
Finally, the map \(u\mapsto [\mathcal {G}_{u}]\) is a homomorphism \(\mathbb {C} \rightarrow Ext_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(\mathbb {C}/\mathfrak {b},\mathfrak {a}\otimes \mathbb {C}^{\times })\), so we only have to prove that \(\mathcal {G}_{u}\) is split if and only if \(u\in \overline{\mathfrak {a}}\overline{\mathfrak {b}}^{-1}\). But one can check easily that \(\mathcal {G}_{u}\) is trivial if and only if \((s{\bar{u}},{\bar{s}}u)\in \ker e_{\mathfrak {a}}=\mathfrak {a}\otimes 1=\{(a,{\bar{a}})|a\in \mathfrak {a}\}\) for every \(s\in \mathfrak {b}\), and this holds if and only if \(u\in \overline{\mathfrak {a}}\overline{\mathfrak {b}}^{-1}\).

Corollary 1.13

Let \(N\ge 3\) be even. Let \(c=c_{\infty }\) be the cusp at infinity. Then the map
$$\begin{aligned} E_{c}\rightarrow Ext_{{\mathcal {O}}_{{\mathcal {K}}}}^{1}(\mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}},{\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C}^{\times }) \end{aligned}$$
(1.75)
sending u to the isomorphism class of the semi-abelian variety above \(u\mod \Lambda \) is the isogeny of multiplication by N.

Proof

In view of the computations above, and the description of a neighborhood of \(E_{c}\) in \({\bar{X}}_{\varGamma }\) given in Proposition 1.7 this map is identified with the canonical map
$$\begin{aligned} \mathbb {C}/N{\mathcal {O}}_{{\mathcal {K}}}\rightarrow \mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}}. \end{aligned}$$
(1.76)

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 semi-abelian schemes of the type considered above, together with level-N structure (\(\mathcal {M}_{\infty ,N} \) structures in the language of [2]). A level-N structure on a semi-abelian variety \(\mathcal {G}\) of type \((\mathfrak {a},\mathfrak {b})\) consists of (i) a level-N structures \(\alpha :N^{-1}{\mathcal {O}}_{{\mathcal {K}}}/{\mathcal {O}}_{{\mathcal {K}}}\simeq \mathfrak {a}\otimes \mu _{N}\) on the toric part (ii) a level-N 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 \(\mathcal {G}[N]\rightarrow B[N]\).

Over \(c=c_{\infty }\), when \(\mathfrak {a}=\mathfrak {b}={\mathcal {O}}_{{\mathcal {K}}}\), there are obvious natural choices for \(\alpha \) and \(\beta \) (independent of u), but the splittings \(\gamma \) in (iii) form a torsor under \({\mathcal {O}}_{{\mathcal {K}}}/N{\mathcal {O}}_{{\mathcal {K}}}\). If we consider the splitting
$$\begin{aligned} \gamma _{u}:N^{-1}{\mathcal {O}}_{{\mathcal {K}}}/{\mathcal {O}}_{{\mathcal {K}}}\ni s\mapsto (s,e_{{\mathcal {O}}_{{\mathcal {K}}}}(s{\bar{u}},{\bar{s}}u))\mod L_{u} \end{aligned}$$
(1.77)
then the tuples \((\mathcal {G}_{u},\alpha ,\beta ,\gamma _{u})\) and \((\mathcal {G}_{v},\alpha ,\beta ,\gamma _{v})\) are isomorphic if and only if \(u\equiv v\mod N{\mathcal {O}}_{{\mathcal {K}}}\), i.e. if and only if u and v represent the same point of \(E_{c}\).

1.7 The basic automorphic vector bundles

1.7.1 Definition and first properties

Recall that we have denoted by \(\pi :{\mathcal {A}}\rightarrow {\bar{S}}\) the universal semi-abelian variety over \({\bar{S}}\) (over the base ring \(R_{0}\)). Let \(\omega _{{\mathcal {A}}}\) be the relative cotangent space at the origin of \({\mathcal {A}}\). If \(e:{\bar{S}}\rightarrow {\mathcal {A}}\) is the zero section,
$$\begin{aligned} \omega _{{\mathcal {A}}}=e^{*}\left( \varOmega _{{\mathcal {A}}/{\bar{S}}}^{1}\right) . \end{aligned}$$
(1.78)
This is a rank 3 vector bundle over \({\bar{S}}\) and the action of \({\mathcal {O}}_{{\mathcal {K}}}\) allows to decompose it according to types. We let
$$\begin{aligned} \mathcal {P}=\omega _{{\mathcal {A}}}(\varSigma ),\quad \mathcal {L}=\omega _{{\mathcal {A}}}({\bar{\varSigma }}). \end{aligned}$$
(1.79)
Then \(\mathcal {P}\) is a plane bundle, and \(\mathcal {L}\) a line bundle.
Over S (but not over the cuspidal divisor \(C={\bar{S}}\backslash S\)), we have the usual identification \(\omega _{{\mathcal {A}}}=\pi _{*}\varOmega _{{\mathcal {A}}/S}^{1}\). The relative de Rham cohomology of \({\mathcal {A}}/S\) is a rank 6 vector bundle sitting in an exact sequence (the Hodge filtration)
$$\begin{aligned} 0\rightarrow \omega _{{\mathcal {A}}}\rightarrow H_{dR}^{1}({\mathcal {A}}/S)\rightarrow R^{1}\pi _{*}{\mathcal {O}}_{{\mathcal {A}}}\rightarrow 0. \end{aligned}$$
(1.80)
Since, for any abelian scheme, \(R^{1}\pi _{*}{\mathcal {O}}_{{\mathcal {A}}}=\omega _{{\mathcal {A}}^{t}}^{\vee }\) (canonical isomorphism, see [31]), and \(\lambda :{\mathcal {A}}\rightarrow {\mathcal {A}}^{t}\) is an isomorphism which reverses CM types, we obtain an exact sequence
$$\begin{aligned} 0\rightarrow \omega _{{\mathcal {A}}}\rightarrow H_{dR}^{1}({\mathcal {A}}/S)\rightarrow \omega _{{\mathcal {A}}}^{\vee }(\rho )\rightarrow 0. \end{aligned}$$
(1.81)
The notation \(\mathcal {M}(\rho )\) means that \(\mathcal {M}\) is a vector bundle with an \({\mathcal {O}}_{{\mathcal {K}}}\) action and in \(\mathcal {M}(\rho )\) the vector bundle structure is that of \(\mathcal {M}\), but the \({\mathcal {O}}_{{\mathcal {K}}}\) action is conjugated. Decomposing according to types, we have two short exact sequences
$$\begin{aligned} 0\rightarrow & {} \mathcal {P}\rightarrow H_{dR}^{1}({\mathcal {A}}/S)(\varSigma )\rightarrow \mathcal {L}^{\vee }(\rho )\rightarrow 0 \\ 0\rightarrow & {} \mathcal {L}\rightarrow H_{dR}^{1}({\mathcal {A}}/S)({\bar{\varSigma }})\rightarrow \mathcal {P}^{\vee }(\rho )\rightarrow 0. \nonumber \end{aligned}$$
(1.82)
The pairing \(\left\langle ,\right\rangle _{\lambda }\) on \(H_{dR}^{1}({\mathcal {A}}/S)\) induced by the polarization is \({\mathcal {O}}_{S}\)-linear, alternating, perfect, and satisfies \(\left\langle \iota (a)x,y\right\rangle _{\lambda }=\left\langle x,\iota ({\bar{a}})y\right\rangle _{\lambda }\). It follows that \(H_{dR}^{1}({\mathcal {A}}/S)(\varSigma )\) and \(H_{dR}^{1}({\mathcal {A}}/S)({\bar{\varSigma }})\) are maximal isotropic subspaces and are set in duality. As \(\omega _{{\mathcal {A}}}\) is also isotropic, this pairing induces pairings
$$\begin{aligned} \mathcal {P}\times \mathcal {P}^{\vee }(\rho )\rightarrow {\mathcal {O}}_{S},\,\,\mathcal {L}\times \mathcal {L}^{\vee }(\rho )\rightarrow {\mathcal {O}}_{S}. \end{aligned}$$
(1.83)
These two pairings are the tautological pairings between a vector bundle and its dual.
Another consequence of this discussion that we wish to record is the canonical isomorphism over S
$$\begin{aligned} \det \mathcal {P}=\mathcal {L}(\rho )\otimes \det \left( H_{dR}^{1}({\mathcal {A}}/S)(\varSigma )\right) . \end{aligned}$$
(1.84)

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 pull-backs to \(\mathfrak {X}\).

To trivialize \(\mathcal {L}_{an}\), we must choose a nowhere vanishing global section over \(\mathfrak {X}\). As usual, we describe it only on the connected component containing the standard cusp, corresponding to \(j=1\) (where \(L=L_{g_{1}}=L_{0}\)). Recalling the “moving lattice model” (1.27) and the coordinates \(\zeta _{1},\zeta _{2},\zeta _{3}\) introduced there, we note that \(\text {d}\zeta _{3}\) is a generator of \(\mathcal {L}_{an}=\omega _{{\mathcal {A}}}({\bar{\varSigma }})\). For reasons that will become clear later (cf Sect. 1.12), we use \(2\pi i\cdot \text {d}\zeta _{3}\) to trivialize \(\mathcal {L}_{an}\) over \(\mathfrak {X}\). Suppose
$$\begin{aligned} \gamma =\left( \begin{array}{lll} a_{1} &{}\quad b_{1} &{}\quad c_{1} \\ a_{2} &{}\quad b_{2} &{}\quad c_{2} \\ a_{3} &{}\quad b_{3} &{}\quad c_{3} \end{array} \right) \in \varGamma \subset SU_{\infty }. \end{aligned}$$
(1.85)
If \(\gamma (z,u)=(z^{\prime },u^{\prime })\), then
$$\begin{aligned} z^{\prime }=\frac{a_{1}z+b_{1}u+c_{1}}{a_{3}z+b_{3}u+c_{3}},\quad u^{\prime }=\frac{a_{2}z+b_{2}u+c_{2}}{a_{3}z+b_{3}u+c_{3}} \end{aligned}$$
(1.86)
and
$$\begin{aligned} \gamma \left( \begin{array}{l} z \\ u \\ 1 \end{array} \right) =j(\gamma ;z,u)\left( \begin{array}{l} z^{\prime } \\ u^{\prime } \\ 1 \end{array} \right) ,\quad j(\gamma ;z,u)=a_{3}z+b_{3}u+c_{3}. \end{aligned}$$
(1.87)

Lemma 1.14

The following relation holds for every \(\gamma \in U_{\infty }\)
$$\begin{aligned} \lambda (z,u)=\lambda (\gamma (z,u))\cdot |j(\gamma ;z,u)|^{2}. \end{aligned}$$
(1.88)

Proof

Let \(v=v(z,u)=\,^{t}(z,u,1)\). Then
$$\begin{aligned} \lambda (z,u)=-(v,v). \end{aligned}$$
(1.89)
As \(v(\gamma (z,u))=j(\gamma ;z,u)^{-1}\cdot \gamma (v(z,u))\) the lemma follows from \((\gamma v,\gamma v)=(v,v)\).

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

For \(x=(z,u)\in \mathfrak {X}\) let
$$\begin{aligned} v_{3}(z,u)=\lambda (z,u)^{-1}\left( \begin{array}{l} z \\ u \\ 1 \end{array} \right) \in \mathcal {W}_{x}. \end{aligned}$$
(1.90)
Then (i) \(v_{3}(z,u)\) is a nowhere vanishing holomorphic section of \(\mathcal {W}\), (ii) \(\left\langle \text {d}\zeta _{3},v_{3}\right\rangle \equiv 1\), (iii) the automorphy factor corresponding to \(\text {d}\zeta _{3}\) is the function \(j(\gamma ;z,u)\).

Proof

Since, by construction, \(\text {d}\zeta _{3}\) is a nowhere vanishing holomorphic section of \(\mathcal {L}\) (over \(\mathfrak {X}\)), (i) follows from (ii). To prove (ii), we transfer \(v_{3}(z,u)\) to the moving lattice model and get \(^{t}(0,0,1)\), which is the dual vector to \(\text {d}\zeta _{3}\). To prove (iii), we compute in \(V_{\mathbb {R}}\) (with the original complex structure!)
$$\begin{aligned} \frac{\gamma _{*}v_{3}(z,u)}{v_{3}(\gamma (z,u))}=\frac{\lambda (\gamma (z,u))}{\lambda (z,u)}j(\gamma ;z,u)=\overline{j(\gamma ;z,u)}^{-1}, \end{aligned}$$
(1.91)
and recall that since \(W_{\gamma (z,u)}\) is precisely the line where the complex structure in \((V_{\mathbb {R}},J_{\gamma (z,u)})\) has been reversed, in \((V_{\mathbb {R}},J_{\gamma (z,u)})\) we have
$$\begin{aligned} \frac{\gamma _{*}v_{3}(z,u)}{v_{3}(\gamma (z,u))}=j(\gamma ;z,u)^{-1}. \end{aligned}$$
(1.92)
Dualizing, we get (\(x=(z,u)\))
$$\begin{aligned} \frac{(\gamma ^{-1})^{*}\text {d}\zeta _{3}|_{x}}{\text {d}\zeta _{3}|_{\gamma (x)}}=j(\gamma ,x). \end{aligned}$$
(1.93)
This concludes the proof.

Consider next the plane bundle \(\mathcal {P}_{an}\). As we will only be interested in scalar-valued modular forms, we do not compute its matrix-valued 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

There is an isomorphism of analytic line bundles over \(X_{\varGamma }\),
$$\begin{aligned} \det \mathcal {P}_{an}\simeq \mathcal {L}_{an}. \end{aligned}$$
(1.94)
Moreover, \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) is a nowhere vanishing holomorphic section of \(\det \mathcal {P}_{an}\) over \(\mathfrak {X}\), and the factor of automorphy corresponding to it is \(j(\gamma ;z,u)\).

Proof

Since a holomorphic line bundle on \(X_{\varGamma }=\varGamma \backslash \mathfrak {X}\) is determined, up to an isomorphism, by its factor of automorphy, and \(j(\gamma ;z,u)\) is the factor of automorphy of \(\mathcal {L}_{an}\) corresponding to \(\text {d}\zeta _{3}\), it is enough to prove the second statement. Let \(\mathcal {U}=\mathcal {V}(\varSigma )\) be the plane bundle dual to \(\mathcal {P}_{an}\). Let
$$\begin{aligned} v_{1}(z,u)=-\lambda (z,u)^{-1}\left( \begin{array}{c} {\bar{u}}z \\ (z-{\bar{z}})/\delta \\ {\bar{u}} \end{array} \right) \end{aligned}$$
(1.95)
and
$$\begin{aligned} v_{2}(z,u)=-\lambda (z,u)^{-1}\left( \begin{array}{c} {\bar{z}}+\delta u{\bar{u}} \\ u \\ 1 \end{array} \right) \end{aligned}$$
(1.96)
(considered as vectors in \((V_{\mathbb {R}},J_{x})=\mathcal {V}_{x}\)). As we have seen in (1.27), these two vector fields are sections of \(\mathcal {U}\) and at each point \(x\in \mathfrak {X}\) form a basis dual to \(\text {d}\zeta _{1}\) and \(\text {d}\zeta _{2}\). It follows that they are holomorphic sections and that \(v_{1}\wedge v_{2}\) is the basis dual to \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\). We must show that the factor of automorphy corresponding to \(v_{1}\wedge v_{2}\) is \(j(\gamma ;z,u)^{-1}\), i.e. that
$$\begin{aligned} \frac{\gamma _{*}(v_{1}\wedge v_{2}(z,u))}{v_{1}\wedge v_{2}(\gamma (z,u))}=j(\gamma ;z,u)^{-1}. \end{aligned}$$
(1.97)
Working in \(V_{\mathbb {R}}=\mathbb {C}^{3}\) (with the original complex structure)
$$\begin{aligned}&\frac{\gamma _{*}(v_{1}\wedge v_{2}(z,u))}{v_{1}\wedge v_{2}(\gamma (z,u))}\cdot \frac{1}{\overline{j(\gamma ;z,u)}} \nonumber \\&\quad =\frac{\gamma _{*}(v_{1}\wedge v_{2}(z,u))}{v_{1}\wedge v_{2}(\gamma (z,u))}\cdot \frac{\gamma _{*}v_{3}(z,u)}{v_{3}(\gamma (z,u))} =\frac{\gamma _{*}(v_{1}\wedge v_{2}\wedge v_{3}(z,u))}{v_{1}\wedge v_{2}\wedge v_{3}(\gamma (z,u))}. \end{aligned}$$
(1.98)
But
$$\begin{aligned} v_{1}\wedge v_{2}\wedge v_{3}(z,u)=\delta \lambda (z,u)^{-1}e_{1}\wedge e_{2}\wedge e_{3}, \end{aligned}$$
(1.99)
because
$$\begin{aligned} \det \left( \begin{array}{ccc} {\bar{u}}z &{}\quad {\bar{z}}+\delta u{\bar{u}} &{}\quad z \\ (z-{\bar{z}})/\delta &{}\quad u &{}\quad u \\ {\bar{u}} &{}\quad 1 &{}\quad 1 \end{array} \right) =\delta \lambda (z,u)^{2}. \end{aligned}$$
(1.100)
As \(\det (\gamma )=1\), this gives
$$\begin{aligned} \frac{\gamma _{*}(v_{1}\wedge v_{2}(z,u))}{v_{1}\wedge v_{2}(\gamma (z,u))}\cdot \frac{1}{\overline{j(\gamma ;z,u)}}=\frac{\lambda (\gamma (z,u))}{\lambda (z,u)}=\frac{1}{j(\gamma ;z,u)\overline{j(\gamma ;z,u)}}, \end{aligned}$$
(1.101)
and the proof is complete.

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

Since \(Pic({\bar{S}}_{{\mathcal {K}}})\subset Pic({\bar{S}}_{\mathbb {C}})\), it is enough to prove the proposition over \(\mathbb {C}\). By GAGA, it is enough to establish the triviality of \(\det \mathcal {P}\otimes \mathcal {L}^{-1}\) in the analytic category. For each connected component \(X_{\varGamma }\) of \(S_{\mathbb {C}}\), the section \((\text {d}\zeta _{1}\wedge \text {d}\zeta _{2})\otimes \text {d}\zeta _{3}^{-1}\) descends from \(\mathfrak {X}\) to \(X_{\varGamma }\), because \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\) have the same factor of automorphy \(j(\gamma ,x)\) (\(\gamma \in \varGamma \), \(x\in \mathfrak {X}\)). This section is nowhere vanishing on \(X_{\varGamma }\) and extends to a nowhere vanishing section on \({\bar{X}}_{\varGamma }\), trivializing \(\det \mathcal {P}\otimes \mathcal {L}^{-1}\). In fact, if c is the standard cusp, \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\) are already well defined and nowhere vanishing sections of \(\det \mathcal {P}\) and \(\mathcal {L}\) in the neighborhood
$$\begin{aligned} \overline{\varGamma _\mathrm{cusp}\backslash \varOmega _{R}}=(\varGamma _\mathrm{cusp}\backslash \varOmega _{R})\cup E_{c} \end{aligned}$$
(1.102)
of \(E_{c}\) (see 1.4.1). This is a consequence of the fact that \(j(\gamma ,x)=1\) for \(\gamma \in \varGamma _\mathrm{cusp}. \)

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

Let R be an \(R_{0}\)-algebra. A modular form of weight \(k\ge 0\) and level \(N\ge 3\) defined over R is an element of the finite R-module
$$\begin{aligned} M_{k}(N,R)=H^{0}\left( {\bar{S}}_{R},\mathcal {L}^{k}\right) . \end{aligned}$$
(1.103)
We usually omit the subscript R, remembering that \({\bar{S}}\) is now to be considered over R. The well-known Koecher principle says that \(H^{0}({\bar{S}},\mathcal {L}^{k})=H^{0}(S,\mathcal {L}^{k})\). See [2], Section 2.2, for an arithmetic proof valid integrally over any \(R_{0}\)-algebra R. A cusp form is an element of the space
$$\begin{aligned} M_{k}^{0}(N,R)=H^{0}\left( {\bar{S}},\mathcal {L}^{k}\otimes {\mathcal {O}}(C)^{\vee }\right) . \end{aligned}$$
(1.104)
As we shall see below (cf Corollary 1.23), if \(k\ge 3\), there is an isomorphism \(\mathcal {L}^{k}\otimes {\mathcal {O}}(C)^{\vee }\simeq \varOmega _{{\bar{S}}}^{2}\otimes \mathcal {L}^{k-3}\). In particular, cusp forms of weight 3 are “the same” as holomorphic 2-forms on \({\bar{S}}\).
An alternative definition (à la Katz) of a modular form of weight k and level N defined over R, is as a “rule” f which assigns to every R-scheme T, and every \(\underline{A}=(A,\lambda ,\iota ,\alpha )\in \mathcal {M}(T)\), together with a nowhere vanishing section \(\omega \in H^{0}(T,\omega _{A/T}({\bar{\varSigma }}))\), an element \(f(\underline{A},\omega )\in H^{0}(T,{\mathcal {O}}_{T})\) satisfying
  • \(f(\underline{A},\lambda \omega )=\lambda ^{-k}f(\underline{A},\omega )\) for every \(\lambda \in H^{0}(T,{\mathcal {O}}_{T})^{\times }\)

  • The “rule” f is compatible with base change \(T^{\prime }/T\).

Indeed, if f is an element of \(M_{k}(N,R)\), then given such an \(\underline{A}\) and \(\omega \), the universal property of S produces a unique morphism \(\varphi :T\rightarrow S\) over R, \(\varphi ^{*}{\mathcal {A}}=A\), and we may let \(f(\underline{A},\omega )=\varphi ^{*}f/\omega ^{k}\). Conversely, given such a rule f we may cover S by Zariski open sets T where \(\mathcal {L}\) is trivialized, and then the sections \(f({\mathcal {A}}_{T},\omega _{T})\omega _{T}^{k}\) (\(\omega _{T}\) a trivializing section over T) glue to give \(f\in M_{k}(N,R)\). While viewing f as a “rule” rather than a section is a matter of language, it is sometimes more convenient to use this language.

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

If \(k\ge 3\) (resp. \(k\ge 6\)), then \(M_{k}^{0}(N,R)\) (resp. \(M_{k}(N,R)\)) is a locally free finite R-module, and the base-change homomorphism
$$\begin{aligned} R^{\prime }\otimes M_{k}^{0}(N,R)\simeq M_{k}^{0}(N,R^{\prime }) \end{aligned}$$
(1.105)
is an isomorphism (resp. base change for \(M_{k}(N,R)\)).

Bellaïche considers only weights divisible by 3, but his proofs generalize to all k (cf remark on the bottom of p. 43 in [2]).

Over \(\mathbb {C}\), pulling back to \(\mathfrak {X}\) and using the trivialization of \(\mathcal {L}\) given by the nowhere vanishing section \(2\pi i\cdot \text {d}\zeta _{3} \), a modular form of weight k is a collection \((f_{j})_{1\le j\le m}\) of holomorphic functions on \(\mathfrak {X}\) satisfying
$$\begin{aligned} f_{j}(\gamma (z,u))=j(\gamma ;z,u)^{k}f_{j}(z,u)\quad \forall \gamma \in \varGamma _{j} \end{aligned}$$
(1.106)
(the Koecher principle means that no condition has to be imposed at the cusps).

1.8 The Kodaira–Spencer isomorphism

Let \(\pi :A\rightarrow S\) be an abelian scheme of relative dimension 3, as in the Picard moduli problem. The Gauss–Manin connection
$$\begin{aligned} \nabla :H_{dR}^{1}(A/S)\rightarrow H_{dR}^{1}(A/S)\otimes _{{\mathcal {O}}_{S}}\varOmega _{S}^{1} \end{aligned}$$
(1.107)
defines the Kodaira–Spencer map
$$\begin{aligned} \text {KS}\in \text {Hom}_{{\mathcal {O}}_{S}}\left( \omega _{A}\otimes _{{\mathcal {O}}_{S}}\omega _{A^{t}},\varOmega _{S}^{1}\right) \end{aligned}$$
(1.108)
as the composition of the maps
(1.109)
and finally using \(\text {Hom}(L,M^{\vee }\otimes N)=\text {Hom}(L\otimes M,N)\). Recall that if A is endowed with an \({\mathcal {O}}_{{\mathcal {K}}}\) action via \(\iota \), then the induced action of \(a\in {\mathcal {O}}_{K}\) on \(A^{t}\) is induced from the action on Pic(A), taking a line bundle \(\mathcal {M}\) to \(\iota (a)^{*}\mathcal {M}\). As the polarization \(\lambda :A\rightarrow A^{t}\) is \({\mathcal {O}}_{S}\)-linear but satisfies \(\lambda \circ \iota (a)=\iota (a^{\rho })\circ \lambda \), it follows that the induced \({\mathcal {O}}_{{\mathcal {K}}}\) action on \(A^{t}\) is of type (1, 2) and hence \(\omega _{A^{t}}^{\vee }\) is of type (1, 2).

Lemma 1.20

The map KS induces maps
$$\begin{aligned} \mathrm{KS}(\varSigma ):\omega _{A}(\varSigma )\rightarrow \omega _{A^{t}}^{\vee }(\varSigma )\otimes _{{\mathcal {O}}_{S}}\varOmega _{S}^{1} \nonumber \\ \mathrm{KS}({\bar{\varSigma }}):\omega _{A}({\bar{\varSigma }})\rightarrow \omega _{A^{t}}^{\vee }({\bar{\varSigma }})\otimes _{{\mathcal {O}}_{S}}\varOmega _{S}^{1} \end{aligned}$$
(1.110)
hence maps, denoted by the same symbols,
$$\begin{aligned} \mathrm{KS}(\varSigma ):\omega _{A}(\varSigma )\otimes _{{\mathcal {O}}_{S}}\omega _{A^{t}}(\varSigma )\rightarrow \varOmega _{S}^{1} \nonumber \\ \mathrm{KS}({\bar{\varSigma }}):\omega _{A}({\bar{\varSigma }})\otimes _{{\mathcal {O}}_{S}}\omega _{A^{t}}({\bar{\varSigma }})\rightarrow \varOmega _{S}^{1}. \end{aligned}$$
(1.111)
The CM-type-reversing isomorphism \(\lambda ^{*}:\omega _{A^{t}}\rightarrow \omega _{A}\) induced by the principal polarization satisfies
$$\begin{aligned} \mathrm{KS}(\varSigma )(\lambda ^{*}x\otimes y)=\mathrm{KS}({\bar{\varSigma }})(\lambda ^{*}y\otimes x) \end{aligned}$$
(1.112)
for all \(x\in \omega _{A^{t}}({\bar{\varSigma }})\) and \(y\in \omega _{A^{t}}(\varSigma )\).

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

If S is the Picard modular surface and \(A={\mathcal {A}}\) is the universal abelian variety, then
$$\begin{aligned} \mathrm{KS}(\varSigma ):\omega _{{\mathcal {A}}}(\varSigma )\otimes _{{\mathcal {O}}_{S}}\omega _{{\mathcal {A}}^{t}}(\varSigma )\rightarrow \varOmega _{S}^{1} \end{aligned}$$
(1.113)
is an isomorphism, and so is \(\mathrm{KS}({\bar{\varSigma }})\).

Proof

This is well known and follows from deformation theory. For a self-contained proof, see [2], Prop. II.2.1.5.

Proposition 1.22

The Kodaira–Spencer map induces a canonical isomorphism of vector bundles over S
$$\begin{aligned} \mathcal {P}\otimes \mathcal {L}\simeq \varOmega _{S}^{1}. \end{aligned}$$
(1.114)

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 semi-abelian \({\mathcal {O}}_{{\mathcal {K}}} \)-threefold of type \((\mathfrak {a},B)\), over \(\delta _{{\mathcal {K}}}\mathfrak {a}\otimes _{{\mathcal {O}}_{{\mathcal {K}}}}B^{t}\). The semi-abelian scheme \({\mathcal {A}}\) over E is the pull-back 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 sub-bundle \(\mathcal {P}_{0}=\omega _{B}\). 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

Working over \({\mathcal {K}}_{N}\), let \(E_{j}\) (\(1\le j\le h\)) be the connected components of C. Then
$$\begin{aligned} \varOmega _{{\bar{S}}}^{2}\simeq \mathcal {L}^{3}\otimes \bigotimes _{j=1}^{h}{\mathcal {O}}(E_{j})^{\vee }. \end{aligned}$$
(1.115)

Proof

By [15] II.6.5, \(\varOmega _{{\bar{S}}}^{2}\simeq \mathcal {L}^{3}\otimes \bigotimes _{j=1}^{h}{\mathcal {O}}(E_{j})^{n_{j}}\) for some integers \(n_{j}\) and we want to show that \(n_{j}=-1\) for all j. By the adjunction formula on the smooth surface \({\bar{S}}\), if we denote by \(K_{{\bar{S}}}\) a canonical divisor, \({\mathcal {O}}(K_{{\bar{S}}})=\varOmega _{{\bar{S}}}^{2}\), then
$$\begin{aligned} 0=2g_{E_{j}}-2=E_{j}\cdot (E_{j}+K_{{\bar{S}}}). \end{aligned}$$
(1.116)
We conclude that
$$\begin{aligned} \deg \left( \varOmega _{{\bar{S}}}^{2}|_{E_{j}}\right) =E_{j}\cdot K_{{\bar{S}}}=-E_{j}\cdot E_{j}>0. \end{aligned}$$
(1.117)
Here \(E_{j}\cdot E_{j}<0\) because \(E_{j}\) can be contracted to a point (Grauert’s theorem). As \(\mathcal {L}|_{E_{j}}\) and \({\mathcal {O}}(E_{i})|_{E_{j}}\) \((i\ne j)\) are trivial, we get
$$\begin{aligned} -E_{j}\cdot E_{j}=n_{j}E_{j} \cdot E_{j}, \end{aligned}$$
(1.118)
hence \(n_{j}=-1\) as desired.

1.10 Fourier–Jacobi expansions

1.10.1 The infinitesimal retraction

We follow the arithmetic theory of Fourier–Jacobi expansions as developed in [2]. Let \({\widehat{S}}\) be the formal completion of \({\bar{S}}\) along the cuspidal divisor \(C={\bar{S}}\backslash S\). We work over \(R_{0}\) and denote by \(C^{(n)}\) the n-th infinitesimal neighborhood of C in \({\bar{S}}\). The closed immersion \(i:C\hookrightarrow {\widehat{S}}\) admits a canonical left inverse \(r:{\widehat{S}}\rightarrow C\), a retraction satisfying \(r\circ i=Id_{C}\). This is not automatic, but rather a consequence of the rigidity of tori, as explained in [2], Proposition II.2.4.2. As a corollary, the universal semi-abelian scheme \({\mathcal {A}}_{/C^{(n)}}\) is the pull-back of \({\mathcal {A}}_{/C}\) via r. The same therefore holds for \(\mathcal {P}\) and \(\mathcal {L}\), namely there are natural isomorphisms \(r^{*}(\mathcal {P}|_{C})\simeq \mathcal {P}|_{C^{(n)}}\) and \(r^{*}(\mathcal {L}|_{C})\simeq \mathcal {L}|_{C^{(n)}}\). As a consequence, the filtration
$$\begin{aligned} 0\rightarrow \mathcal {P}_{0}\rightarrow \mathcal {P}\rightarrow \mathcal {P}_{\mu }\rightarrow 0 \end{aligned}$$
(1.119)
extends canonically to \(C^{(n)}\). Since \(\mathcal {L},\mathcal {P}_{0}\) and \(\mathcal {P}_{\mu }\) are trivial on C, they are trivial over \(C^{(n)}\) as well.

1.10.2 Arithmetic Fourier–Jacobi expansions

We fix an arbitrary Noetherian \(R_{0}\)-algebra R and consider all our schemes over R, without a change in notation. As usual, we let \({\mathcal {O}}_{{\widehat{S}}}=\lim _{\leftarrow }{\mathcal {O}}_{C^{(n)}}\) (a sheaf in the Zariski topology on C). Via \(r^{*}\), this is a sheaf of \({\mathcal {O}}_{C}\)-modules. Choose a global nowhere vanishing section \(s\in H^{0}(C,\mathcal {L})\) trivializing \(\mathcal {L}\). Such a section is unique up to a unit of R on each connected component of C. This s determines an isomorphism
$$\begin{aligned} \mathcal {L}^{k}|_{{\widehat{S}}}\simeq {\mathcal {O}}_{{\widehat{S}}},\quad f\mapsto f/(r^{*}s)^{k} \end{aligned}$$
(1.120)
for each k, hence a ring homomorphism
$$\begin{aligned} FJ:\oplus _{k=0}^{\infty }M_{k}(N,R)\rightarrow H^{0}(C,{\mathcal {O}}_{{\widehat{S}}}). \end{aligned}$$
(1.121)
We call FJ(f) the (arithmetic) Fourier–Jacobi expansion of f. It depends on s in an obvious way.

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 \(-E_{j}^{2}>0\).

Now \(r^{*}\) supplies, for every \(n\ge 2\), a canonical splitting of
$$\begin{aligned} 0\rightarrow \mathcal {I}/\mathcal {I}^{n}\rightarrow {\mathcal {O}}_{{\bar{S}}}/\mathcal {I}^{n}\overset{\curvearrowleft }{\rightarrow }{\mathcal {O}}_{{\bar{S}}}/\mathcal {I}\rightarrow 0. \end{aligned}$$
(1.122)
Inductively, we get a direct sum decomposition
$$\begin{aligned} {\mathcal {O}}_{{\bar{S}}}/\mathcal {I}^{n}\simeq \bigoplus _{m=0}^{n-1}\mathcal {I}^{m}/\mathcal {I}^{m+1} \end{aligned}$$
(1.123)
as \({\mathcal {O}}_{C}\)-modules, hence, since \(\mathcal {I}^{m}/\mathcal {I}^{m+1}\simeq \mathcal {N}^{m}\), an isomorphism
$$\begin{aligned} H^{0}(C,{\mathcal {O}}_{C^{(n)}})\simeq \bigoplus _{m=0}^{n-1}H^{0}(C,\mathcal {N}^{m}),\,\,\,\,f\mapsto \sum _{m=0}^{n-1}c_{m}(f). \end{aligned}$$
(1.124)
This isomorphism respects the multiplicative structure, so is a ring isomorphism. Going to the projective limit, and noting that the \(c_{m}(f)\) are independent of n, we get
$$\begin{aligned} FJ(f)=\sum _{m=0}^{\infty }c_{m}(f)\in \prod _{m=0}^{\infty }H^{0}(C,\mathcal {N}^{m}). \end{aligned}$$
(1.125)

1.10.3 Fourier–Jacobi expansions over \(\mathbb {C}\)

Working over \(\mathbb {C}\), we shall now relate the infinitesimal retraction r to the geodesic retraction, and the powers of the conormal bundle \(\mathcal {N}\) to theta functions. Recall the analytic compactification of \(X_{\varGamma }\) described in Proposition 1.7. Let E be the connected component of \({\bar{X}}_{\varGamma }\backslash X_{\varGamma }\) corresponding to the standard cusp \(c_{\infty }\). As before, we denote by \(E^{(n)}\) its nth infinitesimal neighborhood. The line bundle \(\mathcal {T}|_{E}\) is just the analytic normal bundle to E, and hence we have an isomorphism
$$\begin{aligned} \mathcal {N}_{an}\simeq \mathcal {T}^{\vee } \end{aligned}$$
(1.126)
between the analytification of \(\mathcal {N}=\mathcal {I}/\mathcal {I}^{2}\) and the dual of \(\mathcal {T}\).

Lemma 1.25

The infinitesimal retraction \(r:E^{(n)}\rightarrow E\) coincides with the map induced by the geodesic retraction (1.15).

Proof

The meaning of the lemma is this. The infinitesimal retraction induces a map of ringed spaces
$$\begin{aligned} r_{an}:E_{an}^{(n)}\rightarrow E_{an} \end{aligned}$$
(1.127)
where \(E_{an}\) is the analytic space associated with E with its sheaf of analytic functions \({\mathcal {O}}_{E}^{hol}\), and \(E_{an}^{(n)}\) is the same topological space with the sheaf \({\mathcal {O}}_{{\bar{S}}}^{hol}/\mathcal {I}_{an}^{n}\). The geodesic retraction (sending (zu) to \(u\mod \Lambda \)) is an analytic map \(r_{geo}:E_{an}(\varepsilon )\rightarrow E_{an}\), where \(E_{an}(\varepsilon )\) is our notation for some tubular neighborhood of \(E_{an}\) in \({\bar{S}}_{an}\). On the other hand, there is a canonical map can of ringed spaces from \(E_{an}^{(n)}\) to \(E_{an}(\varepsilon )\). We claim that these three maps satisfy \(r_{geo}\circ can=r_{an}\).

To prove the lemma, note that the infinitesimal retraction \(r:E^{(n)}\rightarrow E\) is uniquely characterized by the fact that the \({\mathcal {O}}_{{\mathcal {K}}}\)-semi-abelian 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 \(2\pi i\cdot \text {d}\zeta _{3}\).

Pulling back the sheaf \(\mathcal {N}_{an}\) from \(E=\mathbb {C}/\Lambda \) to \(\mathbb {C}\), it is clear that \(q=q(z)=e^{2\pi iz/M}\) maps, at each \(u\in \mathbb {C}\), to a generator of \(\mathcal {T}^{\vee }=\mathcal {N}_{an}=\mathcal {I}_{an}/\mathcal {I}_{an}^{2}\), and we denote by \(q^{m}\) the corresponding generator of \(\mathcal {N}_{an}^{m}=\mathcal {I}_{an}^{m}/\mathcal {I}_{an}^{m+1}\). If
$$\begin{aligned} f(z,u)=\sum _{m=0}^{\infty }\theta _{m}(u)e^{2\pi imz/M}=\sum _{m=0}^{\infty }\theta _{m}(u)q^{m} \end{aligned}$$
(1.128)
is the complex analytic Fourier expansion of f at a neighborhood of \(c_{\infty }\), then \(c_{m}(z,u)=\theta _{m}(u)q^{m}\in H^{0}(E,\mathcal {N}_{an}^{m})\) is just the restriction of the section denoted above by \(c_{m}(f) \) to E. The functions \(\theta _{m}\) are classical elliptic theta functions (for the lattice \(\Lambda \)).

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 semi-abelian 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 sub-line-bundle 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 \(\mathcal {P}_{\mu }={\mathcal {O}}_{{\widehat{S}}}\cdot \text {d}\zeta _{2}\mod \mathcal {P}_{0}\).

We shall now pull back these vector bundles to \(\mathfrak {X}\) and compute the Gauss Manin connection \(\nabla \) complex analytically on \(\omega _{{\mathcal {A}}/\mathfrak {X}}\). We write \(\mathcal {P}_{0}={\mathcal {O}}_{\mathfrak {X}}\cdot \text {d}\zeta _{1}\) for \(\mathcal {P}_{0,an}\) etc. dropping the decoration an. Recalling that \({\mathcal {O}}_{{\mathcal {K}}}=\mathbb {Z}\oplus \mathbb {Z}\omega _{{\mathcal {K}}}\), we let
$$\begin{aligned} \alpha _{1}=\left( \begin{array}{l} 0 \\ 1 \\ 1 \end{array} \right) ,\quad \alpha _{2}=\left( \begin{array}{l} 1 \\ 0 \\ u \end{array} \right) ,\quad \alpha _{3}=\left( \begin{array}{c} u \\ -z/\delta \\ z/\delta \end{array} \right) \end{aligned}$$
(1.129)
and
$$\begin{aligned} \alpha _{1}^{\prime }= & {} \iota ^{\prime }(\omega _{{\mathcal {K}}})\alpha _{1}=\left( \begin{array}{c} 0 \\ \omega _{{\mathcal {K}}} \\ {\bar{\omega }}_{{\mathcal {K}}} \end{array} \right) ,\quad \alpha _{2}^{\prime }=\iota ^{\prime }(\omega _{{\mathcal {K}}})\alpha _{2}=\left( \begin{array}{c} \omega _{{\mathcal {K}}} \\ 0 \\ {\bar{\omega }}_{{\mathcal {K}}}u \end{array} \right) ,\nonumber \\ \alpha _{3}^{\prime }= & {} \iota ^{\prime }(\omega _{{\mathcal {K}}})\alpha _{3}=\left( \begin{array}{c} \omega _{{\mathcal {K}}}u \\ -\omega _{{\mathcal {K}}}z/\delta \\ {\bar{\omega }}_{{\mathcal {K}}}z/\delta \end{array} \right) . \end{aligned}$$
(1.130)
These 6 vectors span \(L_{(z,u)}^{\prime }\) over \(\mathbb {Z}\). Let \(\beta _{1},\dots ,\beta _{3}^{\prime }\) be the dual basis to \(\{\alpha _{1},\dots ,\alpha _{3}^{\prime }\}\) in \(H_{dR}^{1}({\mathcal {A}}/{\mathcal {O}}_{\mathfrak {X}}), \) i.e. \(\int _{\alpha _{1}}\beta _{1}=1\) etc. As the periods of the \(\beta _{i}\)’s along the integral homology are constant, the \(\beta \)-basis is horizontal for the Gauss–Manin connection. The first coordinate of the \(\alpha _{i}\) and \(\alpha _{i}^{\prime }\) gives us
$$\begin{aligned} \text {d}\zeta _{1}=0\cdot \beta _{1}+1\cdot \beta _{2}+u\cdot \beta _{3}+0\cdot \beta _{1}^{\prime }+\omega _{{\mathcal {K}}}\cdot \beta _{2}^{\prime }+\omega _{{\mathcal {K}}}u\cdot \beta _{3}^{\prime }, \end{aligned}$$
(1.131)
and we find that
$$\begin{aligned} \nabla (\text {d}\zeta _{1})=\left( \beta _{3}+\omega _{{\mathcal {K}}}\beta _{3}^{\prime }\right) \otimes \text {d}u. \end{aligned}$$
(1.132)
Similarly, we find
$$\begin{aligned} \nabla (\text {d}\zeta _{2})= & {} -\delta ^{-1}\left( \beta _{3}+\omega _{{\mathcal {K}}}\beta _{3}^{\prime }\right) \otimes \text {d}z \\ \nabla (\text {d}\zeta _{3})= & {} \left( \beta _{2}+{\bar{\omega }}_{{\mathcal {K}}}\beta _{2}^{\prime }\right) \otimes \text {d}u+\delta ^{-1}\left( \beta _{3}+{\bar{\omega }}_{{\mathcal {K}}}\beta _{3}^{\prime }\right) \otimes \text {d}z. \nonumber \end{aligned}$$
(1.133)

1.11.2 A computation of KS in the complex model

We go on to compute the Kodaira–Spencer map on \(\mathcal {P}\), i.e. the map denoted KS\((\varSigma )\). For that, we have to take \(\nabla (\text {d}\zeta _{1})\) and \(\nabla (\text {d}\zeta _{2})\) and project them to \(R^{1}\pi _{*}{\mathcal {O}}_{{\mathcal {A}}}(\varSigma )\otimes \varOmega _{\mathfrak {X}}^{1}\). We then pair the result, using the polarization form \(\left\langle ,\right\rangle _{\lambda }\) on \(H_{dR}^{1}({\mathcal {A}})\) (reflecting the isomorphism
$$\begin{aligned} R^{1}\pi _{*}{\mathcal {O}}_{{\mathcal {A}}}(\varSigma )=Lie({\mathcal {A}}^{t})(\varSigma )=\omega _{{\mathcal {A}}^{t}}^{\vee }(\varSigma )\simeq \mathcal {L}^{\vee }(\rho ) \end{aligned}$$
(1.134)
coming from \(\lambda \)), with \(\text {d}\zeta _{3}\).

To perform the computation, we need two lemmas.

Lemma 1.26

The Riemann form on \(L_{x}^{\prime }\), associated with the polarization \(\lambda \), is given in the basis \(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{1}^{\prime },\alpha _{2}^{\prime },\alpha _{3}^{\prime }\) by the matrix
$$\begin{aligned} J=\left( \begin{array}{llllll} &{} &{} &{} &{} &{} 1 \\ &{} &{} &{} &{} -1 &{} \\ &{} &{} &{} 1 &{} &{} \\ &{} &{} -1 &{} &{} &{} \\ &{} 1 &{} &{} &{} &{} \\ -1 &{} &{} &{} &{} &{} \end{array} \right) . \end{aligned}$$
(1.135)

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

These are essentially Riemann’s bilinear relations. For example, if A is the Jacobian of a curve \(\mathcal {C}\) and the basis \(\alpha _{1},\dots ,\alpha _{2g}\) has the standard intersection matrix
$$\begin{aligned} J=\left( \begin{array}{cc} 0 &{}\quad I \\ -I &{}\quad 0 \end{array} \right) \end{aligned}$$
(1.136)
then the lemma follows from the well-known formula for the cup product (\(\xi ,\eta \) being differentials of the second kind on \(\mathcal {C}\))
$$\begin{aligned} \xi \cup \eta =\frac{1}{2\pi i}\sum _{i=1}^{g}\left( \int _{\alpha _{i}}\xi \int _{\alpha _{i+g}}\eta -\int _{\alpha _{i}}\eta \int _{\alpha _{i+g}}\xi \right) . \end{aligned}$$
(1.137)
Using the two lemmas, we get
$$\begin{aligned} \text {KS}(\text {d}\zeta _{1}\otimes \text {d}\zeta _{3})= & {} \left\langle \beta _{3}+\omega _{{\mathcal {K}}}\beta _{3}^{\prime },\text {d}\zeta _{3}\right\rangle _{\lambda }\cdot \text {d}u\\= & {} \begin{array}{c} \left\langle \beta _{3}+\omega _{{\mathcal {K}}}\beta _{3}^{\prime },\,\,\beta _{1}+u\beta _{2}+z\delta ^{-1}\beta _{3}\right. +\\ \left. {\bar{\omega }}_{{\mathcal {K}}}\beta _{1}^{\prime }+{\bar{\omega }}_{{\mathcal {K}}}u\beta _{2}^{\prime }+{\bar{\omega }}_{{\mathcal {K}}}z\delta ^{-1}\beta _{3}^{\prime }\right\rangle _{\lambda }\cdot \text {d}u \end{array} \nonumber \\= & {} -\delta (2\pi i)^{-1}\text {d}u. \nonumber \end{aligned}$$
(1.138)
Similarly,
$$\begin{aligned} \text {KS}(\text {d}\zeta _{2}\otimes \text {d}\zeta _{3})= & {} \left\langle -\delta ^{-1}(\beta _{3}+\omega _{{\mathcal {K}}}\beta _{3}^{\prime }),\text {d}\zeta _{3}\right\rangle _{\lambda }\cdot \text {d}z \\= & {} (2\pi i)^{-1}\text {d}z. \nonumber \end{aligned}$$
(1.139)
We summarize.

Proposition 1.28

Let \(z,u,\zeta _{1},\zeta _{2},\zeta _{3}\) be the standard coordinates in a neighborhood of the cusp \(c_{\infty }\). Then, complex analytically, the Kodaira–Spencer isomorphism
$$\begin{aligned} \mathrm{KS}(\varSigma ):\mathcal {P}\otimes \mathcal {L}\simeq \varOmega _{\mathfrak {X}}^{1} \end{aligned}$$
(1.140)
is given by the formulae
$$\begin{aligned} \mathrm{KS}(\mathrm{d}\zeta _{1}\otimes \mathrm{d}\zeta _{3})=-\delta (2\pi i)^{-1}\mathrm{d}u,\quad \mathrm{KS}(\mathrm{d}\zeta _{2}\otimes \mathrm{d}\zeta _{3})=(2\pi i)^{-1}\mathrm{d}z. \end{aligned}$$
(1.141)

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 sub-bundle \(\mathcal {P}_{0}\otimes \mathcal {L}\) 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 sub-bundle \(\mathcal {P}_{0}\otimes \mathcal {L}\). On the other hand \(\text {d}u\) is holomorphic there, while \(\text {d}z\) has a simple pole along the boundary.

Corollary 1.30

The induced map
$$\begin{aligned} \psi :\varOmega _{\mathfrak {X}}^{1}\rightarrow \mathcal {P}_{\mu }\otimes \mathcal {L} \end{aligned}$$
(1.142)
(\(\mathcal {P}_{\mu }=\mathcal {P}/\mathcal {P}_{0}\)) obtained by inverting the isomorphism \(\mathrm{KS}(\varSigma )\) and dividing \(\mathcal {P}\) by \(\mathcal {P}_{0}\) kills \(\mathrm{d}u\) and maps \(\mathrm{d}z\) to \(2\pi i\cdot \mathrm{d}\zeta _{2}\otimes \mathrm{d}\zeta _{3}\).

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.

From the relation
$$\begin{aligned} \frac{\text {d}q}{q}=\frac{2\pi i}{M}\text {d}z, \end{aligned}$$
(1.143)
we deduce that the map \(\psi \) has a simple zero along the cuspidal divisor.

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 semi-abelian 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 \(c\in {\mathcal {K}}_{N}^{\times }\).

We begin with the vector bundles \(\mathcal {P}\) and \(\mathcal {L}\). Over \(\mathbb {C}\), they yield analytic vector bundles \(\mathcal {P}_{an}\) and \(\mathcal {L}_{an}\) on each \(X_{\varGamma _{j}}\) (\(1\le j\le m\)). Assume for the rest of this section that \(j=1\) and write \(\varGamma =\varGamma _{1}\). Similar results will hold for every j. The vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) are trivialized over the unit ball \(\mathfrak {X}\) by means of the nowhere vanishing sections \(\text {d}\zeta _{3}\in H^{0}(\mathfrak {X},\mathcal {L}_{an})\) and \(\text {d}\zeta _{1},\text {d}\zeta _{2}\in H^{0}(\mathfrak {X},\mathcal {P}_{an})\). These sections do not descend to \(X_{\varGamma }\), but
$$\begin{aligned} \sigma _{an}=\left( \text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\right) \otimes \text {d}\zeta _{3}^{-1}\in H^{0}\left( X_{\varGamma },\det \mathcal {P}\otimes \mathcal {L}^{-1}\right) \end{aligned}$$
(1.144)
does, as the factors of automorphy of \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\) are the same (cf Sect. 1.7.2). Furthermore, this factor of automorphy (i.e. \(j(\gamma ;z,u)\)) is trivial on \(\varGamma _\mathrm{cusp}\), the stabilizer of \(c_{\infty }\) in \(\varGamma \), so \(\text {d}\zeta _{1}\wedge \text {d}\zeta _{2}\) and \(\text {d}\zeta _{3}\) define sections of \(\det \mathcal {P}\) and \(\mathcal {L}\) on \({\widehat{S}}_{\mathbb {C}}\), the formal completion of \({\bar{S}}_{\mathbb {C}}\) along the cuspidal divisor \(E_{c}=p^{-1}(c_{\infty })\subset {\bar{S}}_{\mathbb {C}}\). The same also holds for \(\text {d}\zeta _{1}\) and \(\text {d}\zeta _{2}\mod \left\langle \text {d}\zeta _{1}\right\rangle \) individually (Lemma 1.8). Along \(E_{c},\mathcal {P}\) has a canonical filtration
$$\begin{aligned} 0\rightarrow \mathcal {P}_{0}\rightarrow \mathcal {P}\rightarrow \mathcal {P}_{\mu }\rightarrow 0 \end{aligned}$$
(1.145)
and \(\text {d}\zeta _{1}\) is a generator of \(\mathcal {P}_{0}\). (Compare (1.63) and (1.71) and note that the projection to \(\mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}}=B(\mathbb {C})\) is via the coordinate \(\zeta _{1}\), so \(\text {d}\zeta _{1}\) is a generator of \(\mathcal {P}_{0}|_{E_{c}}=\omega _{B}\).) As we have shown in Sect. 1.10.1, this filtration extends to the formal neighborhood \({\widehat{S}}_{\mathbb {C}}\) of \(E_{c}\). The vector bundles \(\mathcal {P}\) and \(\mathcal {L}\), as well as the filtration on \(\mathcal {P}\), are defined over \({\mathcal {K}}_{N}\) . It makes sense therefore to ask whether certain sections are \({\mathcal {K}}_{N}\)-rational. Recall that the cusp \(c_{\infty }\) is of type \(({\mathcal {O}}_{{\mathcal {K}}},{\mathcal {O}}_{{\mathcal {K}}}).\)

Proposition 1.32

  1. (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.

     
  2. (ii)

    Similarly \(2\pi i\cdot \mathrm{d}\zeta _{2}\) projects (modulo \(\mathcal {P}_{0}\)) to a \({\mathcal {K}}_{N}\)-rational section of \(\mathcal {P}_{\mu }.\)

     
  3. (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 semi-abelian scheme over E. Then \(\mathcal {G}\) is a semi-abelian 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 L-functions associated with Grossencharacters of \({\mathcal {K}}\). They are therefore instrumental in the construction of p-adic L-functions on \({\mathcal {K}}.\) We expect them to appear in the p-adic interpolation of holomorphic Eisenstein series on the group \(\mathbf {G}\), much as powers of \(2\pi i\) (values of \(\zeta (2k)\)) appear in the p-adic interpolation of Eisenstein series on \(GL_{2}(\mathbb {Q}).\)

1.12.2 Rationality of local parameters at the cusps

We keep the assumptions and the notation of the previous section. Analytically, neighborhoods of \(E_{c_{\infty }}\) were described in Sect. 1.4.1 with the aid of the parameters (zu). Let \({\widehat{S}}\) denote the formal completion of \({\bar{S}}_{{\mathcal {K}}_{N}}\) along E. Let \(r:{\widehat{S}}\rightarrow E\) be the infinitesimal retraction discussed in Sect. 1.10.1. If \(i:E\hookrightarrow {\widehat{S}}\) is the closed embedding, then \(r\circ i=Id_{E}.\) If \(\mathcal {I}\) is the sheaf of definition of E, then \(\mathcal {N}=\mathcal {I}/\mathcal {I}^{2}\) is the conormal bundle to E and hence its analytification is the dual of the line bundle \(\mathcal {T}\),
$$\begin{aligned} \mathcal {N}_{an}=\mathcal {T}^{\vee }. \end{aligned}$$
(1.146)
Consider \(r^{*}\mathcal {N}\) on \({\widehat{S}}.\) The retraction allows us to split the exact sequence
$$\begin{aligned} 0\rightarrow \mathcal {N}\rightarrow i^{*}\varOmega _{{\widehat{S}}}^{1}\rightarrow \varOmega _{E}^{1}\rightarrow 0 \end{aligned}$$
(1.147)
using \(\varOmega _{E}^{1}=i^{*}r^{*}\varOmega _{E}^{1}\subset i^{*}\varOmega _{{\widehat{S}}}^{1}\). Thus \(i^{*}\varOmega _{{\widehat{S}}}^{1}=\mathcal {N}\times \varOmega _{E}^{1}.\) The map \(i\circ r:{\widehat{S}}\rightarrow {\widehat{S}}\) induces a sheaf homomorphism \(r^{*}i^{*}\varOmega _{{\widehat{S}}}^{1}\rightarrow \varOmega _{{\widehat{S}}}^{1}\), which becomes the identity if we restrict it to E (i.e. follow it with \(i^{*}\)). By Nakayama’s lemma, it is an isomorphism. It follows that
$$\begin{aligned} \varOmega _{{\widehat{S}}}^{1}=r^{*}i^{*}\varOmega _{{\widehat{S}}}^{1}=r^{*}\mathcal {N}\times r^{*}\varOmega _{E}^{1}. \end{aligned}$$
(1.148)
Let \(x\in E\) and represent it by \(u\in \mathbb {C}\) (modulo \(\Lambda \)). Then \(q=e^{2\pi iz/M}\), where M is the width of the cusp (1.54), is a local analytic parameter on a classical neighborhood \(U_{x}\) of x which vanishes to first order along E. Note that q depends on the choice of u (see Remark below). It follows that \(\text {d}q\), the image of q in \(\mathcal {I}_{an}/\mathcal {I}_{an}^{2}\), is a basis of \(\mathcal {N}_{an}\) (on \(U_{x}\cap E\)). But
$$\begin{aligned} 2\pi i\cdot \text {d}z=M\frac{\text {d}q}{q} \end{aligned}$$
(1.149)
(mod \(\left\langle \text {d}u\right\rangle \)) is independent of u [see (1.55)], so represents a global meromorphic section of \(r^{*}\mathcal {N}_{an}\), with a simple pole along \(E\subset {\widehat{S}}_{\mathbb {C}}.\) By GAGA, this section is (meromorphic) algebraic.

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

The proof relies on the Kodaira–Spencer isomorphism KS\((\varSigma )\) (1.141), which is a \({\mathcal {K}}_{N}\)-rational (even \({\mathcal {K}}\)-rational) algebraic isomorphism between \(\mathcal {P}\otimes \mathcal {L}\) and \(\varOmega _{S}^{1}.\) As we have shown, it extends to a meromorphic homomorphism from \(\mathcal {P}\otimes \mathcal {L}\) to \(\varOmega _{{\bar{S}}}^{1}\) over \({\bar{S}}.\) Over \({\widehat{S}}\) it induces an isomorphism of \(\mathcal {P}_{0}\otimes \mathcal {L}\) onto \(r^{*}\varOmega _{E}^{1}\subset \varOmega _{{\widehat{S}}}^{1}\) carrying the \({\mathcal {K}}_{N}\)-rational section \(\varOmega _{B}\text {d}\zeta _{1}\otimes 2\pi i\text {d}\zeta _{3}\) to \(-\varOmega _{B}\delta \cdot \text {d}u\), proving part (ii) of the proposition. It also carries \(2\pi i\text {d}\zeta _{2}\otimes 2\pi i\text {d}\zeta _{3}\) to \(2\pi i\text {d}z\), but the latter is only meromorphic. We may summarize the situation over \({\widehat{S}}\) by the following commutative diagram with exact rows:
$$\begin{aligned} \begin{array}{lllllllll} 0 &{} \rightarrow &{} {\widehat{\mathcal {I}}}\otimes \mathcal {P}_{0}\otimes \mathcal {L} &{} \rightarrow &{} {\widehat{\mathcal {I}}}\otimes \mathcal {P}\otimes \mathcal {L} &{} \rightarrow &{} {\widehat{\mathcal {I}}}\otimes \mathcal {P}_{\mu }\otimes \mathcal {L} &{} \rightarrow &{} 0 \\ &{} &{} \downarrow &{} &{} \downarrow \text {KS}(\varSigma ) &{} &{} \downarrow &{} &{} \\ 0 &{} \rightarrow &{} r^{*}\varOmega _{E}^{1} &{} \rightarrow &{} \varOmega _{{\widehat{S}}}^{1} &{} \rightarrow &{} r^{*}\mathcal {N} &{} \rightarrow &{} 0 \end{array} . \end{aligned}$$
(1.150)
Let h be a \({\mathcal {K}}_{N}\)-rational local equation of E, i.e. a \({\mathcal {K}}_{N}\)-rational section of \(\mathcal {I}\) in some Zariski open \(U\ \)intersecting E non-trivially, vanishing to first order along \(E\cap U.\) The differential \(\eta =h\cdot (2\pi i\text {d}z)\) is regular on U, and to prove that it is \({\mathcal {K}}_{N}\)-rational we may restrict it to \({\widehat{S}}\) and check rationality there. But in \({\widehat{S}}\) we have a \({\mathcal {K}}_{N}\)-rational product decomposition \(\varOmega _{{\widehat{S}}}^{1}=r^{*}\mathcal {N}\times r^{*}\varOmega _{E}^{1}\) and the projection of \(\eta \) to the second factor is 0, so it is enough to prove rationality of its projection to \(r^{*}\mathcal {N}.\) This projection is the image, under KS\((\varSigma )\), of \(h\cdot (2\pi i\text {d}\zeta _{2}\otimes 2\pi i\text {d}\zeta _{3}\mod \mathcal {P}_{0}\otimes \mathcal {L})\), so our assertion follows from parts (i) and (ii) of the previous proposition. This proves that \(\eta \), hence \(h^{-1}\eta =2\pi i\text {d}z\) is a \({\mathcal {K}}_{N}\)-rational differential. An alternative proof of part (ii) is to note that E is isogenous over \({\mathcal {K}}_{N}\) to B, so up to a \({\mathcal {K}}_{N}\)-multiple has the same period.

Remark 1.1

The parameter q is not a well-defined 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 \(\det \mathcal {P}\simeq \mathcal {L}\)

Let us fix a nowhere vanishing section
$$\begin{aligned} \sigma \in H^{0}\left( S_{{\mathcal {K}}},\det \mathcal {P}\otimes \mathcal {L}^{-1}\right) . \end{aligned}$$
(1.151)
This section is determined up to \({\mathcal {K}}^{\times }.\) From now on, we shall use this section to identify \(\det \mathcal {P}\) with \(\mathcal {L}\) whenever such an identification is needed. From Corollary 1.33, we deduce that when we base change to \(\mathbb {C}\), on each connected component \(X_{\varGamma }\)
$$\begin{aligned} \sigma \sim \varOmega _{B}\cdot \sigma _{an}. \end{aligned}$$
(1.152)

2 Picard modular schemes modulo an inert prime

2.1 The stratification

2.1.1 The three strata

Let p be a rational prime which is inert in \({\mathcal {K}}\) and relatively prime to 2N. Then \(\kappa _{0}=R_{0}/pR_{0}\) is isomorphic to \(\mathbb {F}_{p^{2}}.\) We fix an algebraic closure \(\kappa \) of \(\kappa _{0}\) and consider the characteristic p fiber
$$\begin{aligned} {\bar{S}}_{\kappa }={\bar{S}}\times _{R_{0}}\kappa . \end{aligned}$$
(2.1)
Unless otherwise specified, in this section we let S and \({\bar{S}}\) denote the characteristic p fibers \(S_{\kappa }\) and \({\bar{S}}_{\kappa }.\) We also use the abbreviation \(\omega _{{\mathcal {A}}}\) for \(\omega _{{\mathcal {A}}/{\bar{S}}}\) etc.

Recall that an abelian variety over an algebraically closed field of characteristic p is called supersingular if the Newton polygon of its p-divisible 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

  1. (i)

    There exists a closed reduced 1-dimensional 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 \(\kappa _{0}.\)

     
  2. (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)
     
  3. (iii)
    Assume that N is large enough (depending on p). Then the irreducible components of \(S_\mathrm{ss}\) are non-singular and in fact are all isomorphic to the Fermat curve \(\mathcal {C}_{p}\) given by the equation
    $$\begin{aligned} x^{p+1}+y^{p+1}+z^{p+1}=0. \end{aligned}$$
    (2.3)
    There 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.
     
  4. (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 self-intersections. 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 p-divisible 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 p-divisible group. Let \(\mathfrak {G} \) be the p-divisible group of a supersingular elliptic curve over k (the group denoted by \(G_{1,1}\) in the Manin-Dieudonné classification). The following theorem can be deduced from [4, 39].

Theorem 2.2

  1. (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)
     
  2. (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.

     
While the p-divisible group of a \(\mu \)-ordinary geometric fiber actually splits as a product of its multiplicative, local–local and étale parts, over the whole of \(S_{\mu }\) we only get a filtration
$$\begin{aligned} 0\subset Fil^{2}{\mathcal {A}}(p)\subset Fil^{1}{\mathcal {A}}(p)\subset Fil^{0}{\mathcal {A}}(p)={\mathcal {A}}(p) \end{aligned}$$
(2.5)
by \({\mathcal {O}}_{{\mathcal {K}}}\)-stable p-divisible groups. Here \(gr^{2}=Fil^{2}\) is of multiplicative type, \(gr^{1}=Fil^{1}/Fil^{2}\) is a local–local group and \(gr^{0}=Fil^{0}/Fil^{1}\) is étale, each of height 2 (\({\mathcal {O}}_{{\mathcal {K}}}\)-height 1).

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 \({\bar{S}}_{\mu }\)

Consider the universal semi-abelian variety \({\mathcal {A}}\) over the Zariski open set \({\bar{S}}_{\mu }.\) Over the cuspidal divisor \(C={\bar{S}}\backslash S\), \(\mathcal {P}=\omega _{{\mathcal {A}}}(\varSigma )\) admits a canonical filtration
$$\begin{aligned} 0\rightarrow \mathcal {P}_{0}\rightarrow \mathcal {P}\rightarrow \mathcal {P}_{\mu }\rightarrow 0 \end{aligned}$$
(2.6)
where \(\mathcal {P}_{0}\) is the cotangent space to the abelian part of \({\mathcal {A}}\), and \(\mathcal {P}_{\mu }\) is the \(\varSigma \)-component of the cotangent space to the toric part of \({\mathcal {A}}.\) This filtration exists already in characteristic 0, but when we reduce the Picard surface modulo p, it extends to the whole of \({\bar{S}}_{\mu }.\) Over the non-cuspidal part \(S_{\mu }\), we may set
$$\begin{aligned} \mathcal {P}_{0}=\ker \left( \omega _{{\mathcal {A}}[p]^{0}}\rightarrow \omega _{{\mathcal {A}}[p]^{\mu }}\right) \end{aligned}$$
(2.7)
where \({\mathcal {A}}[p]^{\mu }\) is the p-torsion in \({\mathcal {A}}(p)^{\mu }=Fil^{2}{\mathcal {A}}(p).\) Then \(\mathcal {P}_{\mu }\) is identified with \(\omega _{{\mathcal {A}}[p]^{\mu }}(\varSigma ).\)

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 \(Frob_{{\mathcal {A}}^{t}}:{\mathcal {A}}^{t}\rightarrow ({\mathcal {A}}^{t})^{(p)}.\)

We clearly have \(\omega _{{\mathcal {A}}^{(p)}}=\omega _{{\mathcal {A}}}^{(p)}\), and we let
$$\begin{aligned} F:\omega _{{\mathcal {A}}}^{(p)}\rightarrow \omega _{{\mathcal {A}}},\,\,\,V:\omega _{{\mathcal {A}}}\rightarrow \omega _{{\mathcal {A}}}^{(p)} \end{aligned}$$
(2.8)
be the \({\mathcal {O}}_{{\bar{S}}}\)-linear maps of vector bundles induced by the isogenies \(Frob_{{\mathcal {A}}}\) and \(Ver_{{\mathcal {A}}}\) on the cotangent spaces. We refer to [8] for a discussion how to define V over the whole of \({\bar{S}}\), despite the fact that \(Ver_{{\mathcal {A}}}\) is only defined over S.
Taking \(\varSigma \)-components, we get the map
$$\begin{aligned} V_{\mathcal {P}}:\mathcal {P}=\omega _{{\mathcal {A}}}(\varSigma )\rightarrow \omega _{{\mathcal {A}}}^{(p)}(\varSigma )=\omega _{{\mathcal {A}}}({\bar{\varSigma }})^{(p)}=\mathcal {L}^{(p)}, \end{aligned}$$
(2.9)
and taking the \({\bar{\varSigma }}\)-component, we similarly get
$$\begin{aligned} V_{\mathcal {L}}:\mathcal {L}\rightarrow \mathcal {P}^{(p)}. \end{aligned}$$
(2.10)

Proposition 2.3

Over \({\bar{S}}_{\mu }\) both \(V_{\mathcal {P}}\) and \(V_{\mathcal {L}}\) are of rank 1,
$$\begin{aligned} \mathcal {P}_{0}=\ker V_{\mathcal {P}} \end{aligned}$$
(2.11)
and the image of \(V_{\mathcal {L}}\) is a direct sum complement to \(\mathcal {P}_{0}^{(p)}:\)
$$\begin{aligned} \mathcal {P}^{(p)}=\mathcal {P}_{0}^{(p)}\oplus V(\mathcal {L}). \end{aligned}$$
(2.12)

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

Over \({\bar{S}}_{\mu }\), \(\mathcal {P}_{\mu }\simeq \mathcal {L}^{p},\,\,\mathcal {P}_{0}\simeq \mathcal {L}^{1-p}\), and \(\mathcal {L}^{p^{2}}\simeq \mathcal {L}\). For \(k\ge 1\) odd, \(\mathcal {P}^{(p^{k})}\simeq \mathcal {L}^{p-1}\oplus \mathcal {L}.\) For \(k\ge 2\) even, \(\mathcal {P}^{(p^{k})}\simeq \mathcal {L}^{1-p}\oplus \mathcal {L}^{p}\), but for \(k=0\) we only have an exact sequence
$$\begin{aligned} 0\rightarrow \mathcal {L}^{1-p}\rightarrow \mathcal {P}\rightarrow \mathcal {L}^{p}\rightarrow 0. \end{aligned}$$
(2.13)

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

  1. (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.

     
  2. (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 \(\mathcal {P}_{0}\simeq \mathcal {L}^{1-p}\).

     

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 sub-line-bundle 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

The Hasse invariant is
$$\begin{aligned} h_{{\bar{\varSigma }}}=V_{\mathcal {P}}^{(p)}\circ V_{\mathcal {L}}\in \text {Hom}\left( \mathcal {L},\mathcal {L}^{(p^{2})}\right) . \end{aligned}$$
(2.14)
As \(\mathcal {L}^{(p^{2})}\simeq \mathcal {L}^{p^{2}}\), the Hasse invariant is a global section of \(\mathcal {L}^{p^{2}-1}\), i.e. a modular form of weight \(p^{2}-1\) over \(\kappa \),
$$\begin{aligned} h_{{\bar{\varSigma }}}\in M_{p^{2}-1}(N,\kappa ). \end{aligned}$$
(2.15)
It turns out that \(h_{{\bar{\varSigma }}}\) has a simple zero along the supersingular locus \(S_\mathrm{ss}\). Once again, this requires a little computation with Dieudonné modules. Equivalently, we have the following theorem.

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

Let \(N\ge 3\) as always, and let \(\mathcal {M}\) be the moduli problem of Sect. 1.3.1. Let \(n\ge 1\) and consider the following moduli problem on \(\kappa _{0}\)-algebras:
  • \(\mathcal {M}_{Ig(p^{n})}(R)\) is the set of isomorphism classes of pairs \((\underline{A},\varepsilon )\) where \(\underline{A}\in \mathcal {M}(R)\) and
    $$\begin{aligned} \varepsilon :\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\hookrightarrow A[p^{n}] \end{aligned}$$
    (2.16)
    is a closed immersion of \({\mathcal {O}}_{{\mathcal {K}}}\)-group schemes over R.
It is clear that if \((\underline{A},\varepsilon )\in \mathcal {M}_{Ig(p^{n})}(R)\), then A is fiber-by-fiber \(\mu \)-ordinary and therefore \(\underline{A}\in \mathcal {M}(R)\) defines an R-point of \(S_{\mu }.\) The image of \(\varepsilon \) is then \(A[p^{n}]^{\mu }\), the maximal subgroup scheme of \(A[p^{n}]\) of multiplicative type. It is also clear that the functor \(R\rightsquigarrow \mathcal {M}_{Ig(p^{n})}(R)\) is relatively representable over \(\mathcal {M}\), and therefore as \(N\ge 3\) and \(\mathcal {M}\) is representable, this functor is also representable by a scheme \(Ig_{\mu }(p^{n})\) which maps to \(S_{\mu }.\) See [23] for the notion of relative representability. We call \(Ig_{\mu }(p^{n})\) the Igusa scheme of level \(p^{n}.\)

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

Every \(\mu \)-ordinary abelian variety has a unique finite flat \({\mathcal {O}}_{{\mathcal {K}}}\)-subgroup scheme of multiplicative type \(A[p^{n}]^{\mu }\) of rank \(p^{2n}.\) Such a subgroup scheme is, locally in the étale topology, isomorphic to \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\), and any two isomorphisms differ by a unique automorphism of \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}.\) But \(\varDelta (p^{n})=Aut_{{\mathcal {O}}_{{\mathcal {K}}}}(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}).\) If we let \(\gamma \in \varDelta (p^{n})\) act on the pair \((\underline{A},\varepsilon )\) via
$$\begin{aligned} \gamma ((\underline{A},\varepsilon ))=\left( \underline{A},\varepsilon \circ \gamma ^{-1}\right) \end{aligned}$$
(2.17)
\(\varDelta (p^{n})\) becomes a group of deck transformation and the proof is complete.

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 non-canonically 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

Let \({\mathcal {A}}\) denote the pull-back of the universal semi-abelian variety from \({\bar{S}}_{\mu }\) to \(\overline{Ig}_{\mu }(p^{n}).\) Then \({\mathcal {A}}\) is equipped with a canonical Igusa level structure
$$\begin{aligned} \varepsilon :\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\simeq {\mathcal {A}}[p^{n}]^{\mu }. \end{aligned}$$
(2.18)
Over C and after base change to \(R_{N}/pR_{N}\) the toric part of \({\mathcal {A}}\) is locally Zariski of the form \(\mathfrak {a}\otimes \mathbb {G}_{m}\) and \(\varepsilon \) is then an \({\mathcal {O}}_{{\mathcal {K}}}\)-linear isomorphism between \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\) and \(\mathfrak {a}\otimes \mu _{p^{n}}.\)

2.3.3 A trivialization of \(\mathcal {L}\) over the Igusa surface

From now on, we focus on \(\overline{Ig}_{\mu }=\overline{Ig}_{\mu }(p)\) although similar results hold when \(n>1\), and would be instrumental in the study of p-adic modular forms. The vector bundle \(\omega _{{\mathcal {A}}}\) pulls back to a similar vector bundle over \(\overline{Ig}_{\mu }.\) But there
$$\begin{aligned} \omega _{{\mathcal {A}}}^{\mu }\,{:}{=}\,\omega _{{\mathcal {A}}[p]^{\mu }} \end{aligned}$$
(2.19)
is a rank 2 quotient bundle stable under \({\mathcal {O}}_{{\mathcal {K}}}\) (of type (1, 1)), and the isomorphism \(\varepsilon \) induces an isomorphism
$$\begin{aligned} \varepsilon ^{*}:\omega _{{\mathcal {A}}}^{\mu }\simeq \omega _{\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p}}. \end{aligned}$$
(2.20)
Now \(Lie(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p})=\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes Lie(\mu _{p})=\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes Lie(\mathbb {G}_{m})\) and by duality
$$\begin{aligned} \omega _{\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p}}={\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{\mathbb {G}_{m}}, \end{aligned}$$
(2.21)
with \(1\otimes dT/T\) as a generator (if T is the parameter of \(\mathbb {G}_{m}\)). Here we have used the fact that the \(\mathbb {Z}\)-dual of \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\) is \({\mathcal {O}}_{{\mathcal {K}}}\) via the trace pairing. This is the constant vector bundle \({\mathcal {O}}_{{\mathcal {K}}}\otimes R=R(\varSigma )\oplus R({\bar{\varSigma }}).\)

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 \(\det \mathcal {P}\simeq \mathcal {L}.\)

We can now give an alternative proof to the fact that \(\mathcal {L}^{p^{2}-1}\) and \(\mathcal {P}_{\mu }^{p^{2}-1}\) are trivial on \({\bar{S}}_{\mu }.\) Denote by \({\mathcal {O}}_{Ig}\) the structure sheaf of \(\overline{Ig}_{\mu }.\) By the projection formula, \(\tau _{*}(\tau ^{*}\mathcal {L)}\simeq \mathcal {L}\otimes \tau _{*}{\mathcal {O}}_{Ig}\). Taking determinants, we get
$$\begin{aligned} \det \tau _{*}(\tau ^{*}\mathcal {L)}\simeq \mathcal {L}^{p^{2}-1}\otimes \det \tau _{*}{\mathcal {O}}_{Ig}. \end{aligned}$$
(2.22)
As \(\tau ^{*}\mathcal {L}\simeq {\mathcal {O}}_{Ig}\), we get that \(\mathcal {L}^{p^{2}-1}\simeq {\mathcal {O}}_{{\bar{S}}}\). The same argument works for \(\mathcal {P}_{\mu }\) and for \(\mathcal {P}_{0}.\) The fact that \(\mathcal {P}_{0}^{p+1}\) is already trivial could be deduced by a similar argument had we worked out an analogue of Ig(p) classifying symplectic isomorphisms of \(\mathfrak {G}[p]\) with \(gr^{1}A[p].\) The role of \(\varDelta (p)\) for such a moduli space would be assumed by
$$\begin{aligned} \varDelta ^{1}(p)=\ker (N:({\mathcal {O}}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}})^{\times }\rightarrow \mathbb {F}_{p}^{\times }), \end{aligned}$$
(2.23)
which is a group of order \(p+1.\) We do not go any further in this direction here.

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 \(\overline{Ig}_{\mu }\)

Let a be the canonical nowhere vanishing section of \(\mathcal {L}\) over \(\overline{Ig}_{\mu }\) which is sent to \(e_{{\bar{\varSigma }}}\cdot (1\otimes dT/T) \) under the trivialization
$$\begin{aligned} \varepsilon ^{*}:\mathcal {L}=\omega _{{\mathcal {A}}}^{\mu }({\bar{\varSigma }})\simeq ({\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{\mathbb {G}_{m}})({\bar{\varSigma }})=R({\bar{\varSigma }}). \end{aligned}$$
(2.24)
Here R is any \(R_{0}/pR_{0}\)-algebra over which we choose to work. In other words, \(a=(\varepsilon ^{*})^{-1}(e_{{\bar{\varSigma }}}\cdot 1\otimes dT/T). \) Dually, a is the homomorphism from \(Lie({\mathcal {A}})({\bar{\varSigma }}) \) to \(\delta _{{\mathcal {K}}}^{-1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }})\) arising from \(\varepsilon ^{-1}.\) Let \(a(k)=a^{\otimes k}\in H^{0}(\overline{Ig}_{\mu },\mathcal {L}^{k}).\)

Proposition 2.14

  1. (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)
     
  2. (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

  1. (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 embeddings
    $$\begin{aligned} \delta _{{\mathcal {K}}}^{-1}\otimes \mu _{p}\hookrightarrow A[p] \end{aligned}$$
    (2.27)
    differ by \(\iota (\gamma )\) (2.17). The induced trivializations of \(Lie(A)({\bar{\varSigma }})\) differ by \({\bar{\varSigma }}(\gamma )\) and by duality we get (i).
     
  2. (ii)
    Since over any \(\mathbb {F}_{p}\)-base, \(Ver_{\mathbb {G}_{m}}=1\), we have a 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)
    Using the isomorphism \(Lie({\mathcal {A}})({\bar{\varSigma }})^{(p^{2})}\simeq Lie({\mathcal {A}})({\bar{\varSigma }})^{p^{2}}\), we get the commutative diagram
    $$\begin{aligned} \begin{array}{lll} Lie({\mathcal {A}})({\bar{\varSigma }})^{p^{2}} &{} \overset{h_{{\bar{\varSigma }}}}{\rightarrow } &{} Lie({\mathcal {A}})({\bar{\varSigma }}) \\ \downarrow a(p^{2}) &{} &{} \downarrow a \\ \delta _{{\mathcal {K}}}^{-1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) &{} = &{} \delta _{{\mathcal {K}}}^{-1}\otimes Lie(\mathbb {G}_{m})({\bar{\varSigma }}) \end{array} , \end{aligned}$$
    (2.29)
    from which we deduce that \(h_{{\bar{\varSigma }}}=a(p^{2}-1)\).
     

2.4.2 The compactification \(\overline{Ig}\) of \(\overline{Ig}_{\mu }\)

In this section, we follow the method outlined in [1, Sections 6-9] and [12] for Hilbert modular varieties. Quite generally, let \(L\rightarrow X\) be a line bundle associated with an invertible sheaf \(\mathcal {L}\) on a scheme X. Write \(L^{n}\) for the line bundle \(L^{\otimes n}\) over X. Let \(s:X\rightarrow L^{n}\) be a section. Consider the fiber product
$$\begin{aligned} Y=L\times _{L^{n}}X \end{aligned}$$
(2.30)
where the two maps to \(L^{n}\) are \(\lambda \mapsto \lambda ^{n}\) and s. Let \(p:Y\overset{pr_{2}}{\rightarrow }X\) be the projection which factors also as \(Y\overset{pr_{1}}{\rightarrow }L\rightarrow L^{n}\rightarrow X\) (since \(X\overset{s}{\rightarrow }L^{n}\rightarrow X\) is the identity). Consider
$$\begin{aligned} p^{*}L=L\times _{X}(L\times _{L^{n}}X). \end{aligned}$$
(2.31)
This line bundle on Y has a tautological section \(t:Y\rightarrow p^{*}L\),
$$\begin{aligned} t:y=(\lambda ,x)\mapsto (\lambda ,y)=(\lambda ,(\lambda ,x)) \end{aligned}$$
(2.32)
Here \(s(x)=\lambda ^{n}\) and
$$\begin{aligned} t^{n}(y)=(\lambda ^{n},y)=(s(x),y)=p^{*}s(y) \end{aligned}$$
(2.33)
so t is an nth root of \(p^{*}s.\) Moreover, Y has the universal property with respect to extracting nth roots from s: If \(p_{1}:Y_{1}\rightarrow X\), and \(t_{1}\in \varGamma (Y_{1},p_{1}^{*}L)\) is such that \(t_{1}^{n}=p_{1}^{*}s\), then there exists a unique morphism \(h:Y_{1}\rightarrow Y\) covering the two maps to X such that \(t_{1}=h^{*}t.\)

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.

Applying this in our example with \(n=p^{2}-1\), we define the complete Igusa surface of level p, \(\overline{Ig}=\overline{Ig}(p)\) as
$$\begin{aligned} \overline{Ig}=\mathcal {L}\times _{\mathcal {L}^{p^{2}-1}}{\bar{S}} \end{aligned}$$
(2.34)
where the map \(\overline{S}\rightarrow \mathcal {L}^{p^{2}-1}\) is \(h_{{\bar{\varSigma }}}.\) From the universal property and part (ii) of Proposition 2.14 we get a map of \({\bar{S}}\)-schemes
$$\begin{aligned} \overline{Ig}_{\mu }\rightarrow \overline{Ig}. \end{aligned}$$
(2.35)
This map is an isomorphism over \({\bar{S}}_{\mu }\) because both schemes are étale torsors for \(\varDelta (p)=({\mathcal {O}}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}})^{\times }\) and the map respects the action of this group. We summarize the discussion in the following theorem (for the last point, consult [32], Proposition 2, p.198).

Theorem 2.15

The morphism \(\tau :\overline{Ig}\rightarrow {\bar{S}}\) satisfies the following properties:
  1. (i)

    It is finite flat of degree \(p^{2}-1\), étale over \({\bar{S}}_{\mu }\), totally ramified over \(S_\mathrm{ss}\).

     
  2. (ii)

    \(\varDelta (p)\) acts on \(\overline{Ig}\) as a group of deck transformations and the quotient is \({\bar{S}}.\)

     
  3. (iii)

    Let \(s_{0}\in S_\mathrm{gss}(\mathbb {{\bar{F}}}_{p}).\) Then there exist local parameters uv 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.

     
  4. (iv)
    Let \(s_{0}\in S_\mathrm{ssp}(\mathbb {{\bar{F}}}_{p}).\) Then there exist local parameters uv 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 \(\tau \), then
    $$\begin{aligned} {\widehat{{\mathcal {O}}}}_{Ig,{\tilde{s}}_{0}}=\mathbb {{\bar{F}}}_{p}[[w,u,v]]/\left( w^{p^{2}-1}-u^{p+1}-v^{p+1}\right) \end{aligned}$$
    (2.36)
    In particular, \({\tilde{s}}_{0}\) is a normal singularity of Ig.
     

2.4.3 Irreducibility of Ig

So far we have avoided the delicate question of whether \(\overline{Ig}\) is “relatively irreducible”, i.e. whether \(\tau ^{-1}(T)\) is irreducible if \(T\subset {\bar{S}}\) is an irreducible (equivalently, connected) component. Using an idea of Katz, and following the approach taken by Ribet [33], the irreducibility of \(\tau ^{-1}(T)\) could be proven for any level \(p^{n}\) if we could prove the following:
  • 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 \(Gal(\mathbb {{\bar{F}}}_{q}/\mathbb {F}_{q^{r}})\) in
    $$\begin{aligned} Aut\left( Isom_{\mathbb {{\bar{F}}}_{q}}(\delta _{{\mathcal {K}}}^{-1}\otimes \mu _{p^{n}},A[p^{n}]^{\mu })\right) =({\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}})^{\times } \end{aligned}$$
    (2.37)
    contains \(\gamma .\)
See also the discussion in 5.2.5. Instead, we shall give a different argument valid for the case \(n=1.\)

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

The Galois group \(\varDelta (p)=({\mathcal {O}}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}})^{\times }\) acts on the coordinate ring \(H^{0}(Ig_{\mu },{\mathcal {O}})\) and we let \(H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)}\) be the subspace where it acts via the character \({\bar{\varSigma }}^{k}\). Then
$$\begin{aligned} H^{0}(Ig_{\mu },{\mathcal {O}})=\bigoplus _{k=0}^{p^{2}-2}H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)} \end{aligned}$$
(3.1)
and each \(H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)}\) is free of rank 1 over \(H^{0}(S_{\mu },{\mathcal {O}})=H^{0}(Ig_{\mu },{\mathcal {O}})^{(0)}.\)
For any \(0\le k\), the map \(f\mapsto f/a(k)\) is an embedding
$$\begin{aligned} M_{k}(N,\kappa _{0})\hookrightarrow H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)}. \end{aligned}$$
(3.2)

Lemma 3.1

Fix \(0\le k<p^{2}-1.\) Then we have a surjective homomorphism
$$\begin{aligned} \bigoplus _{n\ge 0}M_{k+n(p^{2}-1)}(N,\kappa _{0})\twoheadrightarrow H^{0}(Ig_{\mu },{\mathcal {O}})^{(k)}. \end{aligned}$$
(3.3)

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

The resulting ring homomorphism
$$\begin{aligned} r:\bigoplus _{k\ge 0}M_{k}(N,\kappa _{0})\twoheadrightarrow H^{0}(Ig_{\mu },\mathcal {O)} \end{aligned}$$
(3.4)
obtained by dividing a modular form of weight k by a(k) is surjective, respects the \(\mathbb {Z}/(p^{2}-1)\mathbb {Z}\)-grading on both sides, and its kernel is the ideal generated by \((h_{{\bar{\varSigma }}}-1)\).

Proof

We only have to prove that anything in \(\ker (r)\) is a multiple of \(h_{{\bar{\varSigma }}}-1\), the rest being clear. Since r respects the grading, we may assume that for some \(k\ge 0\) we have \(f_{j}\in M_{k+j(p^{2}-1)}(S,\kappa _{0})\) and \(f=\sum _{j=0}^{m}f_{j}\in \ker (r)\), i.e.
$$\begin{aligned} \sum _{j=0}^{m}a(k)^{-1}h_{{\bar{\varSigma }}}^{-j}f_{j}=0. \end{aligned}$$
(3.5)
But then \(f_{m}=-h_{{\bar{\varSigma }}}^{m}\left( \sum _{j=0}^{m-1}h_{{\bar{\varSigma }}}^{-j}f_{j}\right) \), so \(\sum _{j=0}^{m}f_{j}=\sum _{j=0}^{m-1}(1-h_{{\bar{\varSigma }}}^{m-j})f_{j}\) belongs to \((1-h_{{\bar{\varSigma }}}).\)
As a result, we get that
$$\begin{aligned} Ig_{\mu }^{*}=\text {Spec}\left( \bigoplus _{k\ge 0}M_{k}(N,\kappa _{0})/(h_{{\bar{\varSigma }}}-1)\right) \end{aligned}$$
(3.6)
and
$$\begin{aligned} S_{\mu }^{*}=\text {Spec}\left( \bigoplus _{k\ge 0}M_{k(p^{2}-1)}(N,\kappa _{0})/(h_{{\bar{\varSigma }}}-1)\right) . \end{aligned}$$
(3.7)

3.1.2 Fourier–Jacobi expansions modulo p

The arithmetic Fourier–Jacobi expansion (1.125) depended on a choice of a nowhere vanishing section s of \(\mathcal {L}\) along the boundary \(C={\bar{S}}\backslash S\) of \({\bar{S}}.\) As the boundary \({\tilde{C}}=\overline{Ig}_{\mu }\backslash Ig_{\mu }\) is (non-canonically) identified with \(\varDelta (p)\times C\), we may “compute” the Fourier–Jacobi expansion on the Igusa surface rather than on S. But on the Igusa surface, a is a canonical choice for such an s. We may therefore associate a canonical Fourier–Jacobi expansion
$$\begin{aligned} {\widetilde{FJ}}(f)=\sum _{m=0}^{\infty }c_{m}(f)\in \prod _{m=0}^{\infty }H^{0}({\tilde{C}},\mathcal {N}^{m}) \end{aligned}$$
(3.8)
along the boundary of Ig, with every
$$\begin{aligned} f\in M_{*}(N,R)=\bigoplus _{k=0}^{\infty }M_{k}(N,R) \end{aligned}$$
(3.9)
(R a \(\kappa _{0}\)-algebra). The following proposition becomes almost a tautology.

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 (“q-expansion 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

Let \(f\in M_{k}(N,R).\) Then \(0\le \omega (f)\le k\) and
$$\begin{aligned} \omega (f)\equiv k \mod (p^{2}-1). \end{aligned}$$
(3.10)
Let \(\omega (f)=k-(p^{2}-1)n.\) Then n is the order of vanishing of f along \(S_\mathrm{ss}.\) Equivalently, \(k-\omega (f)\) is the order of vanishing of the pull-back of f to Ig along \(Ig_\mathrm{ss}\). In addition, \(\omega (f^{m})=m\omega (f).\)

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 pre-image 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 pre-image of \(Z^{\prime } \) under \(\tau .\)

Let \(f\in H^{0}(S,\mathcal {L}^{k}).\) Then \(\tau ^{*}f/a^{k}\in H^{0}(Ig_{\mu },{\mathcal {O}})\) has a pole of order at most k along \({\tilde{Z}}\), and the Galois group acts on it via \({\bar{\varSigma }}^{k}.\) Let
$$\begin{aligned} \eta _{f}=d(\tau ^{*}f/a^{k})\in H^{0}(Ig_{\mu },\varOmega _{Ig}^{1})=H^{0}(Ig_{\mu },\tau ^{*}\varOmega _{S}^{1}). \end{aligned}$$
(3.11)
The Kodaira–Spencer isomorphism KS\((\varSigma )\) is an isomorphism
$$\begin{aligned} \text {KS}(\varSigma ):\mathcal {P}\otimes \mathcal {L}\simeq \varOmega _{S}^{1}. \end{aligned}$$
(3.12)
Let
$$\begin{aligned} \psi =(V_{\mathcal {P}}\otimes 1)\circ \text {KS}(\varSigma )^{-1}:\varOmega _{S}^{1}\rightarrow \mathcal {L}^{(p)}\otimes \mathcal {L}\simeq \mathcal {L}^{p+1}. \end{aligned}$$
(3.13)
We denote by \(\psi \) also the map induced on the base change of these vector bundles by \(\tau ^{*}\) to Ig and consider \(\psi (\eta _{f}).\) As \(\varDelta (p)\) still acts on \(\psi (\eta _{f})\) via \({\bar{\varSigma }}^{k}\), its action on \(a^{k}\psi (\eta _{f})\) is trivial, so this section descends to \(S_{\mu }.\) We define
$$\begin{aligned} \varTheta (f)=a^{k}\psi (\eta _{f})\in H^{0}(S_{\mu },\mathcal {L}^{k+p+1}). \end{aligned}$$
(3.14)
A priori, this extends only to a meromorphic modular form of weight \(k+p+1\), as it may have poles along Z.

3.2.2 The main theorem

For the formulation of the next theorem, we need to define what we mean by the standard cuspidal component of \({\bar{S}}\) or \(\overline{Ig}.\) Since its definition involves a transition back and forth between \(\mathbb {C}\) and \(\kappa \), we need to fix, besides the embedding of \(R_{N}\) in \(\mathbb {C}\) also a homomorphism
$$\begin{aligned} R_{N}\rightarrow \kappa \end{aligned}$$
(3.15)
extending the map \(R_{0}\rightarrow \kappa _{0}\subset \kappa \), and we let \(\mathfrak {P}\) be its kernel (a prime above p).
Recall that according to [2, 28] the cuspidal scheme \(C={\bar{S}}\backslash S\) classifies \({\mathcal {O}}_{{\mathcal {K}}}\)-semi-abelian varieties with level N structure. The standard component of C over \(\mathbb {C}\) is the component which classifies extensions of the elliptic curve \(\mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}}\) by the \({\mathcal {O}}_{{\mathcal {K}}}\)-torus \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C}^{\times }\) (thus sits over a cusp of type \(({\mathcal {O}}_{{\mathcal {K}}},{\mathcal {O}}_{{\mathcal {K}}})\) in \(S_{\mathbb {C}}^{*}\)), together with a level-N structure \((\alpha ,\beta ,\gamma )\) (see [2], I.4.2 and Sect. 1.6.2), where
$$\begin{aligned} \alpha :{\mathcal {O}}_{{\mathcal {K}}}/N{\mathcal {O}}_{{\mathcal {K}}}={\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {Z}/N\mathbb {Z} \rightarrow {\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {C}^{\times } \end{aligned}$$
(3.16)
is given by \(1\otimes (a\mapsto \exp (2\pi ia/N))\) and
$$\begin{aligned} \beta :{\mathcal {O}}_{{\mathcal {K}}}/N{\mathcal {O}}_{{\mathcal {K}}}=N^{-1}{\mathcal {O}}_{{\mathcal {K}}}/{\mathcal {O}}_{{\mathcal {K}}}\rightarrow \mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}} \end{aligned}$$
(3.17)
is the canonical embedding. (The splitting \(\gamma \) varies along the component.) The standard component of C over \(R_{N}\) is the one which becomes this component after base change to \(\mathbb {C}\). The standard component of C over \(\kappa \) is the reduction modulo \(\mathfrak {P}\) of the standard component of C over \(R_{N}\). Finally, \(\overline{Ig}\) maps to \({\bar{S}}\) (over \(\kappa \)) and the cuspidal components mapping to a given component E of C are classified by the embedding of \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p}\) in the toric part of \({\mathcal {A}}.\) Since the toric part of the universal semi-abelian variety over the standard component is \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m}, \) we may define the standard cuspidal component of \(\overline{Ig}\) to be the component where the map
$$\begin{aligned} \varepsilon :\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p}\rightarrow {\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m} \end{aligned}$$
(3.18)
is the natural embedding. Here we use the fact that
$$\begin{aligned} \delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p}={\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p} \end{aligned}$$
(3.19)
since \(\delta _{{\mathcal {K}}}\) is invertible in \({\mathcal {O}}_{{\mathcal {K}}}/p{\mathcal {O}}_{{\mathcal {K}}}.\) Let \({\tilde{E}}\subset {\tilde{C}}=\) \(\overline{Ig}\backslash Ig\) be this standard component.

Theorem 3.5

  1. (i)

    The operator \(\varTheta \) maps \(H^{0}(S,\mathcal {L}^{k}) \) to \(H^{0}(S,\mathcal {L}^{k+p+1}).\)

     
  2. (ii)
    The effect of \(\varTheta \) on Fourier–Jacobi expansions is a “Tate twist”. More precisely, let
    $$\begin{aligned} {\widetilde{FJ}}(f)=\sum _{m=0}^{\infty }c_{m}(f) \end{aligned}$$
    (3.20)
    be the canonical Fourier–Jacobi expansion of f along \({\tilde{E}}\) (thus \(c_{m}(f)\in H^{0}({\tilde{E}},\mathcal {N}^{m})\)). Then
    $$\begin{aligned} {\widetilde{FJ}}(\varTheta (f))=M^{-1}\sum _{m=0}^{\infty }mc_{m}(f). \end{aligned}$$
    (3.21)
    Here M (equal to \(N|D_{{\mathcal {K}}}|\) or \(2^{-1}N|D_{{\mathcal {K}}}|\)) is the width of the cusp.
     
  3. (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)
     
  4. (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 Q-th 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 m-th FJ coefficient at level N becomes the Qm-th 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

Let A be the universal semi-abelian variety over E. Its toric part is \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m}\), hence, taking \({\bar{\varSigma }}\)-component of the cotangent space at the origin
$$\begin{aligned} \mathcal {L}|_{{\tilde{E}}}=\omega _{A/{\tilde{E}}}({\bar{\varSigma }})=(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{\mathbb {G}_{m}})({\bar{\varSigma }}) \end{aligned}$$
(3.23)
admits the canonical section \(e_{{\bar{\varSigma }}}\cdot (1\otimes dT/T).\) Tracing back the definitions and using (1.69), this section becomes, under the base change \(R_{N}\hookrightarrow \mathbb {C}\), just \(2\pi id\zeta _{3}.\) On the other hand, when we reduce it modulo \(\mathfrak {P}\) and use the Igusa level structure \(\varepsilon \) at the standard cusp, it pulls back to the section “with the same name” \(e_{{\bar{\varSigma }}}\cdot (1\otimes dT/T)\), because along \({\tilde{E}}\) (3.18) induces the identity on cotangent spaces. The lemma follows from the fact that, by definition, \(\varepsilon ^{*}a=e_{{\bar{\varSigma }}}\cdot (1\otimes dT/T)\) too.

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

Let \(\sigma _{2}\) (resp. \(\sigma _{3}\)) be the \({\mathcal {K}}_{N}\)-rational section of \(\mathcal {P}_{\mu }\) (resp. \(\mathcal {L}\)) along E, which over \(\mathbb {C}\) becomes the section \(2\pi id\zeta _{2}\) (resp. \(2\pi id\zeta _{3}\)). We have just seen that modulo \(\mathfrak {P}\), when we identify \({\tilde{E}}\) with E (via the covering map \(\tau :\overline{Ig}\rightarrow {\bar{S}}\)), \(\sigma _{3}\) reduces to a. To conclude, we must show that the map
$$\begin{aligned} V:\mathcal {P}/\mathcal {P}_{0}=\mathcal {P}_{\mu }\simeq \mathcal {L}^{(p)} \end{aligned}$$
(3.24)
carries \(\sigma _{2}\) to \(\sigma _{3}^{(p)}.\) This will map, under \(\mathcal {L}^{(p)}\simeq \mathcal {L}^{p}\), to \(a^{p}.\) Along E, the line bundles \(\mathcal {P}_{\mu }\) and \(\mathcal {L}\) are just the \(\varSigma \)- and \({\bar{\varSigma }}\)-parts of the cotangent space at the origin of the torus \({\mathcal {O}}_{{\mathcal {K}}}\otimes \mathbb {G}_{m}\), and \(\sigma _{2}\) and \(\sigma _{3}\) are the sections
$$\begin{aligned} \sigma _{2}=e_{\varSigma }\cdot (1\otimes dT/T),\quad \sigma _{3}=e_{{\bar{\varSigma }}}\cdot (1\otimes dT/T). \end{aligned}$$
(3.25)
Since in characteristic p, \(V=Ver^{*}:\omega _{\mathbb {G}_{m}}\rightarrow \omega _{\mathbb {G}_{m}}^{(p)}\) maps dT / T to \((dT/T)^{(p)}\), for the \({\mathcal {O}}_{{\mathcal {K}}}\)-torus, \(V(\sigma _{2})=\sigma _{3}^{(p)}\), and we are done.
To prove part (ii) of the main theorem, we argue as follows. Let \(g=f/a^{k}\) be the function on \(\overline{Ig}_{\mu }\) obtained by trivializing the line bundle \(\mathcal {L}.\) We have to study the FJ expansion along \({\tilde{E}}\) of \(\psi (dg)/a^{p+1}\), where \(\psi \) is the map defined in (3.13). For that purpose, we may restrict to a formal neighborhood of \({\tilde{E}}\). This formal neighborhood is isomorphic, under the covering map \(\tau :\) \(\overline{Ig}_{\mu }\rightarrow {\bar{S}}_{\mu }\), to the formal neighborhood \({\widehat{S}}\) of E in S. We may therefore regard dg as an element of \(\varOmega _{{\widehat{S}}}^{1}.\) Now
$$\begin{aligned} \psi :\varOmega _{{\widehat{S}}}^{1}\rightarrow \mathcal {P}_{\mu }\otimes \mathcal {L} \end{aligned}$$
(3.26)
is a homomorphism of \({\mathcal {O}}_{{\widehat{S}}}\)-modules defined over \(R_{N}\) so, having restricted to \({\widehat{S}}\), we may study the effect of \(\psi \) on FJ expansions by embedding \({\widehat{S}}_{\mathbb {C}}\) in a tubular neighborhood \({\bar{S}}(\varepsilon )\) of E and using complex analytic Fourier–Jacobi expansions. We are thus reduced to a complex-analytic computation, near the standard cusp at infinity.
Let
$$\begin{aligned} g(z,u)=\sum _{m=0}^{\infty }\theta _{m}(u)q^{m} \end{aligned}$$
(3.27)
where \(q=e^{2\pi iz/M}\) and \(\theta _{m}\) is a theta function, so that \(\theta _{m}(u)q^{m}\) is a section of \(\mathcal {N}^{m}\) along E (now over \(\mathbb {C}\)). Then
$$\begin{aligned} \text {d}g=2\pi iM^{-1}\sum _{m=0}^{\infty }m\theta _{m}(u)q^{m}\text {d}z+\sum _{m=0}^{\infty }\theta _{m}^{\prime }(u)q^{m}\text {d}u. \end{aligned}$$
(3.28)
According to Corollary 1.30, \(\psi (\text {d}u)=0\), and \(\psi (\text {d}z)=2\pi i\text {d}\zeta _{2}\otimes \text {d}\zeta _{3}.\) It follows that
$$\begin{aligned} \psi (\text {d}g)=M^{-1}\sum _{m=0}^{\infty }m\theta _{m}(u)q^{m}\cdot 2\pi i\text {d}\zeta _{2}\otimes 2\pi i\text {d}\zeta _{3}. \end{aligned}$$
(3.29)
Recalling that in characteristic p, \(2\pi i\text {d}\zeta _{2}\otimes 2\pi i\text {d}\zeta _{3}\) reduced to \(a^{p+1}\), the proof of part (ii) of the theorem is now complete. For the convenience of the reader, we summarize the transitions between complex and p-adic maps in the following diagram:
$$\begin{aligned} \begin{array}{lllllll} /\text { }\kappa &{} &{} \varOmega _{{\bar{S}}_{\mu }/\kappa }^{1} &{} \overset{\text {KS}(\varSigma )^{-1}}{\rightarrow } &{} \mathcal {P}\otimes \mathcal {L} &{} &{} \\ &{} &{} \cap &{} &{} \downarrow \mod \mathcal {P}_{0} &{} &{} \\ /\kappa &{} &{} \varOmega _{{\widehat{S}}/\kappa }^{1} &{} \overset{\psi }{\rightarrow } &{} \mathcal {P}_{\mu }\otimes \mathcal {L} &{} \overset{V\otimes 1}{\simeq } &{} \mathcal {L}^{p+1} \\ &{} &{} \uparrow \mod \mathfrak {P} &{} &{} \uparrow &{} &{} \\ /R_{N} &{} &{} \varOmega _{{\widehat{S}}/R_{N}}^{1} &{} \overset{\psi }{\rightarrow } &{} \mathcal {P}_{\mu }\otimes \mathcal {L} &{} &{} \\ &{} &{} \downarrow \otimes _{R_{N}}\mathbb {C} &{} &{} \downarrow &{} &{} \\ /\mathbb {C} &{} &{} \varOmega _{{\widehat{S}}/\mathbb {C}}^{1} &{} \overset{\psi }{\rightarrow } &{} \mathcal {P}_{\mu }\otimes \mathcal {L} &{} &{} \\ &{} &{} \cup &{} &{} \uparrow \mod \mathcal {P}_{0} &{} &{} \\ /\mathbb {C} &{} &{} \varOmega _{{\bar{S}}(\varepsilon )/\mathbb {C}}^{1} &{} \overset{\text {KS}(\varSigma )_{an}^{-1}}{\rightarrow } &{} \mathcal {P}\otimes \mathcal {L} &{} &{} \end{array} . \end{aligned}$$
(3.30)
We next turn to part (i).

3.4 A study of the theta operator along the supersingular locus

3.4.1 De Rham cohomology in characteristic p

We continue to consider the Picard surface S over \(\kappa \) and recall some facts about de Rham cohomology in characteristic p. Let \(U=\text {Spec}(R)\hookrightarrow S\) be a closed point \(s_{0}\) (\(R= \kappa =\) \({\mathcal {O}}_{S,s_{0}}/\mathfrak {m}_{S,s_{0}}\)), a nilpotent thickening of a closed point, or an affine open subset of S. We consider the restriction of the universal abelian scheme to R and denote it by A / R. Let \(A^{(p)}=R\otimes _{\phi ,R}A\) be its base change with respect to the map \(\phi (x)=x^{p}.\) Let
$$\begin{aligned} D=H_{dR}^{1}(A/R), \end{aligned}$$
(3.31)
a locally free R-module of rank 6. The de Rham cohomology of \(A^{(p)}\) is
$$\begin{aligned} D^{(p)}=R\otimes _{\phi ,R}D. \end{aligned}$$
(3.32)
The R-linear Frobenius and Verschiebung morphisms \(Frob:A\rightarrow A^{(p)}\), \(Ver:A^{(p)}\rightarrow A\) induce (by pull-back) linear maps
$$\begin{aligned} F:D^{(p)}\rightarrow D,\,\,V:D\rightarrow D^{(p)}. \end{aligned}$$
(3.33)
Both F and V are everywhere of rank 3, which implies that their kernel and image are locally free direct summands. Moreover, \(\text {Im} F=\ker V\) and \(\text {Im} V=\ker F=\omega _{A^{(p)}/R}.\) The maps F and V preserve the types \(\varSigma ,{\bar{\varSigma }}\), but note that \(D^{(p)}(\varSigma )=D({\bar{\varSigma }})^{(p)}\) etc.
The principal polarization on A induces one on \(A^{(p)}\), and these polarizations induce symplectic forms
$$\begin{aligned} \left\langle ,\right\rangle :D\times D\rightarrow R,\,\,\left\langle ,\right\rangle ^{(p)}:D^{(p)}\times D^{(p)}\rightarrow R \end{aligned}$$
(3.34)
where the second form is just the base change of the first. For \(x\in D^{(p)},y\in D\), we have
$$\begin{aligned} \left\langle Fx,y\right\rangle =\left\langle x,Vy\right\rangle ^{(p)}. \end{aligned}$$
(3.35)
In addition, for \(a\in {\mathcal {O}}_{{\mathcal {K}}}\)
$$\begin{aligned} \left\langle \iota (a)x,y\right\rangle =\left\langle x,\iota ({\bar{a}})y\right\rangle . \end{aligned}$$
(3.36)
As \(VF=FV=0\), the first relation implies that \(\text {Im} F\) and \(\text {Im} V\) are isotropic subspaces. So is \(\omega _{A/R}.\)
The Gauss–Manin connection is an integrable connection
$$\begin{aligned} \nabla :D\rightarrow \varOmega _{R}^{1}\otimes D. \end{aligned}$$
(3.37)
It is a priori defined (e.g. in [24]) when R is smooth over \(\kappa \), but we can define it by base change also when R is a nilpotent thickening of a point of S (see [25], where R is a local Artinian ring).
We shall need to deal only with the first infinitesimal neighborhood of a point, \(R={\mathcal {O}}_{S,s_{0}}/\mathfrak {m}_{S,s_{0}}^{2}.\) In this case, D has a basis of horizontal sections. Indeed, \(R=\kappa [u,v]/(u^{2},uv,v^{2})\) where u and v are local parameters at \(s_{0}\), and
$$\begin{aligned} \varOmega _{R}^{1}=(R\text {d}u+R\text {d}v)/\left\langle u\text {d}u,\,v\text {d}v,\,u\text {d}v+v\text {d}u\right\rangle \end{aligned}$$
(p is odd). If \(x\in D\) and
$$\begin{aligned} \nabla x=\text {d}u\otimes x_{1}+\text {d}v\otimes x_{2} \end{aligned}$$
(3.38)
then \({\tilde{x}}=x-ux_{1}-vx_{2}\) is horizontal, so the horizontal sections span D over R by Nakayama’s lemma. It follows that if \(D_{0}=D^{\nabla }\) is the space of horizontal sections,
$$\begin{aligned} R\otimes _{\kappa }D_{0}=D, \end{aligned}$$
(3.39)
\(\nabla =d\otimes 1\) and we can identify \(D_{0}=H_{dR}^{1}(A_{s_{0}}/ \kappa )\), i.e. every de Rham class at \(s_{0}\) has a unique extension to a horizontal section \(x\in H_{dR}^{1}(A/R).\)

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 \(\nabla (Vx)=0.\)

The pairing \(\left\langle ,\right\rangle \) is horizontal for \(\nabla \), i.e.
$$\begin{aligned} \text {d}\left\langle x,y\right\rangle =\left\langle \nabla x,y\right\rangle +\left\langle x,\nabla y\right\rangle . \end{aligned}$$
(3.40)

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

There exists a basis \(e_{1},e_{2},f_{3},f_{1},f_{2},e_{3}\) of \(D_{0}\) with the following properties. Denote by \(e_{1}^{(p)}=1\otimes e_{1}\in D_{0}^{(p)}\) etc.
  1. (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 \(\varSigma ).\)

     
  2. (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)
     
  3. (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 \(\mathcal {L}.\)

     
  4. (iv)

    \(\ker (V)\) is spanned by \(e_{1},f_{2},e_{3}.\) Hence \(\mathcal {P}_{0}=\mathcal {P}\cap \ker (V)\) is spanned by \(e_{1}.\)

     
  5. (v)

    \(Ve_{2}=f_{3}^{(p)},\,Vf_{3}=e_{1}^{(p)},\,Vf_{1}=e_{2}^{(p)}.\)

     
  6. (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 \(R\otimes _{\kappa }D_{0}.\)

Grothendieck tells us that A / R is completely determined by \(A_{0}\) and by the Hodge filtration \(\omega _{A/R}\subset D=R\otimes _{\kappa }D_{0}.\) Since A is the universal infinitesimal deformation of \(A_{0}\), we may choose the coordinates u and v so that
$$\begin{aligned} \mathcal {P}=\text {Span}_{R}\{e_{1}+ue_{3},e_{2}+ve_{3}\}. \end{aligned}$$
(3.42)
The fact that \(\omega _{A/R}\) is isotropic implies then that
$$\begin{aligned} \mathcal {L}=\text {Span}_{R}\{f_{3}-uf_{1}-vf_{2}\}. \end{aligned}$$
(3.43)
Consider the abelian scheme \(A^{(p)}.\) It is not the universal deformation of \(A_{0}^{(p)}\) over R. In fact, the map \(\phi :R\rightarrow R \) factors as
$$\begin{aligned} R\overset{\pi }{\rightarrow }\kappa \overset{\phi }{\rightarrow } \kappa \overset{i}{\rightarrow }R, \end{aligned}$$
(3.44)
and therefore \(A^{(p)}\), unlike A, is constant: \(A^{(p)}=\text {Spec}(R)\times _{\text {Spec}(\kappa )}A_{0}^{(p)}.\) As with D, \(D^{(p)}=R\otimes _{\mathbb {\kappa }}D_{0}^{(p)}\), \(\nabla =d\otimes 1\), but this time the basis of horizontal sections can be obtained also from the trivialization of \(A^{(p)}\), and \(\omega _{A^{(p)}/R}=\text {Span}_{R}\{e_{1}^{(p)},e_{2}^{(p)},f_{3}^{(p)}\}.\)

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).

The Hodge filtration nevertheless varies, so we conclude that
$$\begin{aligned} \mathcal {P}_{0}=\mathcal {P}\cap \ker (V)=\text {Span}_{R}\{e_{1}+ue_{3}\}. \end{aligned}$$
(3.45)
The condition \(V(\mathcal {L})=\mathcal {P}_{0}^{(p)}\), which is the “equation” of the closed subscheme \(Z^{\prime }\cap \text {Spec}(R)\) (see Theorem 2.9) means
$$\begin{aligned} V(f_{3}-uf_{1}-vf_{2})=e_{1}^{(p)}-ue_{2}^{(p)}\in R\cdot e_{1}^{(p)} \end{aligned}$$
(3.46)
and this holds if and only if \(u=0.\) We have proved the following lemma.

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

We keep the assumptions of the previous subsections and compute what the Gauss–Manin connection does to \(\mathcal {P}_{0}.\) A typical element of \(\mathcal {P}_{0}\) is \(g(e_{1}+ue_{3})\) for some \(g\in R.\) Then
$$\begin{aligned} \nabla (g(e_{1}+ue_{3}))=\text {d}g\otimes (e_{1}+ue_{3})+g\text {d}u\otimes e_{3}. \end{aligned}$$
(3.47)
Note that when we divide by \(\omega _{A/R}\) and project \(H_{dR}^{1}(A/R)\) to \(H^{1}(A,{\mathcal {O}})\), \(e_{1}+ue_{3}\) dies, and the image \(\overline{e_{3}}\) of \(e_{3}\) becomes a basis for the line bundle that we called \(\mathcal {L}^{\vee }(\rho )=H^{1}(A,{\mathcal {O}})(\varSigma )\). Recall the definition of \(\psi \) given in (3.13), but note that this definition only makes sense over \(\text {Spec}({\mathcal {O}}_{S,s_{0}})\) or its completion, where \(\text {KS}(\varSigma )\) is an isomorphism, and can be inverted.

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

Let \(i:Z^{\prime }\hookrightarrow S\) be the locally closed embedding. We must show that in a suitable Zariski neighborhood of \(s_{0}\), where \(u=0\) is the local equation of \(Z^{\prime }\), \(i^{*}\psi (\text {d}u)=0.\) It is enough to show that the image of \(\psi (\text {d}u)\) in the fiber at every point s of \(Z^{\prime }\) near \(s_{0}\), vanishes. All points being alike, it is enough to do it at \(s_{0}.\) In other words, we denote by \(\psi _{0}\) the map
$$\begin{aligned} \psi _{0}:\varOmega _{S,s_{0}}^{1}\rightarrow \mathcal {P}_{\mu }\otimes \mathcal {L}|_{s_{0}}\simeq \mathcal {L}^{p+1}|_{s_{0}}. \end{aligned}$$
(3.48)
and show that \(\psi _{0}(\text {d}u)=0.\) We may now work over \(\text {Spec}(R)\), where \(R={\mathcal {O}}_{S,s_{0}}/\mathfrak {m}^{2}\). It is enough to show that in the diagram
$$\begin{aligned} \begin{array}{lll} \mathcal {P}_{R}\otimes \mathcal {L}_{R} &{} \overset{\text {KS}(\varSigma )}{\rightarrow } &{} \varOmega _{R}^{1} \\ \downarrow &{} &{} \downarrow \\ \mathcal {P}_{s_{0}}\otimes \mathcal {L}_{s_{0}} &{} \simeq &{} \varOmega _{S,s_{0}}^{1} \end{array} \end{aligned}$$
(3.49)
KS\((\varSigma )\) maps the line sub-bundle \(\mathcal {P}_{0,R}\otimes \mathcal {L}_{R}\) onto \(R\text {d}u\). Once we have passed to the infinitesimal neighborhood \(\text {Spec}(R)\), we can replace the local parameters uv by any two formal parameters for which \(u=0\) defines \(Z^{\prime }\cap \text {Spec}(R).\) We may therefore assume, in view of Lemma 3.10, that u and v have been chosen as in Sect. 3.4.3. But then (3.47) shows that the restriction of KS\((\varSigma )\) to \(Z^{\prime }\), i.e. the homomorphism \(i^{*}\text {KS}(\varSigma )\), maps \(i^{*}\mathcal {P}_{0}\) onto \(i^{*}R\cdot \text {d}u\otimes \overline{e_{3}}\). This concludes the 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

A priori, the definition that we have given for \(\varTheta (f)\) produces a meromorphic section of \(\mathcal {L}^{k+p+1}\) which is holomorphic on the \(\mu \)-ordinary part \(S_{\mu }\) but may have a pole along \(Z=S_\mathrm{ss}\). Since S is a non-singular surface, it is enough to show that \(\varTheta (f)\) does not have a pole along \(Z^{\prime }=S_\mathrm{gss}\), the non-singular part of the divisor Z. Consider the degree \(p^{2}-1\) covering \(\tau :Ig\rightarrow S\), which is finite, étale over \(S_{\mu }\) and totally ramified along Z. Let \(s_{0}\in Z^{\prime }\) and let \({\tilde{s}}_{0}\in Ig\) be the closed point above it. Let uv be formal parameters at \(s_{0}\) for which \(Z^{\prime }\) is given by \(u=0\), as in Theorem 2.15. As explained there, we may choose formal parameters wv at \({\tilde{s}}_{0}\) where \(w^{p^{2}-1}=u\) (and v is the same function v pulled back from S to Ig). It follows that in \(\varOmega _{Ig}^{1}\) we have
$$\begin{aligned} \text {d}u=-w^{p^{2}-2}\text {d}w. \end{aligned}$$
(3.50)
We now follow the steps of our construction. Dividing f by \(a^{k}\), we get a function \(g=f/a^{k}\) on Ig with a pole of order at most k along \({\tilde{Z}}\), the supersingular divisor on Ig, whose local equation is \(w=0.\) In \({\widehat{{\mathcal {O}}}}_{Ig,{\tilde{s}}_{0}}\) we may write
$$\begin{aligned} g=\sum _{l=-k}^{\infty }g_{l}(v)w^{l}. \end{aligned}$$
(3.51)
Then
$$\begin{aligned} \text {d}g= & {} \sum _{l=-k}^{\infty }lg_{l}(v)w^{l-1}\text {d}w+\sum _{l=-k}^{\infty }w^{l}g_{l}^{\prime }(v)\text {d}v \nonumber \\= & {} -\sum _{l=-k}^{\infty }lg_{l}w^{l-(p^{2}-1)}\text {d}u+\sum _{l=-k}^{\infty }w^{l}g_{l}^{\prime }(v)\text {d}v. \end{aligned}$$
(3.52)
Applying the map \(\psi \) (extended \({\mathcal {O}}_{Ig}\)-linearly from S to Ig), and noting that \(\psi (\text {d}u)\) has a zero along \(Z^{\prime }\), hence a zero of order \(p^{2}-1\) along \({\tilde{Z}}^{\prime }\), we conclude that \(\psi (\text {d}g)\) has a pole of order k (at most) along \({\tilde{Z}}^{\prime }.\) Finally, \(\varTheta (f)=a^{k}\cdot \psi (\text {d}g)\) becomes holomorphic along \({\tilde{Z}}^{\prime }\), and also descends to S. It is therefore a holomorphic section of \(\mathcal {P}_{\mu }\otimes \mathcal {L}^{k+1}\simeq \mathcal {L}^{k+p+1}.\)

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 \(p-1\), to make the section holomorphic and hence a total increase by \(p+1=2+(p-1)\) 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

In part (ii) of Theorem 3.5, we have described the way \(\varTheta \) acts on Fourier–Jacobi expansions at the standard cusp. A similar formula holds at all the other cusps. We deduce from it that modular forms in the image of \(\varTheta \) have vanishing FJ coefficients in degrees divisible by p. Moreover, for such a form \(f\in \text {Im}(\varTheta )\), \(\varTheta ^{p-1}(f)\) and f have the same FJ expansions, and hence the same filtration. Note also that if \(r(f_{1})=r(f_{2})\), then \(r(\varTheta (f_{1}))=r(\varTheta (f_{2})).\) We may therefore define unambiguously
$$\begin{aligned} \varTheta (r(f))=r(\varTheta (f)). \end{aligned}$$
(4.1)
As we clearly have
$$\begin{aligned} \omega (\varTheta (f))=\omega (f)+p+1-a\left( p^{2}-1\right) \end{aligned}$$
(4.2)
for some \(a\ge 0\) we deduce the following result.

Proposition 4.1

Let \(f\in M_{k}(N,\mathbb {\kappa })\) be a modular form modulo p, and assume that \(r(f)\in \text {Im}(\varTheta ).\) Then
$$\begin{aligned} r(f)=r\left( \varTheta ^{p-1}(f)\right) . \end{aligned}$$
(4.3)
There exists a unique index \(0\le i\le p-2\) such that
$$\begin{aligned} \omega (\varTheta ^{i+1}(f))=\omega \left( \varTheta ^{i}(f)\right) +p+1-\left( p^{2}-1\right) . \end{aligned}$$
(4.4)
For any other i in this range
$$\begin{aligned} \omega (\varTheta ^{i+1}(f))=\omega \left( \varTheta ^{i}(f)\right) +p+1. \end{aligned}$$
(4.5)
This is reminiscent of the “theta cycles” for classical (i.e. elliptic) modular forms modulo p, see [19, 21, 34]. Recall that if f is a \(\mod p\) modular form of weight k on \(\varGamma _{0}(N)\) with q-expansion \(\sum a_{n}q^{n}\) (\(a_{n}\in \mathbb {{\bar{F}}}_{p}\)), then \(\theta (f) \) is a \(\mod p\) modular form of weight \(k+p+1\) with q-expansion \(\sum na_{n}q^{n}\) (Katz denotes \(\theta (f)\) by \(A\theta (f)\)). One has \(\omega (\theta (f))<\omega (f)+p+1\) if and only if \(\omega (f)\equiv 0\mod p\). In such a case, we say that the filtration “drops” and we have
$$\begin{aligned} \omega (\theta (f))=\omega (f)+p+1-a(p-1) \end{aligned}$$
(4.6)
for some \(a>0.\) As a corollary, \(\omega (f)\) can never equal \(1\mod p\) for an \(f\in \text {Im}(\theta ).\) Assume now that \(f\in \text {Im}(\theta )\) is a “low point” in its “theta cycle”, namely, \(\omega (f)\) is minimal among all \(\omega (\theta ^{i}(f)).\) Then \(\omega (\theta ^{i+1}(f))<\omega (\theta ^{i}(f))+p+1\) for one or two values of \(i\in [0,p-2]\), which are completely determined by \(\omega (f) \mod p\) [19].
This is not true anymore for Picard modular forms. Not only is the drop in the theta cycle unique, but the question of when exactly it occurs is mysterious and deserves further study. We make the following elementary observation showing that whether a drop in the filtration occurs in passing from f to \(\varTheta (f)\) can not be determined by \(\omega (f)\) modulo p alone. Let f and k be as in Proposition 4.1.
  1. (1)

    If \(k\le p^{2}-1\), then \(\omega (f)=k.\)

     
  2. (2)

    If \(k<p+1\), then \(\omega (\varTheta ^{i}(f))=k+i(p+1)\) for \(0\le i\le p-2, \) so the drop occurs at the last step of the theta cycle, i.e. at weight \(k+(p-2)(p+1)\), which is congruent to \(k-2\) modulo p.

     
  3. (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 \(\varTheta ^{p-2}(f)\) to \(\varTheta ^{p-1}(f)\), or in passing from \(\varTheta ^{p-1}(f)\) 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 Swinnerton-Dyer in terms of q-expansions, 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 \(\varTheta \).

Let us quickly repeat Gross’ construction as outlined in the introduction. Let X be the open modular curve X(N) over \(\mathbb {{\bar{F}}}_{p}\) (\(N\ge 3,\,p\not \mid N\)) and \(I_\mathrm{ord}\) the Igusa curve of level p lying over \(X_\mathrm{ord}=X\backslash X_\mathrm{ss}\), the ordinary part of X. Let \({\bar{X}}\) and \({\bar{I}}_\mathrm{ord}\) be the curves obtained by adjoing the cusps to X and \(I_\mathrm{ord}\), respectively. Let \(\mathcal {L}=\omega _{E/X}\) be the cotangent bundle of the universal elliptic curve, extended over the cusps as usual. Classical modular forms of weight k and level N are sections of \(\mathcal {L}^{k}\) over \({\bar{X}}.\) Let a be the tautological nowhere vanishing section of \(\mathcal {L}\) over \({\bar{I}}_\mathrm{ord}.\) Given a modular form f of weight k, we consider \(r(f)=\tau ^{*}f/a^{k}\) where \(\tau :{\bar{I}}_\mathrm{ord}\rightarrow {\bar{X}}\) is the covering map, and apply the inverse of the Kodaira–Spencer isomorphism \(\text {KS}:\mathcal {L}^{2}\rightarrow \varOmega _{I_\mathrm{ord}}^{1}\) to get a section \(\text {KS}^{-1}(\text {d}r(f))\) of \(\mathcal {L}^{2}\) over \({\bar{I}}_\mathrm{ord}.\) When multiplied by \(a^{k}\) it descends to \({\bar{X}}_\mathrm{ord}\), and when this is multiplied further by \(h=a^{p-1}\), the Hasse invariant for elliptic modular forms, it extends holomorphically over \(X_\mathrm{ss}\) to an element
$$\begin{aligned} \theta (f)=a^{k+p-1}\text {KS}^{-1}(\text {d}r(f))\in H^{0}\left( {\bar{X}},\mathcal {L}^{k+p+1}\right) . \end{aligned}$$
(4.7)

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).

Embed \(SL_{2}(\mathbb {R})=SU(1,1)\) in \(G_{\infty }^{\prime }\) via
$$\begin{aligned} \left( \begin{array}{ll} a &{}\quad b \\ c &{}\quad d \end{array} \right) \mapsto \left( \begin{array}{lll} a &{} &{}\quad b \\ &{}\quad 1 &{} \\ c &{} &{}\quad d \end{array} \right) . \end{aligned}$$
(4.8)
This embedding induces an embedding of symmetric spaces \(\mathfrak {H}\hookrightarrow \mathfrak {X}\), \(z\mapsto \,^{t}(z,0).\) One can easily compute that the intersection of \(\varGamma \), the stabilizer of the lattice \(L_{0}\) in \(G_{\infty }^{\prime }\), with \(SL_{2}(\mathbb {R})\), is the subgroup of \(SL_{2}(\mathbb {Z})\) given by
$$\begin{aligned} \varGamma ^{0}(D)=\left\{ \left( \begin{array}{ll} a &{}\quad b \\ c &{}\quad d \end{array} \right) :\,D|b\right\} . \end{aligned}$$
(4.9)
Let \(E_{0}=\mathbb {C}/{\mathcal {O}}_{{\mathcal {K}}}\), endowed with the canonical principal polarization and CM type \(\varSigma .\) For \(z\in \mathfrak {H}\), let \(\Lambda _{z}=\mathbb {Z}+\mathbb {Z}z\) and \(E_{z}=\mathbb {C}/\Lambda _{z}.\) Let \(M_{z}\) be the cyclic subgroup of order D of \(E_{z}\) generated by \(D^{-1}z\mod \Lambda _{z}.\) Using the model (1.27) of the abelian variety \(A_{z}\) associated with the point \(^{t}(z,0)\in \mathfrak {X}\), we compute that
$$\begin{aligned} A_{z}\simeq E_{0}\times ({\mathcal {O}}_{{\mathcal {K}}}\otimes E_{z})/(\delta _{{\mathcal {K}}}\otimes M_{z}) \end{aligned}$$
(4.10)
with the obvious \({\mathcal {O}}_{{\mathcal {K}}}\)-structure. The group \(\delta _{{\mathcal {K}}}\otimes M_{z}\) is a cyclic subgroup of \({\mathcal {O}}_{{\mathcal {K}}}\otimes E_{z}\) of order D, generated by \(\delta _{{\mathcal {K}}}^{-1}\otimes z\mod \mathcal {O}_{{\mathcal {K}}}\otimes \Lambda _{z}.\) The principal polarization on \(A_{z}\) provided by the complex uniformization is the product of the canonical polarization of \(E_{0}\) and the principal polarization of \({\mathcal {O}}_{{\mathcal {K}}}\otimes E_{z}/\delta _{{\mathcal {K}}}\otimes M_{z}\) obtained by descending the polarization
$$\begin{aligned} \lambda _{can}:{\mathcal {O}}_{{\mathcal {K}}}\otimes E_{z}\rightarrow \delta _{{\mathcal {K}}}^{-1}\otimes E_{z}=({\mathcal {O}}_{{\mathcal {K}}}\otimes E_{z})^{t} \end{aligned}$$
(4.11)
of degree \(D^{2}\), modulo the maximal isotropic subgroup \(\delta _{{\mathcal {K}}}\otimes M_{z}\) of \(\ker (\lambda _{can}).\)

It is now clear that over any \(R_{0}\)-algebra R we have the same moduli-theoretic construction, sending a pair (EM) where M is a cyclic subgroup of degree D to A(EM), 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

From now on, we work over \(\mathbb {{\bar{F}}}_{p}.\) The modular interpretation of the embedding \(j:{\bar{X}}\rightarrow {\bar{S}}\) allows us to complete it to a diagram
$$\begin{aligned} \begin{array}{lll} {\bar{I}}_\mathrm{ord} &{} \overset{j}{\rightarrow } &{} \overline{Ig}_{\mu } \\ \tau \downarrow &{} &{} \downarrow \tau \\ {\bar{X}}_\mathrm{ord} &{} \overset{j}{\rightarrow } &{} {\bar{S}}_{\mu } \end{array} . \end{aligned}$$
(4.12)
Note that \(j(X_\mathrm{ss})\subset S_\mathrm{ssp}\), i.e. the embedded modular curve cuts the supersingular locus at superspecial points.

Lemma 4.2

The pull-back \(j^{*}\mathcal {\omega }_{{\mathcal {A}}/S}\) decomposes as a product \(\omega _{E_{0}}\times ({\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{E/X}).\) Under this isomorphism,
$$\begin{aligned} j^{*}\mathcal {L}= & {} ({\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{E/X})({\bar{\varSigma }}) \\ j^{*}\mathcal {P}_{0}= & {} \omega _{E_{0}} \nonumber \\ j^{*}\mathcal {P}_{\mu }\simeq & {} ({\mathcal {O}}_{{\mathcal {K}}}\otimes \omega _{E/X})(\varSigma ). \nonumber \end{aligned}$$
(4.13)

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

Identify \(j^{*}\mathcal {L}\) with \(\omega _{E/X}\) (\({\mathcal {O}}_{{\mathcal {K}}}\) acting via \({\bar{\varSigma }}\)). Then for \(f\in H^{0}({\bar{S}},\mathcal {L}^{k})=M_{k}(N,\mathbb {{\bar{F}}}_{p})\)
$$\begin{aligned} \theta (j^{*}(f))=j^{*}(\varTheta (f)). \end{aligned}$$
(4.14)

Proof

We abbreviate \(I_\mathrm{ord}\) by I and \(Ig_{\mu }\) by Ig. The pull-back via j of the tautological section a of \(\mathcal {L}\) over Ig is the tautological section a of \(j^{*}\mathcal {L=\omega }_{E/X}.\) We therefore have
$$\begin{aligned} j^{*}(\text {d}r(f))=\text {d}r(j^{*}(f)) \end{aligned}$$
(4.15)
(\(r(f)=\tau ^{*}f/a^{k}\) is the function on Ig denoted earlier also by g). It remains to check the commutativity of the following diagram
$$\begin{aligned} \begin{array}{lllll} \varOmega _{Ig}^{1} &{} \overset{\text {KS}(\varSigma )^{-1}}{\rightarrow } &{} \mathcal {P}\otimes \mathcal {L} &{} \overset{V\otimes 1}{\rightarrow } &{} \mathcal {L}^{p+1} \\ \downarrow j_{0}^{*} &{} &{} &{} &{} \downarrow j^{*} \\ \varOmega _{I}^{1} &{} \overset{\text {KS}^{-1}}{\rightarrow } &{} j^{*}\mathcal {L}^{2} &{} \overset{\times h}{\rightarrow } &{} j^{*}\mathcal {L}^{p+1} \end{array} . \end{aligned}$$
(4.16)
Here \(j_{0}^{*}\) is the map \(j^{*}\varOmega _{Ig}^{1}\rightarrow \varOmega _{I}^{1} \) on differentials whose kernel is the conormal bundle of I in Ig . For that we have to compare the Kodaira–Spencer maps on S and on X. As we have seen in the lemma, \(\mathcal {P}/\mathcal {P}_{0}=\mathcal {P}_{\mu } \) pulls back under j to \(\mathcal {L}(\rho )\) (the line bundle \(\mathcal {L}\) with the \({\mathcal {O}}_{{\mathcal {K}}}\) action conjugated). But, \(\text {KS}(\varSigma )(\mathcal {P}_{0}\otimes \mathcal {L})\) maps under \(j^{*}\) to the conormal bundle, so we obtain a commutative diagram
$$\begin{aligned} \begin{array}{lll} \varOmega _{Ig}^{1} &{} \overset{\text {KS}(\varSigma )}{\leftarrow } &{} \mathcal {P}\otimes \mathcal {L} \\ \downarrow j_{0}^{*} &{} &{} \downarrow \mod \mathcal {P}_{0} \\ \varOmega _{I}^{1} &{} \overset{\text {KS}}{\leftarrow } &{} j^{*}\mathcal {L}(\rho )\otimes j^{*}\mathcal {L} \end{array} . \end{aligned}$$
(4.17)
The commutativity of the diagram
$$\begin{aligned} \begin{array}{lll} \mathcal {P}_{\mu } &{} \overset{V}{\rightarrow } &{} \mathcal {L}^{(p)} \\ \downarrow &{} &{} \downarrow \\ j^{*}\mathcal {L}(\rho ) &{} \overset{\times h}{\rightarrow } &{} j^{*}\mathcal {L}^{(p)} \end{array} \end{aligned}$$
(4.18)
follows from the definition of the Hasse invariant h on X. Identifying \(\mathcal {L}^{(p)}\) with \(\mathcal {L}^{p}\) as usual and tensoring the last diagram with \(\mathcal {L}\) provides the last piece of the puzzle.

Remark 4.1

The proposition follows, of course, also from the effect of \(\theta \) and \(\varTheta \) on q-expansions, once we compare FJ expansions on \({\bar{S}}\) to q-expansions 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 p-adic modular forms

We shall be very brief, since from now on the development follows closely the classical case of p-adic 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 \(p^{m}\)

Let \(m\ge 1\), and write \(R_{m}=R_{0}/p^{m}R_{0}={\mathcal {O}}_{{\mathcal {K}}}/p^{m}{\mathcal {O}}_{{\mathcal {K}}}.\) Let
$$\begin{aligned} S^{(m)}=S\times _{\text {Spec}(R_{0})}\text {Spec}(R_{m}) \end{aligned}$$
(5.1)
so that \(S^{(1)}=S_{\kappa _{0}}\) is the special fiber, and use a similar notation for the complete surface \({\bar{S}}^{(m)}.\) Write \(S_{\mu }^{(m)}\) (resp. \({\bar{S}}_{\mu }^{(m)}\)) for the Zariski open subset of points whose image in \({\bar{S}}^{(1)}\) lies in \(S_{\mu }^{(1)}\) (resp. in \({\bar{S}}_{\mu }^{(1)}\)).
The generic fiber (in the sense of Raynaud) of the formal scheme
$$\begin{aligned} \underset{\rightarrow }{\lim }\,{\bar{S}}_{\mu }^{(m)} \end{aligned}$$
(5.2)
is a rigid analytic space which we shall denote by \({\bar{S}}_{\mu }^{rig}.\) We shall refer to its complement in \({\bar{S}}^{rig}\) (the rigid analytic space associated with \({\bar{S}}\)) as the supersingular tube. Its \(\mathbb {C}_{p}\)-points are the points of \({\bar{S}}(\mathbb {C}_{p})\) whose reduction modulo p lies in \(S_\mathrm{ss}(\mathbb {{\bar{F}}}_{p}).\)

5.1.2 p-Adic modular forms of integral weight k

The vector bundles \(\mathcal {P}\) and \(\mathcal {L}\) induce vector bundles on \({\bar{S}}^{(m)}\) and \({\bar{S}}_{\mu }^{rig}\) which we shall denote by the same symbols (the latter in the rigid analytic category). Let \(k\in \mathbb {Z}\) (k may be negative). Let R be a topological \({\mathcal {K}}_{p}\)-algebra. We define a p -adic modular form of weight k and tame level N over R to be an element f of
$$\begin{aligned} M_{k}^{p}(N;R)\,{:}{=}\,H^{0}\left( {\bar{S}}_{\mu }^{rig}\widehat{\otimes }_{{\mathcal {K}}_{p}}R,\mathcal {L}^{k}\right) . \end{aligned}$$
(5.3)
Note that \(M_{k}^{p}(N;R)=R\widehat{\otimes }_{{\mathcal {K}}_{p}}M_{k}^{p}(N;{\mathcal {K}}_{p}).\) A p-adic modular form f is said to be overconvergent if there exist finitely many \({\mathcal {K}}_{p}\)-affinoids \(X_{i}\) contained in the supersingular tube and a section of \(\mathcal {L}^{k}\) over \(({\bar{S}}^{rig}\backslash \bigcup X_{i})\widehat{\otimes }_{{\mathcal {K}}_{p}}R\) which restricts to f. We denote the subspace of overconvergent modular forms by \(M_{k}^{oc}(N;R).\)

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 \({\bar{S}}^{rig}{\widehat{\otimes }}_{{\mathcal {K}}_{p}}R.\)

The space \(M_{k}^{p}(N;{\mathcal {K}}_{p})\) is a p-adic Banach space whose unit ball is given by
$$\begin{aligned} M_{k}^{p}(N;{\mathcal {O}}_{p})=\underset{\leftarrow }{\,\lim \,}H^{0}\left( {\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k}\right) . \end{aligned}$$
(5.4)

5.1.3 q-Expansion 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 q-expansion 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 fiber-by-fiber \(\mu \)-ordinary, hence \(A[p^{n}]^{\mu }\), the largest R-subgroup 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 \(p^{n}\)

Fix \(m\ge 1\) and \(n\ge 1\) and consider the moduli problem associating with an \(R_{m}\)-algebra R \(\mu \)-ordinary tuples \(\underline{A}\in S_{\mu }^{(m)}(R) \) together with an isomorphism of finite flat group schemes over R
$$\begin{aligned} \varepsilon =\varepsilon _{n}^{(m)}:\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\simeq A[p^{n}]^{\mu }. \end{aligned}$$
(5.5)
This moduli problem is representable by a scheme \(Ig(p^{n})_{\mu }^{(m)}\), and the map “forget \(\varepsilon ''\) is a finite étale cover
$$\begin{aligned} \tau =\tau _{n}^{(m)}:Ig(p^{n})_{\mu }^{(m)}\rightarrow S_{\mu }^{(m)} \end{aligned}$$
(5.6)
of degree \((p^{2}-1)p^{2(n-1)}.\) It extends to a finite étale cover \(\overline{Ig}(p^{n})_{\mu }^{(m)}\) of \({\bar{S}}_{\mu }^{(m)}.\) The group
$$\begin{aligned} \varDelta (p^{n})=\left( {\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}}\right) ^{\times }=Aut_{{\mathcal {O}}_{{\mathcal {K}}}}\left( \delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{n}}\right) \end{aligned}$$
(5.7)
acts on the covering \(\tau \) as a group of deck transformations via
$$\begin{aligned} \gamma (\underline{A},\varepsilon )=(\underline{A},\varepsilon \circ \gamma ^{-1}), \end{aligned}$$
(5.8)
and the pre-image of the cuspidal divisor C is non-canonically isomorphic to \(\varDelta (p^{n})\times C.\) These constructions satisfy the usual compatibilities in m and n.

5.2.3 The trivialization of \(\mathcal {L}\) when \(m\le n\)

Assume now that \(m\le n.\) In this case, multiplication by \(p^{n}\) is 0 on R, so the inclusion of \(A[p^{n}]\) in A induces an isomorphism between the cotangent spaces at the origin \(\omega _{A[p^{n}]/R}\) and \(\omega _{A/R}.\) To see it note that if \(\mathcal {G}\) is either \(A[p^{n}]\) or A, its Lie algebra, by definition, is the finite flat R-module
$$\begin{aligned} Lie(\mathcal {G})=\ker \left( \mathcal {G}(R[\epsilon ])\rightarrow \mathcal {G}(R)\right) . \end{aligned}$$
(5.9)
Here \(R[\epsilon ]\) is the ring of dual numbers over R. It follows that
$$\begin{aligned} Lie(A[p^{n}])=Lie(A)[p^{n}]=Lie(A), \end{aligned}$$
(5.10)
and dualizing we get \(\omega _{A/R}=\omega _{A[p^{n}]/R}.\)

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.

The group \(\varDelta (p^{n})\) acts on a via the character
$$\begin{aligned} {\bar{\varSigma }}^{-1}:\varDelta (p^{n})=({\mathcal {O}}_{{\mathcal {K}}}/p^{n}{\mathcal {O}}_{{\mathcal {K}}})^{\times }\rightarrow ({\mathcal {O}}_{{\mathcal {K}}}/p^{m}{\mathcal {O}}_{{\mathcal {K}}})^{\times }=R_{m}^{\times }. \end{aligned}$$
(5.11)
From now on we take \(n=m\) and use a to trivialize \(\mathcal {L}\) along \({\tilde{C}}=\tau ^{-1}(C)\), the cuspidal divisor in \(\overline{Ig}(p^{m})_{\mu }^{(m)}.\) If \(f\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k})\), then \(\tau ^{*}f/a^{k}\) is a function on \(\overline{Ig}(p^{m})_{\mu }^{(m)}\) and we may attach to it a canonical FJ expansion
$$\begin{aligned} {\widetilde{FJ}}(f)=\sum _{j=0}^{\infty }c_{j}(f) \end{aligned}$$
(5.12)
where \(c_{j}(f)\in H^{0}({\tilde{C}},\mathcal {N}^{j})\) as before. This FJ expansion does not depend on any choice (but is defined along \({\tilde{C}}\) and not along C).

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 \(k_{1}\equiv k_{2}\mod (p^{2}-1)p^{m-1}.\)

Proof

Let \({\tilde{T}}\) be an irreducible component of \(\overline{Ig}(p^{m})_{\mu }^{(m)}.\) Then, \(\tau \) being finite étale, \(\tau ({\tilde{T}})\) is both open and closed in \({\bar{S}}_{\mu }^{(m)}\), so must be an irreducible component T of \({\bar{S}}_{\mu }^{(m)}\). It follows that \(\tau ({\tilde{T}})\) meets C, hence \({\tilde{T}}\) meets \({\tilde{C}}\), and the q-expansion principle holds in \(\overline{Ig}(p^{m})_{\mu }^{(m)}.\) We therefore have an equality
$$\begin{aligned} \tau ^{*}f_{1}/a^{k_{1}}=\tau ^{*}f_{2}/a^{k_{2}} \end{aligned}$$
(5.13)
between functions on \(\overline{Ig}(p^{m})_{\mu }^{(m)}.\) Since the left-hand side is not divisible by p by assumption, so is the right-hand side. The group \(\varDelta (p^{m})\) acts on the left-hand side via \({\bar{\varSigma }}^{k_{1}}\) and on the right-hand side via \({\bar{\varSigma }}^{k_{2}}.\) But these two characters are equal if and only if \(k_{1}\equiv k_{2}\mod (p^{2}-1)p^{m-1}\), because the exponent of the group \(\varDelta (p^{m})\) is \((p^{2}-1)p^{m-1}.\)

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 \(FJ(f_{1})=FJ(f_{2})\), \(k_{1}\equiv k_{2}\mod (p^{2}-1)p^{m-1}.\)

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 \({\tilde{C}}\simeq \varDelta (p^{m})\times C\) (non-canonically). 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

Let \(f_{i}\in M_{k_{i}}^{p}(N;{\mathcal {O}}_{p})\) (\(i=1,2\)) and assume that \(f_{1}\) is not divisible by p. Trivialize \(\mathcal {L}|_{C}\) by fixing an \({\mathcal {O}}_{{\mathcal {K}}}\)-isomorphism of the p-divisible group of the toric part of the universal semi-abelian variety \({\mathcal {A}}|_{C}\) with \(\delta _{{\mathcal {K}}}^{-1}{\mathcal {O}}_{{\mathcal {K}}}\otimes \mu _{p^{\infty }}\), and using this isomorphism to identify \(\mathcal {L}|_{C}=\omega _{{\mathcal {A}}/C}({\bar{\varSigma }})\) with \({\mathcal {O}}_{C}.\) Suppose that with this trivialization
$$\begin{aligned} FJ(f_{1})\equiv FJ(f_{2})\mod p^{m}. \end{aligned}$$
(5.14)
Then \(k_{1}\equiv k_{2}\mod \left( p^{2}-1\right) p^{m-1}.\)

5.2.5 Irreducibility of the Igusa tower and the big Igusa tower

It is possible to define an even larger Igusa tower \((BigIg(p^{n}))_{n\ge 1}\) over \(\kappa =\mathbb {{\bar{F}}}_{p}\), of which \((Ig(p^{n}))_{n\ge 1}\) is a quotient. If R is a \(\kappa \)-algebra and \(\underline{A}\in S_{\mu }(R)\), then \(A[p^{n}]\) admits a filtration as in 2.1.2. One can define \(BigIg(p^{n})\) as the moduli space of \(\mu \)-ordinary tuples \(\underline{A}\), equipped with \({\mathcal {O}}_{{\mathcal {K}}}\)-isomorphisms
(5.15)
This would be, in the language of [17], the GU-Igusa tower. If we insist that the isomorphisms respect the pairings induced on these group schemes by the polarization and Cartier duality (\(gr^{0}\) and \(gr^{2}\) are dual to each other, \(gr^{1}\) is self-dual), we would get the U-Igusa tower. Both these towers are reducible, by the reasoning of [16, 8.4.1] or [17], and by the description of the connected components of the characteristic 0 fiber of the Shimura variety given in 1.3.3. The SU-Igusa tower, which is irreducible, turns out to be our tower \((Ig(p^{n})).\) It is also the quotient of \((BigIg(p^{n}))\) under the map “forget \(\varepsilon ^{0}\) and \(\varepsilon ^{1}\)”. Thus there is no real advantage in studying the tower BigIg.

5.3 p-adic modular forms of p-adic weights

5.3.1 The space of p-adic weights

Let
$$\begin{aligned} \mathfrak {X}_{p}=\underset{\leftarrow }{\,\lim }\,\mathbb {Z}/(p^{2}-1)p^{m-1}\mathbb {Z}, \end{aligned}$$
(5.16)
This is the space of p-adic weights. If \(k\in \mathfrak {X}_{p}\), then \({\bar{\varSigma }}^{k}\) is a well-defined locally \(\mathbb {Q}_{p}\)-analytic homomorphism of \({\mathcal {O}}_{p}^{\times }\) to itself, but note that not every such homomorphism is a \({\bar{\varSigma }}^{k}\) for some k from \(\mathfrak {X}_{p}.\)

5.3.2 p-adic modular forms à la Serre

We work with \({\bar{S}}\) (hence also the cuspidal divisor C) over the base \({\mathcal {O}}_{p}\), the p-adic 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 p-divisible group of the toric part of the universal semi-abelian 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.

Let \(k\in \mathfrak {X}_{p}.\) The space \(M_{k}^\mathrm{Serre}(N;{\mathcal {K}}_{p})\) will be a subspace of the Banach algebra
$$\begin{aligned} \mathcal {FJ}_{p}={\mathcal {K}}_{p}\otimes _{{\mathcal {O}}_{p}}\prod _{j=0}^{\infty }H^{0}(C,\mathcal {N}^{j}). \end{aligned}$$
(5.17)
It will consist of all the \(f\in \mathcal {FJ}_{p}\) for which there exists a sequence \((f_{\nu })\), \(f_{\nu }\in M_{k_{\nu }}^{p}(N;{\mathcal {K}}_{p})\), \((k_{\nu }\in \mathbb {Z})\), with \(FJ(f_{\nu })\) converging to f, and \(k_{\nu }\) converging in \(\mathfrak {X}_{p}\) to k. As we have seen, if the sequence \((FJ(f_{\nu }))\) converges, the \(k_{\nu }\) have to converge in \(\mathfrak {X}_{p}.\) We shall denote by \(M_{k}^\mathrm{Serre}(N;{\mathcal {O}}_{p})\) the intersection of \(M_{k}^\mathrm{Serre}(N;{\mathcal {K}}_{p})\) with \(\prod _{j=0}^{\infty }H^{0}(C,\mathcal {N}^{j}).\)

Proposition 5.7

  1. (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 p-adic modular forms by allowing limits of p-adic modular forms of varying weights, if the weights converge to an integral k.

     
  2. (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.

     
  3. (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 \(M_{k_{1}+k_{2}}^\mathrm{Serre}\).

     
  4. (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 \(H_{{\bar{\varSigma }}}^{-1}\in M_{1-p^{2}}^{p}(N;{\mathcal {O}}_{p})\) is a p-adic 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 \(f_{\nu }H_{{\bar{\varSigma }}}^{(k-k_{\nu })/(p^{2}-1)}\) 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 p-adic modular form of integral weight is the p-adic limit of classical forms of varying weights, and the same is therefore true for Serre modular forms of p-adic weight. Points (iii) and (iv) are obvious.

5.3.3 p-Adic modular forms à la Katz

We now explain Katz’ point of view of the same objects. Let
$$\begin{aligned} V_{n}^{(m)}=H^{0}\left( \overline{Ig}(p^{n})_{\mu }^{(m)},{\mathcal {O}}\right) \end{aligned}$$
(5.18)
be the ring of regular functions on \(\overline{Ig}(p^{n})_{\mu }^{(m)}.\) Let
$$\begin{aligned} V^{(m)}=\text { }\underset{\rightarrow }{\lim }\,V_{n}^{(m)},\,\,\,\,\,\,\,V=\,\underset{\leftarrow }{\lim }\,V^{(m)}. \end{aligned}$$
(5.19)
We call V the space of Katz p -adic modular forms (of all weights). Let
$$\begin{aligned} \gamma \in \varDelta ={\mathcal {O}}_{p}^{\times }=\,\underset{\leftarrow }{\lim }\,\varDelta (p^{n}) \end{aligned}$$
(5.20)
act on \(V^{(m)}\) and on V as usual, \(\gamma (f)=f\circ \gamma ^{-1}\), and recall that \(\gamma ^{-1}(\underline{A},\varepsilon _{n}^{(m)})=(\underline{A},\varepsilon _{n}^{(m)}\circ \gamma ).\) Thus
$$\begin{aligned} \gamma (f)(\underline{A},\varepsilon )=f(\underline{A},\varepsilon \circ \gamma ) \end{aligned}$$
(5.21)
(i.e. \(\gamma \) acts by “right translation”). Let \(k\in \mathfrak {X}_{p}\) and define
$$\begin{aligned} M_{k}^\mathrm{Katz}(N;{\mathcal {O}}_{p})=V({\bar{\varSigma }}^{k})=\left\{ f\in V|\,\gamma (f)={\bar{\varSigma }}^{k}(\gamma )\cdot f\,\,\,\,\,\forall \gamma \in \varDelta \right\} . \end{aligned}$$
(5.22)
We similarly define \(M_{k}^\mathrm{Katz}(N;R_{m})=V^{(m)}({\bar{\varSigma }}^{k}).\)
By the irreducibility of the Igusa tower and the q-expansion principle the FJ expansion map
$$\begin{aligned} V\rightarrow \mathcal {FJ}_{p}({\mathcal {O}}_{p}) \end{aligned}$$
(5.23)
is injective. It depends on our choice of the splitting of \({\tilde{C}}\rightarrow C\).

Proposition 5.8

For \(k\in \mathfrak {X}_{p}\), there is a natural isomorphism
$$\begin{aligned} M_{k}^\mathrm{Serre}(N;{\mathcal {O}}_{p})\simeq M_{k}^\mathrm{Katz}(N;{\mathcal {O}}_{p}). \end{aligned}$$
(5.24)

Proof

Given \(k\in \mathbb {Z}\) and \(f\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k})\), the functions \((\tau _{n}^{(m)})^{*}f/(a_{n}^{(m)})^{k}\in V_{n}^{(m)}\) for all \(n\ge m\), and these functions satisfy the obvious compatibility in n, so they define
$$\begin{aligned} f^\mathrm{Katz}\in V^{(m)}({\bar{\varSigma }}^{k}). \end{aligned}$$
(5.25)
If \(k\in \mathbb {Z}\), this gives, by going to the inverse limit over m, a map
$$\begin{aligned} f\mapsto f^\mathrm{Katz},\,\,\,\,\,\,\,M_{k}^{p}(N;{\mathcal {O}}_{p})\rightarrow M_{k}^\mathrm{Katz}(N;{\mathcal {O}}_{p}). \end{aligned}$$
(5.26)
This map is an isomorphism, which can be enhanced to include p-adic weights \(k\in \mathfrak {X}_{p}\) as follows. If \(k_{\nu }\in \mathbb {Z}\), \(k_{\nu }\rightarrow k\in \mathfrak {X}_{p}\) and if \(f_{\nu }\in M_{k_{\nu }}^{p}(N;{\mathcal {O}}_{p})\) are such that \(FJ(f_{\nu })\) converge to \(f\in M_{k}^\mathrm{Serre}(N;{\mathcal {O}}_{p})\), then reducing modulo \(p^{m}\) for a fixed m, \((f_{\nu }^{(m)})^\mathrm{Katz}\in V^{(m)}({\bar{\varSigma }}^{k_{\nu }}).\) But for a fixed m, for all large enough \(\nu \),
$$\begin{aligned} V^{(m)}({\bar{\varSigma }}^{k_{\nu }})=V^{(m)}({\bar{\varSigma }}^{k}), \end{aligned}$$
(5.27)
and the sequence \(FJ(f_{\nu }^{(m)})\) stabilizes, so taking the limit over \(\nu \) we get a well defined \((f^{(m)})^\mathrm{Katz}\in V^{(m)}({\bar{\varSigma }}^{k}).\) Finally, an inverse limit over m gives \(f^\mathrm{Katz}\in M_{k}^\mathrm{Katz}(N;{\mathcal {O}}_{p}).\) It is by now standard that this gives an isomorphism between \(M_{k}^\mathrm{Serre}(N;{\mathcal {O}}_{p})\) and \(M_{k}^\mathrm{Katz}(N;{\mathcal {O}}_{p}).\) As we have seen earlier, when \(k\in \mathbb {Z}\), this is also the same as \(M_{k}^{p}(N;{\mathcal {O}}_{p}).\)

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 p-Adic modular forms of p-adic bi-weights

5.4.1 The space of bi-weights

A new feature of p-adic 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 scalar-valued p-adic modular forms, we sometimes need to consider classical vector-valued 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 p-adic theory.

The space \(\mathfrak {X}_{p}\) of p-adic weights can be written as \(\mathbb {Z}/(p^{2}-1)\mathbb {Z\times Z}_{p}\), and when we decompose it in such a way we write
$$\begin{aligned} k=(w,j)=\left( \omega (k),\left\langle k\right\rangle \right) \end{aligned}$$
(5.28)
for the two components. The space of bi-weights \(\mathfrak {X}_{p}^{(2)}\) is, by definition, the quotient of \(\mathfrak {X}_{p}^{2}\) modulo the relation
$$\begin{aligned} ((w_{1},j_{1}),(w_{2},j_{2}))\equiv ((0,j_{1}),(pw_{1}+w_{2},j_{2}))\equiv ((pw_{2}+w_{1},j_{1}),(0,j_{2})). \end{aligned}$$
(5.29)
If \(k_{1}\) and \(k_{2}\) are in \(\mathfrak {X}_{p}\), then the character \({\bar{\varSigma }}^{k_{1}}\varSigma ^{k_{2}}:\varDelta \rightarrow {\mathcal {O}}_{p}^{\times }\) depends only on the image of \((k_{1},k_{2})\) in \(\mathfrak {X}_{p}^{(2)}.\) Here \(\varDelta =\,\underset{\leftarrow }{\lim }\,\varDelta (p^{n})\) is also \({\mathcal {O}}_{p}^{\times }\), but in the rôle of the Galois group of the Igusa tower. The image of \(\mathbb {Z}^{2}\) is dense in \(\mathfrak {X}_{p}^{2}\), hence also in \(\mathfrak {X}_{p}^{(2)}.\)

5.4.2 The line bundle \(\mathcal {L}^{(k_{1},k_{2})}\) over \({\bar{S}}_{\mu }^{rig}\) and p-adic modular forms of integral bi-weights

Let \(m\ge 1.\) The plane bundle \(\mathcal {P}\) admits a canonical filtration
$$\begin{aligned} 0\rightarrow \mathcal {P}_{0}\rightarrow \mathcal {P}\rightarrow \mathcal {P}_{\mu }\rightarrow 0 \end{aligned}$$
(5.30)
over \({\bar{S}}_{\mu }^{(m)}\) defined by choosing any \(n\ge m\) and setting
$$\begin{aligned} \mathcal {P}_{0}=\ker (\omega _{{\mathcal {A}}[p^{n}]^{0}}\rightarrow \omega _{{\mathcal {A}}[p^{n}]^{\mu }}),\,\,\mathcal {P}_{\mu }=\omega _{{\mathcal {A}}[p^{n}]^{\mu }}(\varSigma ) \end{aligned}$$
(recall \(\omega _{{\mathcal {A}}}=\omega _{{\mathcal {A}}[p^{n}]^{0}}\)). We also recall that \(\mathcal {L}=\omega _{{\mathcal {A}}[p^{n}]^{\mu }}({\bar{\varSigma }}).\)
If \(m=1\) we showed that over \({\bar{S}}_{\mu }^{(1)}\), \(\mathcal {L}\simeq \mathcal {P}_{\mu }^{p}\) and \(\mathcal {P}_{\mu }\simeq \mathcal {L}^{p}.\) This is no longer true for general m and we let for \((k_{1},k_{2})\in \mathbb {Z}^{2}\)
$$\begin{aligned} \mathcal {L}^{(k_{1},k_{2})}=\mathcal {L}^{k_{1}}\otimes \mathcal {P}_{\mu }^{k_{2}}. \end{aligned}$$
(5.31)
Going to the limit over m, this defines a rigid analytic line bundle over \({\bar{S}}_{\mu }^{rig}.\)
We define the space of p -adic modular forms of bi-weight \((k_{1},k_{2})\) and level N over \({\mathcal {O}}_{p}\) as
$$\begin{aligned} M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p})=\,\underset{\leftarrow }{\lim }\,H^{0}\left( {\bar{S}}_{\mu }^{(m)},\,\mathcal {L}^{(k_{1},k_{2})}\right) . \end{aligned}$$
(5.32)
This is the unit ball of the p-adic Banach space
$$\begin{aligned} M_{k_{1},k_{2}}^{p}(N;{\mathcal {K}}_{p})={\mathcal {K}}_{p}\otimes _{{\mathcal {O}}_{p}}M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p})=H^{0}\left( {\bar{S}}_{\mu }^{rig},\mathcal {L}^{(k_{1},k_{2})}\right) . \end{aligned}$$
(5.33)

5.4.3 The trivialization of \(\mathcal {L}^{(k_{1},k_{2})}\) over the Igusa tower

As before, fix m, let \(m\le n\) and consider the isomorphism
$$\begin{aligned} \varepsilon ^{*}:\tau ^{*}\omega _{{\mathcal {A}}[p^{n}]^{\mu }}\simeq {\mathcal {O}}_{{\mathcal {K}}}\otimes {\mathcal {O}}_{\overline{Ig}(p^{n})_{\mu }^{(m)}} \end{aligned}$$
(5.34)
induced by the Igusa level structure \(\varepsilon =\varepsilon _{n}^{(m)}\). Taking \({\bar{\varSigma }}\) and \(\varSigma \)-types, it induces trivializations
$$\begin{aligned} \tau ^{*}\mathcal {L}\simeq {\mathcal {O}}_{\overline{Ig}(p^{n})_{\mu }^{(m)}},\quad \tau ^{*}\mathcal {P}_{\mu }\,\simeq {\mathcal {O}}_{\overline{Ig}(p^{n})_{\mu }^{(m)}} \end{aligned}$$
(5.35)
and we let \(a=a_{n}^{(m)}\) and \({\bar{a}}={\bar{a}}_{n}^{(m)}\) be the sections corresponding to 1. Of course, the trivialization of \(\mathcal {L}\) is the one that we have met before.

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 p-Adic modular forms of p-adic bi-weights

The yoga of p-adic weights, either à la Serre or à la Katz, allows us now to define the space
$$\begin{aligned} M_{k_{1},k_{2}}^{p}(N;{\mathcal {K}}_{p}) \end{aligned}$$
(5.36)
of p-adic modular forms of any bi-weight \((k_{1},k_{2})\in \mathfrak {X}_{p}^{(2)}.\) If we follow Serre, we define them as elements of the Banach space \(\mathcal {FJ}_{p}\) via limits of p-adic modular forms of integral bi-weights. If we follow Katz, we have
$$\begin{aligned} M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p})=V\left( {\bar{\varSigma }}^{k_{1}}\varSigma ^{k_{2}}\right) . \end{aligned}$$
(5.37)
We let the reader complete the details, which are identical to the case of a single weight treated before.

5.5 The theta operator for p-adic modular forms

We are finally able to define the operator \(\varTheta \) on p-adic modular forms. Compare [22, V.5.8]. Let \(f\in M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p}).\) Assume first that \(k_{1}\) and \(k_{2}\) are from \(\mathbb {Z}\), and reduce modulo \(p^{m}\), to get \(f\in H^{0}({\bar{S}}_{\mu }^{(m)},\mathcal {L}^{k_{1}}\otimes \mathcal {P}_{\mu }^{k_{2}}).\) Take any \(n\ge m\), pull back to \(\overline{Ig}(p^{n})_{\mu }^{(m)}\), divide by \(a^{k_{1},k_{2}}\) and consider
$$\begin{aligned} \eta _{f}=\text {d}\left( \tau ^{*}f/a^{k_{1},k_{2}}\right) \in H^{0}\left( \overline{Ig}(p^{n})_{\mu }^{(m)},\varOmega _{Ig}^{1}\right) . \end{aligned}$$
(5.38)
Apply \(\text {KS}^{-1}\) to \(\eta _{f}.\) This results in a section of \(\mathcal {L}\otimes \mathcal {P}.\) As explained before, when we project this section to \(\mathcal {L}\otimes \mathcal {P}_{\mu }\), we get a section that is holomorphic along \({\tilde{C}}\) and even vanishes there (recall KS had a pole along the cuspidal divisor). Multiply back by \(a_{k_{1},k_{2}}\) and use Galois descent to descend the resulting section to \(S_{\mu }^{(m)}.\)
We may now take the limit over m to get our \(\varTheta \), if \((k_{1},k_{2})\in \mathbb {Z}^{2}.\) A further limit over weights, as in the proof of Proposition 5.8, allows us to extend the definition to \((k_{1},k_{2})\in \mathfrak {X}_{p}^{(2)}.\) Using Katz’ approach, where the process of dividing and multiplying back by \(a^{k_{1},k_{2}}\) is already built into the isomorphism with \(V({\bar{\varSigma }}^{k_{1}}\varSigma ^{k_{2}})\), \(\varTheta \) is nothing but the map
$$\begin{aligned} \varTheta :f\mapsto (1\otimes pr_{\mu })\circ \text {KS}^{-1}\circ d(f) \end{aligned}$$
(5.39)
sending \(V({\bar{\varSigma }}^{k_{1}}\varSigma ^{k_{2}})\) to \(V({\bar{\varSigma }}^{k_{1}+1}\varSigma ^{k_{2}+1}).\) Here \(pr_{\mu }:\mathcal {P}\rightarrow \mathcal {P}_{\mu }\) is the projection defined over \({\bar{S}}_{\mu }^{rig}.\)

Theorem 5.9

Let \((k_{1},k_{2})\in \mathfrak {X}_{p}^{(2)}.\) The operator,
$$\begin{aligned} \varTheta :M_{k_{1},k_{2}}^{p}(N;{\mathcal {O}}_{p})\rightarrow M_{k_{1}+1,k_{2}+1}^{p}(N;{\mathcal {O}}_{p}) \end{aligned}$$
(5.40)
defined by the above formula, satisfies the following properties (and is uniquely determined by its effect on q-expansions).
  1. (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 operator
    $$\begin{aligned} \varTheta :M_{k}(N;\kappa )\rightarrow M_{k+p+1}(N;\kappa ) \end{aligned}$$
    (5.41)
    on \(\mod p\) modular forms.
     
  2. (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.

Footnotes
1

The Steinitz class of a finite projective \({\mathcal {O}}_{{\mathcal {K}}}\)-module is the class of its top exterior power as an invertible module.

 
2

No confusion should arise from the use of the letter N to denote both the level and the unipotent radical of P.

 

Notes

Declarations

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Hebrew University
(2)
McGill University

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