A theta operator on Picard modular forms modulo an inert prime

We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.

The p-adic theory of modular curves and modular forms, as developed in the early 70's by Serre, Katz, Mazur, Manin and many others, inspired over the last thirty years an extensive study of higher dimensional Shimura varieties, with spectacular applications to almost every area in number theory and representation theory.
In the course of these vast generalizations, some of the more special phenomena pertaining to modular curves did not find appropriate analogues, and are only beginning to get addressed in recent years. In this work we study what we believe to be the simplest Shimura varieties beyond modular curves (and Shimura curves), namely Picard modular surfaces. For arithmetic questions over quadratic imaginary fields they play a role similar to the role played by modular curves over Q. Our goal is to explore the arithmetic and geometry of these surfaces, and of modular forms over them, concentrating on aspects that have been only recently discovered, or appear to be new. Our hope is that staying within the bounds of U (2, 1) will keep the notation simple, and our tool-box relatively light, yet at the same time will allow for the exposition of new principles.
Our work relies on important work of Larsen, Bellaiche, Vollaard, Bültel, and Wedhorn. The modulo-p geometry of Picard modular surfaces, and, more generally, Shimura varieties asscoiated to unitary groups, has seen important contributions recently also by Goldring and Nicole, Koskivirta, Kudla, Rapoport, Terstiege, Howard and Pappas, and maybe others of which we are anaware. For completeness, we have included more than usual background material. We hope that we have not neglected to give appropriate credit when citing others' results.
We have chosen to work with a prime p which is inert in the underlying quadratic imaginary field, as this is the more interesting case, both from the algebro-geometric point of view, where the geometry modulo p is more challenging, and from the representation theoretic point of view, where the unitary group is non-split. We do not touch upon questions of automorphic L-functions or representation theory. For these see the comprehensive volume [L-R] and the references therein. We do not touch upon Galois representations either, although eventually we hope that our results will be relevant to both these areas.
This paper concerns the algebraic geometry of Picard modular schemes and modular forms modulo p. In a subsequent work we hope to treat p-adic modular forms in the style of Serre and Katz, the canonical subgroup, overconvergence, and further topics. We shall now explain the main results and the structure of the paper.
Chapters 1-2 summarize known results, and serve as a systematic introduction on which chapters 3-4 and future work will be based. In particular, sections 1.4 and 2.4 rely on the theses of Larsen [La1] and Bellaiche [Bel]. We recommend the latter for its very readable exposition. Sections 1.2.4, 1.6, 2.1, 2.5 and 2.6 contain results or computations for which we did not find adequate references in the literature. Chapter 2 introduces the two basic automorphic vector bundles P (a plane bundle) and L (a line bundle). Modular forms are sections of vector bundles from the tensor algebra generated by these two, but we contend ourselves with scalar valued modular forms, which are sections of L k (k being the weight).
Chapter 3 contains new results. There are three strata to the Picard surface S modulo p: the µ-ordinary (open and dense) locus S µ , the general supersingular (one dimensional) locus S gss , and the superspecial points S ssp (see [Bu-We], [V]). After recalling this, we find various relations between P and L that are peculiar to characteristic p. We study the Verschiebung homomorphism V L from L to P (p) and V P from P to L (p) . We find that outside the superspecial points, both maps have rank one, but that Im(V L ) = ker (V (p) P ) is the defining equation of the general supersingular stratum S gss . We prove that the line bundle P 0 = ker(V P ), defined over S µ ∪S gss , does not extend across the superspecial points. Using these relations we define a Hasse invariant hΣ, a section of L p 2 −1 , whose zero divisor is precisely the supersingular locus S ss = S gss ∪ S ssp . Although the same definition has been already given by Goldring and Nicole in [Go-Ni] for all unitary Shimura varieties, and recently generailzed to all Shimura varieties of Hodge type by Koskivirta and Wedhorn [Ko-We], in the special case of Picard surfaces our analysis goes deeper.
Following the definition of hΣ we define on S ss a secondary Hasse invariant h ssp , a section of L p 3 +1 | Sss , whose zeroes are the superspecial points S ssp (albeit with high multiplicity). This secondary Hasse invariant is closely related to recent work of Boxer [Bo], and points at a general phenomenon that deserves further exploration. It seems reasonable that Boxer's method will generalize to Shimura varieties of PEL type and will produce invariants which agree with the partial Hasse invariants appearing in [Gor]. As an application of our analysis of the secondary Hasse invariant we derive a striking formula, expressing the number of irreducible components of the supersingular stratum in terms of the Euler number ofS C . By well-known results, due in this case to Holzapfel, this number is given essentially by an L-value.
As for modular curves, one can introduce an Igusa scheme Ig(p n ) of any level p n . We focus on Ig = Ig(p), but note in passing that the higher Ig(p n ) will become intsrumental in the theory of p-adic modular forms, deferred to future work. Although initially defined only over the µ-ordinary stratum, Ig can be compactified over the whole of S (and over the cuspidal divisors inS as well) by "extracting a p 2 − 1 root of hΣ". The line bundle L is trivialized over the µ-ordinary locus Ig µ , and this allows us to regard modular forms as functions on Ig, with prescribed poles along Ig ss , the supersingular locus. This is the geometric basis for developing an analogue of Serre's theory of "modular forms modulo p" in our context. In particular, we study their Fourier-Jacobi expansions (complex FJ expansions are q-expansions with theta functions as coefficients, but here we employ the arithmetic analysis of Bellaiche, explained in Section 2.4). This leads to the notion of a "filtration" of a modular form mod p, as in the case of elliptic modular forms.
Chapter 4 seems to be entirely new. The theory becomes richer once we bring in the Gauss-Manin connection and the Kodaira-Spencer isomorphism. We define a theta operator on modular forms mod p (in weight k) as follows. We first divide our modular form f by the k-th power of the canonical section a(1) of L on Ig, to get a function g = r(f ) on Ig, with a pole of order k along Ig ss . We apply the (inverse of the) Kodaira-Spencer isomorphism to dg to get a section of P ⊗ L, which we map via V P ⊗ 1 to get a meromorphic section of L p+1 . Multiplying by a(1) k allows us to descend back to S, so we obtain a meromorphic modular form Θ(f ) of weight k + p + 1. This construction is motivated by [An-Go], although substantial new phenomena appear in the present case. We study this operator. On the one hand, near the cusps we relate the retraction of a formal neighborhood of the cuspidal divisor, which was introduced by Bellaiche in [Bel], to the complex computations of section 1.6 and 2.6. This allows us to show that the effect of Θ on FJ expansions, is, as in the classical case, a "Tate twist". On the other hand, we study Θ(f ) along the supersingular locus, and show that, thanks to the fact that we have divided out by P 0 = ker V P , Θ(f ) is in fact everywhere holomorphic! We derive some interesting consequences for "theta cycles", where there are similarities, but also surprising deviations from the classical theory. We end the paper with a comparison of our theta operator with the Serre-Katz theta operator on modular curves, embedded in S. Theta operators on classical modular forms have been instrumental in studying congruences between them, with applications to Galois representations. We expect the same to be true for Picard modular forms.
1. Background 1.1. The unitary group and its symmetric space.
1.1.1. Notation. Let K be an imaginary quadratic field, contained in C. We denote by Σ : K ֒→ C the inclusion and byΣ : K ֒→ C its complex conjugate. We use the following notation: • d K -the square free integer such that K = Q( √ d K ). • D K -the discriminant of K, equal to d K if d K ≡ 1 mod 4 and 4d K if d K ≡ 2, 3 mod 4. • δ K = √ D K -the square root with positive imaginary part, a generator of the different of K, sometimes simply denoted δ.
•ā -the complex conjugate of a ∈ K.
We fix an integer N ≥ 3 (the "tame level") and let R 0 = O K [1/2d K N ]. This is our base ring. If R is any R 0 -algebra and M is any R-module with O K -action, then M becomes an O K ⊗ R-module and we have a canonical type decomposition Then M (Σ) (resp. M (Σ)) is the part of M on which O K acts via Σ (resp.Σ). The same notation will be used for sheaves of modules on R-schemes, endowed with an O K action. If M is locally free, we say that it has type (p, q) if M (Σ) is of rank p and M (Σ) is of rank q. We denote by (1.3) T = res K Q G m the non-split torus whose Q-points are K × , and by ρ the non-trivial automorphism of T, which on Q-points induces ρ(a) =ā. The group G m embeds in T and the homomorphism a → a · ρ(a) from T to itself factors through a homomorphism (1.4) N : T → G m , the norm homomorphism. Its kernel ker(N ) is denoted T 1 .
1.1.2. The unitary group. Let V = K 3 and endow it with the hermitian pairing

p-ADIC PROPERTIES OF PICARD MODULAR SCHEMES AND MODULAR FORMS
We identify V R with C 3 (K acting via the natural inclusion Σ). It then becomes a hermitian space of signature (2, 1). Conversely, any 3-dimensional hermitian space over 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 (1.6) G = GU (V, (, )) be the general unitary group of V, regarded as an algebraic group over Q. For any Q-algebra A we have We write G = G(Q), G ∞ = G(R) and G p = G(Q p ). A similar notational convention will apply to any algebraic group over Q without further ado. If p splits in K, Q p ⊗K ≃ Q 2 p and G p becomes isomorphic to GL 3 (Q p )×Q × p . The isomorphism depends on the embedding of K in Q p , i.e. on the choice of a prime above p in K. For a non-split prime p the group G p , like G ∞ , is of (semisimple) rank 1.
As µ is determined by g we often abuse notation and write g for the pair (g, µ) and µ(g) for the multiplier µ. It is a character of algebraic groups over Q, µ : G →G m . Another character is det : G → T, defined by det(g, µ) = det(g). If we let (1.8) ν = µ −1 det : G → T then both µ and det are expressible in terms of ν, namely µ = ν · (ρ • ν) and det = ν 2 ·(ρ•ν). The first relation is a consequence of the relation det ·(ρ•det) = µ 3 , and the second is a consequence of the first and the definition of ν. The groups (1.9) U = ker µ, SU = ker ν = ker µ ∩ ker (det) are the unitary and the special unitary group, respectively. We also introduce an alternating Q-linear pairing , : V × V → Q defined by u, v = Im δ (u, v). We then have the formulae (1.10) au, v = u,āv , 2(u, v) = u, δv + δ u, v .
We call , the polarization form, for a reason that will become clear soon.
1.1.3. The hermitian symmetric domain. The group G ∞ = G(R) acts on P 2 C = P(V R ) by projective linear transformations and preserves the open subdomain X of negative definite lines (in the metric (, )). If we switch to coordinates in which the hermitian quadratic form (u, u) assumes the standard shape xx+yȳ−zz, it becomes evident that X is biholomorphic to the open unit ball in C 2 , hence is connected, and G ∞ acts on it transitively. We shall nevertheless stick to the coordinates introduced above. 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 (1.11) λ(z, u) def = Im δ (z) − uū > 0.
One refers to the realization of X as the set of points (z, u) ∈ C 2 satisfying this inequality as a Siegel domain of the second kind. It is convenient to think of the point x 0 = (δ/2, 0) as the "center" of X.
If we let K ∞ be the stabilizer of x 0 in G ∞ , then K ∞ is compact modulo center (K ∞ ∩U(R) is compact and isomorphic to U (2)×U (1)). Since G ∞ acts transitively on X, we may identify X with G ∞ /K ∞ .
The space X carries a Riemannian metric which is invariant under the action of G ∞ , the Bergmann metric. It may be described as follows. Switching once again to coordinates on C 3 in which the hermitian quadratic form (u, u) becomes xx + yȳ − zz, the real symmetric bilinear form Re(u, v) has signature (4, 2), and the manifold given by the equation xx + yȳ − zz = −1 becomes a circle bundle over X. The restriction of Re(u, v) to the tangent bundle of this manifold has signature (4, 1) and when we take the quotient by the circle action we get an invariant Riemannian metric on X.
The usual upper half plane embeds in X (holomorphically and isometrically) as the set of points where u = 0.
1.1.4. The cusps of X. The boundary ∂X of X is the set of points (z, u) where Im δ (z) = uū, together with a unique point "at infinity" c ∞ represented by the line t (1 : 0 : 0). The lines represented by ∂X are the isotropic lines in V R . The set of cusps CX is the set of K-rational isotropic lines, or, equivalently, the set of points on ∂X with coordinates in K. If s ∈ K and r ∈ Q we write (1.12) c r s = (r + δss/2, s). Then CX = {c r s |r ∈ Q, s ∈ K} ∪ {c ∞ }. The group G = G(Q) acts transitively on the cusps.
The stabilizer of a cusp is a Borel subgroup 1 in G ∞ . Since G acts transitively on the cusps, we may assume that our cusp is c ∞ . It is then easy to check that its stabilizer P ∞ has the form P (1.14) The matrix tm(α, β) belongs to U ∞ if and only if t = 1, and to SU ∞ if furthermore β =ᾱ/α. The group N ∞ is contained in SU ∞ . The group P = P ∞ ∩ G consists of the same matrices with K-rational entries. Since N = N ∞ ∩ G still acts transitively on the set of finite cusps c r s , we conclude that G acts doubly transitively on CX. Of particular interest to us will be the geodesics connecting an interior point (z, u) to a cusp c ∈ CX. If (z, u) = n(u, r)m(d, 1)x 0 (recall x 0 = t (δ/2 : 0 : 1)) where d is real and positive (i.e. r = Rez and d = λ(z, u)) then the geodesic connecting (z, u) to c ∞ can be described by the formula The same geodesic extends in the opposite direction for 0 < t ≤ d, and if u and r lie in K, it ends there in the cusp c r u . We shall call γ r u (t) the geodesic retraction of X to the cusp c ∞ . As 0 < t < ∞ these parallel geodesics exhaust X, they converge to c ∞ as t → ∞, and they pass through (z, u) precisely when r = Rez and t = λ(z, u).
The points (z, u) and (z ′ , u ′ ) lie on the same geodesic if and only if u = u ′ and Re(z) = Re(z ′ ).
1.2.1. Lattices and their arithmetic groups.
Equivalently, L is its own O K -dual with respect to the hermitian pairing (, ). We assume also that the Steinitz class 2 of L as an O K -module is [O K ], or, what amounts to the same, that L is a free O K -module. When we introduce the Shimura variety later on, we shall relax this last assumption, but the resulting scheme will be disconnected (over C).
Fix an integer N ≥ 1 and let This Γ is a discrete subgroup of G ∞ , contained in U ∞ . It is easy to see that if N ≥ 3 then Γ is torsion free, acts freely and faithfully on X, and is contained in SU ∞ . From now on we assume that this is the case. If g ∈ G and µ(g) = 1 (i.e. g ∈ U ) the lattice gL is another lattice of the same sort and the discrete group corresponding to it is gΓg −1 . Since U acts transitively on the cusps, this reduces the study of Γ\X near a cusp to the study of a neighborhood of the standard cusp c ∞ (at the price of changing L and Γ).
It is important to know the classification of lattices L as above (self-dual and O K -free). Let e 1 , e 2 , e 3 be the standard basis of K 3 . Let These two lattices are self-dual and of course, O K -free. The following theorem is based on the local-global principle and a classification of lattices over Q p by Shimura [Sh1]. La1], p.25). For any lattice L as above there exists a g ∈ U such that gL = L 0 or gL = L 1 . If D K is odd, L 0 and L 1 are equivalent. If D K is even, they are inequivalent.
Indeed, if D K is even, L 0 ⊗ Q p and L 1 ⊗ Q p are U p -equivalent for every p = 2, but not for p = 2.
1.2.2. Picard modular surfaces and the Baily-Borel compactification. We denote by X Γ the complex surface Γ\X. Since the action of Γ is free, X Γ is smooth. We describe a topological compactification of X Γ . A standard neighborhood of the cusp c ∞ is an open set of the form (1.20) The set C Γ = Γ\CX is finite, and we write c Γ = Γc. We let X * Γ be the disjoint union of X Γ and C Γ . We topologize it by taking Γ\Ω R ∪ {c ∞,Γ } as a basis of neighborhoods 2 The Steinitz class of a finite projective O K -module is the class of its top exterior power as an invertible module. at c ∞,Γ . If c = g(c ∞ ) where g ∈ U, we take g(g −1 Γg\Ω R ) ∪ {c Γ } instead. The following theorem is well-known.
Theorem 1.2. (Satake, Baily-Borel) X * Γ is projective and the singularities at the cusps are normal. In other words, there exists a normal complex projective surface S * Γ and a homeomorphism ι : is an isomorphism of complex manifolds. S * Γ is uniquely determined up to isomorphism. 1.2.3. The universal abelian variety over X Γ . With x ∈ X and with our choice of L we shall now associate a PEL-structure x with its dual abelian variety induced by an ample line bundle), is an embedding of CM type (2, 1) (i.e. the action of ι(a) on the tangent space of A x at the origin induces the representation 2Σ +Σ) such that the Rosati involution induced by λ x preserves ι(O K ) and is given by ι(a) → ι(ā), (4) α x : N −1 L/L ≃ A x [N ] is a full level N structure, compatible with the O K -action and the polarization. The latter condition means that if we denote by , λ the Weil "e N -pairing" on A Let W x be the negative definite complex line in V R = C 3 defined by x, and W ⊥ x its orthogonal complement, a positive definite plane. Let J x be the complex structure which is multiplication by i on W ⊥ x and by −i on W x . Let A x = (V R , J x )/L. Then the polarization form , is a Riemann form on L. One only has to verify that u, J x v + i u, v is a positive definite hermitian form. But up to the factor |δ|/2 this is the same as (u, v) on W ⊥ x and −(u, v) on W x . The Riemann form determines a principal polarization on A x as usual. The action of O K is derived from the underlying K structure of V. As we have changed the complex structure on W x , the CM type is now (2,1). Finally the level N structure α x is the identity map.
If γ ∈ Γ then γ induces an isomorphism between A x and A γ(x) . Conversely, if A x and A x ′ are isomorphic structures, it is easy to see that x ′ and x must belong to the same Γ-orbit. It follows that points of X Γ are in a bijection with PEL structures of the above type for which the triple is isomorphic to (L, ι, , ) (here ι refers to the O K action on L), with the further condition that α x is compatible with the isomorphism between L and H 1 (A x , Z) in the sense that we have a commutative diagram 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 X. In other words, we want to construct a 5-dimensional complex manifold A, together with a holomorphic map A → 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 C 3 is fixed, but the lattice varies. For simplicity we assume from now on that L = L 0 is spanned over O K by δe 1 , e 2 and e 3 . The case of L 1 can be handled similarly.
Let C 3 be given the usual complex structure, and let a ∈ O K act on it via the matrix is a complex linear isomorphism between C 3 and (V R , J x ). In fact, it sends Ce 1 +Ce 2 linearly to W ⊥ x and Ce 3 conjugate-linearly to W x . It intertwines the ι ′ action of O K on C 3 with its ι-action on (V R , J x ). It furthermore sends L ′ x to L. In fact, an easy computation shows that it sends the three generating vectors of L ′ x to δe 1 , e 2 and e 3 , respectively. We conclude that T x induces an isomorphism Consider the differential forms dζ 1 , dζ 2 and dζ 3 . As their periods along any l ∈ L ′ x vary holomorphically in z and u, the five coordinates ζ 1 , ζ 2 , ζ 3 , z, u form a local system of coordinates on the family A ′ → X. Identifying A ′ with A allows us to put the desired complex structure on the family A. Alternatively, we may define A ′ as the quotient of C 3 × X by ζ → ζ + l(z, u) where l(z, u) varies over the holomorphic lattice-sections.
The model A ′ has another advantage, that will become clear when we examine the degeneration of the universal abelian variety at the cusp c ∞ . It suffices to note at this point that the first two of the three generating vectors of L ′ x depend only on u.
(4) α : is an isomorphism of O K -group schemes over R which is compatible with the polarization in the sense that there exists an isomorphism ν N : Z/N Z ≃ µ N of group schemes over R such that In addition we require that for every multiple N ′ of N, locallyétale over Spec(R), there exists a similar level N ′ -structure α ′ , restricting to α on N −1 L/L. One says that α is locallyétale symplectic liftable ( [Lan], 1.3.6.2). In view of Lemma 1.1, the last condition of symplectic liftability is void if D K is odd, while if D K is even it is equivalent to the following condition ([Bel], I.3.1): • For any geometric point η : R → k (k algebraically closed field, necessarily of characteristic different from 2), the O K ⊗ Z 2 polarized module (T 2 A η , , λ ) is isomorphic to (L ⊗ Z 2 , , ) under a suitable identification of lim ← µ 2 n (k) with 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 α can exist only if the group schemes Z/N Z and µ N become isomorphic over R, but the isomorphism ν N is then determined by α. 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 ( [Lan], I.4.1.11).
Theorem 1.3. The functor R → M(R) is represented by a smooth quasi-projective scheme S over 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 (A, λ, ι, α) of the above type over S. We call A the universal abelian scheme over S. For every R 0 -algebra R and PEL structure in 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.
1.3.2. The Shimura variety Sh K . We briefly recall the interpretation of the Picard modular surface as a canonical model of a Shimura variety. The symmetric domain X can be interpreted as a G ∞ -conjugacy class of homomorphisms (1.29) h : S = res C R G m → G turning (G, X) into a Shimura datum in the sense of Deligne [De]. The reflex field associated to this datum turns out to be K. Let K ∞ be the stabilizer of . Then the Shimura variety Sh K is a complex quasi-projective variety whose complex points are isomorphic, as a complex manifold, to the double coset space The theory of Shimura varieties provides a canonical model for Sh K over K. The following important theorem complements the one on the representability of the functor M.
Theorem 1.4. The canonical model of Sh K is the generic fiber S K of S.
Let us explain only how to associate to a point of Sh K (C) a point in S(C). For that we have to associate an element of M(C) to g ∈ G(A), and show that the structures associated to g and to γgk (γ ∈ G, k ∈ K) are isomorphic. Let Note that J x depends only on g ∞ K ∞ and L g only on g f K 0 f , so A g depends only on gK 0 .
Letμ(g) be the unique positive rational number such that for every prime p, (1.32) ord pμ (g) = ord p µ(g p ).
Such a rational number exists since µ(g p ) is a p-adic unit for almost all p and Q has class number 1. We claim that , g =μ(g) −1 , : L g × L g → Q induces a principal polarization λ g on A g . That this is a (rational) Riemann form follows from the fact that (u, v) Jx = u, J x v + i u, v is hermitian positive definite. That , g is indeed Z-valued and L g is self-dual follows from the choice ofμ(g) since locally at p the dual of g p L p under , : We conclude that there exists a unique polarization λ g : for every u, v ∈ A g [l] = l −1 L g /L g and every l ≥ 1. This polarization is principal.
Since g f commutes with the K-structure on V A , L g is still an O K -lattice, hence ι g is defined.
Finally α g is derived from (1.34) We note that α g depends only on gK because K f ⊂ K 0 f is the principal level-N subgroup, and that it lifts to level N ′ structure for any multiple N ′ of N, by the same formula. The isomorphism ν N,g between Z/N Z and µ N (C) that makes (1.28) work is self-evident (see (1.49)). Let A g ∈ M(C) be the structure just constructed.
Let now γ ∈ G(Q). Then the action of γ on V induces an isomorphism between the tuples A g and A γg . Indeed, γ : V R → V R intertwines the complex structures x g and x γg , and carries L g to L γg , so induces an isomorphism of the abelian varieties, which clearly commutes with the PEL structures.
This shows that A g depends solely on the double coset of g in G(Q)\G(A)/K. One is left now with two tasks which we do not do in this survey: (i) Proving that if A g ≃ A g ′ then g and g ′ belong to the same double coset, and that every A ∈ M(C) is obtained in this way, (ii) Identifying the canonical model of Sh K over K with S K .
1.3.3. The connected components of Sh K . Recall that G ′ = SU = ker(ν : G → T). Since G ′ is simple and simply connected, strong approximation holds and Here K × = ν(G(Q)), and it follows that is injective. We now claim (see also Theorem 2.4 and 2.5 of [De]) that is a bijection. For ν is surjective ( [De] (0.2)) and continuous ( where U K is the product of local units at all the finite primes and Cl K is the class group.
1.3.4. The cl and ν N invariants of a connected component. The norm N : . Using the lattice L as an integral structure in V, we see that G comes from a group scheme G Z over Z, whose points in any ring A are We likewise get that µ is a homomorphism from G Z to G m . The diagram (1.44) This shows that µ(K f ) ⊂Ẑ × (N ), the product of local units congruent to 1 mod N. But To conclude, we have shown the existence of two maps from the set of connected components: These two maps are independent: together they map π 0 (G(Q)\G(A)/K) onto On the other hand, they have a non-trivial common kernel, which grows with N, as is evident from the interpretation of K × \K × f /ν(K f ) as the Galois group of a certain class field extension of K. The map cl gives the restriction to the Hilbert class field, while the map ν N gives the restriction to the cyclotomic field Q(µ N ). We have singled out cl and ν N , because when N ≥ 3, they have an interpretation in terms of the complex points of Sh K .
) is (essentially) the ν N,g that appears in the definition of α g (see 1.3.1).
Proof. (i) cl ([g]) is the class of the ideal (ν(g f )) associated to the idele ν(g f ) ∈ K × f . This ideal is in the same class as (det(g f )), because µ(g f ) ∈ Q × f , so (µ(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. ( We then take the result modulo N, so Now the definition of the tuple (A g , λ g , ι g , α g ) is such that if u, v ∈ N −1 L/L then 1.3.5. The complex uniformization. Recall that X = G ∞ /K ∞ and that it was equipped with a base point x 0 (coresponding to (z, u) = (δ K /2, 0) in the Siegel domain of the second kind). Let 1 = g 1 , . . . , g m ∈ G(A f ) (m = #(K × \K × f /ν(K f ))) be representatives of the connected components of G(Q)\G(A)/K, and define congruence groups sending Γ j x to [x, g j ] is an isomorphism.
Note that Γ 1 = Γ is the principal level-N congruence subgroup in G Z (Z), the stabilizer of L. Similarly, Γ j is the principal level-N congruence subgroup in the stabilizer of L gj , 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 gj . As N ≥ 3, det(γ) = 1 and µ(γ) = 1 for all γ ∈ Γ j , for every j. Indeed, on the one hand these are in K × and Q × + respectively. On the other hand, they are local units which are congruent to 1 mod N everywhere. It follows that Γ j are subgroups of G ′ (Q) = SU(Q).
We get a similar decomposition to connected components (as an algebraic surface) 1.4.1. The smooth compactification of X Γ . We begin by working in the complex analytic category and follow the exposition of [Cog]. The Baily-Borel compactification X * Γ 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 O × K , that this smooth compactification is canonical. As all cusps are equivalent (if we vary the lattice L or Γ), it is enough, as usual, to study the smooth compactification at c ∞ . In [Cog] this is described for an arbitrary L (not even O K -free), but for simplicity we write it down only for L = L 0 .
As N ≥ 3, elements of Γ stabilizing c ∞ lie in N ∞ . 3 The computations, which we omit, are somewhat simpler if N is even, an assumption made for the rest of this section. Let Lemma 1.6. Let N ≥ 3 be even. The matrix n(s, r) ∈ Γ cusp if and only if: Let M = N |D K | in case (i) and M = 2 −1 N |D K | in case (ii). This is the width of the cusp c ∞ . Let if R is large enough, two points of it are Γ-equivalent if and only if they are Γ cuspequivalent. A sufficiently small punctured neighborhood of c ∞ in X * Γ therefore looks like Γ cusp \Ω R . As where the action of s ∈ Λ is via (1.57) [s] : (t, u) → (e 2πiδs(u+s/2)/M t, u + s).
It is a line bundle over E via the second projection. We denote the class of (t, u) modulo the action of Λ by [t, u]. (This condition is invariant under the action of Λ.) Let T ′ R be the punctured disk bundle obtained by removing the zero section from T R . Then the map (z, u) → (q(z), u) induces an analytic isomorphism between Γ cusp \Ω R and T ′ R . Proof. This follows from the discussion so far and the fact that λ(z, u) > R is equivalent to the above condition on t = q(z) ( [Cog], Prop. 2.1).
To get a smooth compactificationX Γ of X Γ (as a complex surface), we glue the disk bundle T R to X Γ along T ′ R . In other words, we complete T ′ R by adding the zero section, which is isomorphic to E. The same procedure should be carried out at any other cusp of C Γ .
Note that the geodesic (1.15) connecting (z, u) ∈ X to the cusp c ∞ projects in X Γ to a geodesic which meets E transversally at the point u mod Λ. We caution that this geodesic in X Γ depends on (z, u) and c ∞ and not only on their images modulo Γ.
The line bundle T is the inverse of an ample line bundle on E. In fact, T ∨ is the N -th (resp. 2N -th) power of one of the four basic theta line bundles if d K ≡ 1 mod 4 (resp. d K ≡ 2, 3 mod 4). A basic theta function of the lattice Λ satisfies, for u ∈ C and s ∈ Λ, (1.59) θ(u + s) = α(s)e 2πs(u+s/2)/|δ|N 2 θ(u) where α : Λ → ±1 is a quasi-character (see [Mu], p.25). Recalling the relation between M and N , and the assumption that N was even, we easily get the relation between T and the theta line bundles.
1.4.2. The smooth compactification of S. The arithmetic compactificationS of the Picard surface S (over R 0 ) is due to Larsen [La1] (see also [Bel] and [Lan]). We summarize the results in the following theorem. We mention first that as S C has a canonical model S over R 0 , its Baily-Borel compactification S * C has a similar model S * over R 0 , and S embeds in S * as an open dense subscheme.  (ii) As a complex manifold, there is an isomorphism . Let R N be the integral closure of R 0 in the ray class field K N of conductor N over K. Then the connected components of C RN are geometrically irreducible, and are indexed by the cusps of S * RN over which they sit. Furthermore, each component E ⊂ C RN is an elliptic curve with complex multiplication by O K .
We call C the cuspidal divisor. If c ∈ S * C − S C is a cusp, we denote the complex elliptic curve p −1 (c) by E c . Bear in mind that while E c is in principle definable over the Hilbert class field K 1 , no canonical model of it over that field is provided byS. On the other hand, E c does come with a canonical model over K N , and even over R N .
We refer to [La1] and [Bel] for a moduli-theoretic interpretation of C as a moduli space for semi-abelian schemes with a suitable action of O K and a "level-N structure". Unfortunately these references do not give such a moduli interpretation tō S. While they do construct a universal semi-abelian scheme overS (see the next section), the level-N structure over S does not extend to a flat level-N structure overS in the ordinary sense, and the notion has to be modified over the boundary. What is evidently missing is the construction of a "Tate-Picard" object similar to the "generalized elliptic curve" which was constructed over the complete modular curve by Deligne and Rapoport in [De-Ra].
1.4.3. Change of level. Assume that N ≥ 3 is even, and N ′ = QN. We then obtain a covering map X Γ(N ′ ) → X Γ(N ) where by Γ(N ) we denote the group previously denoted by Γ. Near any of the cusps, the analytic model allows us to analyze this map locally. Let E ′ be an irreducible cuspidal component ofX Γ(N ′ ) mapping to the irreducible component E ofX Γ(N ) . The following is a consequence of the discussion in the previous sections.
Proposition 1.9. The map E ′ → E is a multiplication-by-Q isogeny, henceétale of degree Q 2 . When restricted to a neighborhood of E ′ , the coveringX Γ(N ′ ) →X Γ(N ) is of degree Q 3 , and has ramification index Q along E, in the normal direction to E.
Corollary 1.10. The pull-back to E ′ of the normal bundle T (N ) of E is the Qth power of the normal bundle T (N ′ ) of E ′ .
1.5. The universal semi-abelian scheme A.
1.5.1. The universal semi-abelian scheme overS. As Larsen and Bellaiche explain, the universal abelian scheme π : A → S extends canonically to a semi-abelian scheme π : A →S. The polarization λ extends over the boundary C =S − S to a principal polarization λ of the abelian part of A. The action ι of O 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 RN , mapping (over C and under the projection p) to the cusp c ∈ S * C . Then there exist (1) a principally polarized elliptic curve B defined over R N , with complex multiplication by O K and CM type Σ, and (2) an ideal a of O K , such that every fiber A x of A over E is an O K -group extension of B by the O K -torus a ⊗ G m . Both B (with its polarization) and the ideal class [a] ∈ Cl K are uniquely determined by the cusp c. Only the extension class in the category of O 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 A x is of type (2, 1), as is the case at an interior point x ∈ S. If we extend scalars to C, the isomorphism type of B is given by another ideal class [b] (i.e. B(C) ≃ C/b). In this case we say that the cusp c is of type (a, b).
The above discussion defines a homomorphism (of fppf sheaves over Spec(R N )) . As we shall see soon, the Ext group is represented by an elliptic curve with CM by O K , defined over R N , and this map is an isogeny.
1.5.2. O K -semi-abelian schemes of type (a, 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 O K and CM type Σ, and a an ideal of O K . Consider a semi-abelian scheme G over R, endowed with an O K action ι : O K → End(G), and a short exact sequence We call all this data a semi-abelian scheme of type (a, B) (over R). The group classifying such structures is Ext 1 and that this construction yields an isomorphism Here we have used the canonical identification a * = δ −1 K a −1 (via the trace pairing). Although (δ 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 O K derived from its action on B, isΣ rather than Σ.
Thus over δ K a ⊗ OK B t there is a universal semi-abelian scheme G(a, B) of type (a, B), and any G as above, over any base R ′ /R, is obtained from G(a, B) by pullback (specialization) with respect to a unique map The universal semi-abelian variety G(a, B) will now be denoted G(a, b). In 1.6.2 below we give a complex analytic model of this G(a, b).
1.6. Degeneration of 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 ≥ 3 is even, and that the cusp is the standrad cusp at infinity c = c ∞ . In this case we have shown that E c = C/Λ, where Λ = N O K , and we have given a neighborhood of E c inX Γ the structure of a disk bundle in a line bundle T . See Proposition 1.7. Consider the geodesic (1.15) connecting (z, u) to c ∞ . Consider the universal abelian scheme in the moving lattice model (cf (1.27)). Of the three vectors used to span L ′ x over O 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 O K -multiples). We conclude that A ′ x degenerates to Making the change of variables (ζ ′ 1 , ζ ′ 2 , ζ ′ 3 ) = (ζ 1 , ζ 2 +ūζ 1 , ζ 3 ) does not alter the O K action and gives the more symmetric model with kernel a ⊗ 1. As usual we identify a ⊗ C with C(⊀) ⊕ C(Σ), sending a ⊗ ζ → (aζ,āζ). We now note that if we use this identification to identify C 3 with C ⊕ (O K ⊗ C) (an identification which is compatible with the O K action) then the ι ′ (O K )-span of the vector t (0, 1, 1) is just the kernel of e OK . We conclude that This clearly gives G u the structure of an O K -semi-abelian variety of type (O K , O K ), i.e. an extension 1.6.2. The analytic uniformization of the universal semi-abelian variety of type (a, b). We now compare the description that we have found for the degeneration of A along the geodesic connecting (z, u) to c ∞ with the analytic description of the universal semi-abelian variety of type (a, b).
Proposition 1.11. Let a and b be two Then G u is a semi-abelian variety of type (a, b), any complex semi-abelian variety of this type is a G u , and G u ≃ G v if and only if u − v ∈ ab −1 .
Proof. That G u is a semi-abelian variety of type (a, b) is obvious. That any abelian variety of this type is a G u follows by passing to the universal cover C 2 (Σ) ⊕ C(Σ), and noting that by a change of variables in the Σ-andΣ-isotypical parts, we may assume that the lattice by which we divide is of the form , so we only have to prove that G u is split if and only if u ∈ ab −1 . But one can check easily that G u is trivial if and only if (sū,su) ∈ ker e a = a ⊗ 1 = {(a,ā)|a ∈ a} for every s ∈ b, and this holds if and only if u ∈ ab −1 .
Corollary 1.12. Let N ≥ 3 be even. Let c = c ∞ be the cusp at infinity. Then the map sending u to the isomorphism class of the semi-abelian variety above u mod Λ is the isogeny of multiplication by N .
Proof. In view of the computations above, and the description of a neighborhood of E c inX Γ given in Proposition 1.7 this map is identified with the canonical map The extra data carried by u ∈ E c , which is forgotten by the map of the corollary, comes from the level N structure. As mentioned before, according to [La1] and [Bel] 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 (M ∞,N structures in the language of [Bel]). A level-N structure on a semi-abelian variety G of type (a, b) consists of (i) a level-N structures α : if and only if u and v represent the same point of E c .

The basic automorphic vector bundles
2.1. The vector bundles P and L.
2.1.1. Definition and first properties. Recall our running assumptions and notation. The tame level N ≥ 3, S is the Picard modular scheme over the base ring R 0 ,S is its smooth compactification, and A is the universal semi-abelian scheme overS (an abelian scheme over S) constructed by Larsen and Bellaiche.
Let ω A be the relative cotangent space at the origin of A. If e :S → A is the zero section, ). This is a rank 3 vector bundle overS and the action of O K allows to decompose it according to types. We let Then P is a plane bundle, and L a line bundle. Over S (but not over the cuspidal divisor C =S − S) we have the usual identification ω A = π * Ω 1 A/S . The relative de Rham cohomology of A/S is a rank 6 vector bundle sitting in an exact sequence (the Hodge filtration) [Mu]), and λ : A → A t is an isomorphism which reverses CM types, we obtain an exact sequence The notation M(ρ) means that M is a vector bundle with an O K action and in M(ρ) the vector bundle structure is that of M but the O K action is conjugated. Decomposing according to types, we have two short exact sequences dR (A/S) induced by the polarization is O S -linear, alternating, perfect, and satisfies ι(a)x, y λ = x, ι(ā)y λ . It follows that H 1 dR (A/S)(Σ) and H 1 dR (A/S)(Σ) are maximal isotropic subspaces, and are set in duality. As ω A is also isotropic, this pairing induces pairings These two pairings (in this order) 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 2.1.2. The factors of automorphy corresponding to L and P. The formulae below can be deduced also from the matrix calculations in the first few pages of [Sh2]. Let Γ = Γ j be one of the groups used in the complex uniformization of S C , cf Section 1.3.5. Via the analytic isomorphism X Γ ≃ S with the jth connected component, the vector bundles P and L are pulled back to X Γ and then to the symmetric space X, where they can be trivialized, hence described by means of factors of automorphy. Let us denote by P an and L an the two vector bundles on X Γ , in the complex analytic category, or their pull-backs to X.
To trivialize L an we must choose a nowhere vanishing global section over X. As usual, we describe it only on the connected component containing the standard cusp, corresponding to j = 1 (where L = L g1 = L 0 ). Recalling the "moving lattice model" (1.27) and the coordinates ζ 1 , ζ 2 , ζ 3 introduced there, we note that dζ 3 is a generator of L an = ω A (Σ). For reasons that will become clear later (cf Section 2.6) we use 2πi · dζ 3 to trivialize L an over X. Suppose Lemma 2.1. The following relation holds for every γ ∈ U ∞ As v(γ(z, u)) = j(γ; z, u) −1 ·γ(v(z, u)) the lemma follows from (γv, γv) = (v, v).
the automorphy factor corresponding to dζ 3 is the function j(γ; z, u).
Proof. Since, by construction, dζ 3 is a nowhere vanishing holomorphic section of L (over 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 dζ 3 . To prove (iii) we compute in V R (with the original complex structure!) and recall that since W γ(z,u) is precisely the line where the complex structure in (V R , J γ(z,u) ) has been reversed, in (V R , J γ(z,u) ) we have Dualizing, we get (x = (z, u)) This concludes the proof.
Consider next the plane bundle P an . As we will only be interested in scalarvalued modular forms, we do not compute its matrix-valued factor of automorphy (but see [Sh2]). It is important to know, however, that the line bundle det P an gives nothing new.
Proposition 2.3. There is an isomorphism of analytic line bundles over X Γ , Moreover, dζ 1 ∧ dζ 2 is a nowhere vanishing holomorphic section of det P an over X, and the factor of automorphy corresponding to it is j(γ; z, u).
Proof. Since a holomorphic line bundle on X Γ = Γ\X is determined, up to an isomorphism, by its factor of automorphy, and j(γ; z, u) is the factor of automorphy of L an corresponding to dζ 3 , it is enough to prove the second statement. Let U = V(Σ) be the plane bundle dual to P an . Let As we have seen in (1.27), these two vector fields are sections of U and at each point x ∈ X form a basis dual to dζ 1 and dζ 2 . It follows that they are holomorphic sections, and that v 1 ∧ v 2 is the basis dual to dζ 1 ∧ dζ 2 . We must show that the factor of automorphy corresponding to As det(γ) = 1, this gives and the proof is complete.
2.1.3. The relation det P ≃ L overS K . The isomorphism between det P and L is in fact algebraic, and even extends to the generic fiberS K of the smooth compactification.
Proposition 2.4. One has det P ≃ L overS K .
Proof. Since P ic(S K ) ⊂ P ic(S C ) it is enough to prove the proposition over C. By GAGA, it is enough to establish the triviality of det P ⊗L −1 in the analytic category.
descends from X to X Γ , because dζ 1 ∧dζ 2 and dζ 3 have the same factor of automorphy j(γ, x) (γ ∈ Γ, x ∈ X). This section is nowhere vanishing on X Γ , and extends to a nowhere vanishing section onX Γ , trivializing det P ⊗ L −1 . In fact, if c is the standard cusp, dζ 1 ∧ dζ 2 and dζ 3 are already well-defined and nowhere vanishing sections of det P and L in the neighborhood . This is a consequence of the fact that j(γ, x) = 1 for γ ∈ Γ cusp . An alternative proof is to use Theorem 4.8 of [Ha]. In our case it gives a functor V → [V] from the category of G(C)-equivariant vector bundles on the compact dual P 2 C of Sh K to the category of vector bundles with G(A f )-action on the inverse system of Shimura varieties Sh K . Here P 2 C = G(C)/H(C), where H(C) is the parabolic group stabilizing the line C· t (δ/2, 0, 1) in G(C) = GL 3 (C) × C × , and the irreducible V are associated with highest weight representations of the Levi factor L(C) of H(C). It is strightforward to check that det P and L are associated with the same character of L(C), up to a twist by a character of G(C), which affects the G(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 P and L are isomorphic also algebraically.
We de not know if det P and L are isomorphic as algebraic line bundles over S. This would be equivalent, by (2.7), to the statement that for every PEL structure (A, λ, ι, α) ∈ M(R), for any R 0 -algebra R, det(H 1 dR (A/R)(Σ)) 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, λ) over R, det H 1 dR (A/R) is trivial, follows at once from the Hodge filtration (2.4). Our result, however, suffices to guarantee the following corollary, which is all that we will be using in the sequel.
Corollary 2.5. For any characteristic p geometric point Spec(k) → Spec(R 0 ), we have det P ≃ L onS k . A similar statement holds for morphisms SpecW (k) → Spec(R 0 ).
Proof. SinceS is a regular scheme, det P ⊗ L −1 ≃ O(D) for a Weil divisor D supported on vertical fibers over R 0 . Since any connected component Z ofS k is irreducible, we can modify D so that D and Z are disjoint, showing that det P ⊗ L −1 | Z is trivial. The second claim is proved similarly.
2.1.4. Modular forms. Let R be an R 0 -algebra. A modular form of weight k ≥ 0 and level N ≥ 3 defined over R is an element of the finite R-module We usually omit the subscript R, remembering thatS is now to be considered over R. The well-known Koecher principle says that H 0 (S, L k ) = H 0 (S, L k ). See [Bel], Section 2.2, for an arithmetic proof that works integrally over any R 0 -algebra R.
A cusp form is an element of the space In particular, cusp forms of weight 3 are "the same" as holomorphic 2-forms onS.
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 ev- , then given such an A and ω, the universal property of S produces a unique morphism ϕ : T → S over R, ϕ * A = A, and we may let f (A, ω) = ϕ * f /ω k . Conversely, given such a rule f we may cover S by Zariski open sets T where L is trivialized, and then the sections . While viewing f as a "rule" rather than a section is a matter of language, it is sometimes more convenient to use this language.
Theorem 2.6. If k ≥ 3 (resp. k ≥ 6) then M 0 k (N, R) (resp. M k (N, R)) is a locally free finite R-module, and the base-change homomorphism Bellaiche considers only weights divisible by 3, but his proofs generalize to all k (cf remark on the bottom of p.43 in [Bel]).
Over C, pulling back to X and using the trivialization of L given by the nowehere vanishing section 2πi · dζ 3 , a modular form of weight k is a collection (f j ) 1≤j≤m of holomorphic functions on X satisfying (the Koecher principle means that no condition has to be imposed at the cusps).
2.2. The Kodaira Spencer isomorphism. Let π : A → S be an abelian scheme of relative dimension 3, as in the Picard moduli problem. The Gauss-Manin connection as the composition of the maps hence maps, denoted by the same symbols, 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 [Fa-Ch], Prop. 9.1 on p.81 (in the Siegel modular case).
, are vector bundles of rank 2.
Lemma 2.8. If S is the Picard modular surface and A = A is the universal abelian variety, then is an isomorphism, and so is KS(Σ).
Proposition 2.9. The Kodaira-Spencer map induces a canonical isomorphism of vector bundles over S We refer to Corollary 2.16 for an extension of this result toS.
Corollary 2.10. There is an isomorphism of line bundles L 3 ≃ Ω 2 S . Proof. Take determinants and use det P ≃ L. We emphasize that while KS(Σ) is canonical, the identification of det P with 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 ω E of the universal elliptic curve, becomes isomorphic to Ω 1 Y (N ) . Over C, as the isomorphism between L 3 and Ω 2 S takes dζ ⊗3 3 to a constant multiple of dz ∧ du (see Corollary 2.18), the differential form corresponding to a modular form ( f j ) 1≤j≤m of weight 3, is (up to a constant) (f j (z, u)dz ∧ du) 1≤j≤m .

2.3.
Extensions to the boundary of S.
2.3.1. The vector bundles P and L over C. Let E ⊂ C RN be a connected component of the cuspidal divisor (over the integral closure R N of R 0 in the ray class field K N ). As we have seen, E is an elliptic curve with CM by Since the toric part and the abelian part of G are constant, L, P 0 and P µ = P/P 0 are all trivial line bundles when restricted to E. It can be shown that P itself is not trivial over E.
Proposition 2.11. Working over K N , let E j (1 ≤ j ≤ h) be the connected components of C. Then Proof. By [Hart] II.6.5, Ω 2 S ≃ L 3 ⊗ h j=1 O(E j ) nj for some integers n j and we want to show that n j = −1 for all j. By the adjunction formula on the smooth surfacē S, if we denote by KS a canonical divisor, O(KS) = Ω 2 S , then We conclude that Here E j .E j < 0 because E j can be contracted to a point (Grauert's theorem). As hence n j = −1 as desired.
2.4.1. The infinitesimal retraction. We follow the arithmetic theory of Fourier-Jacobi expansions as developed in [Bel]. Let S be the formal completion ofS along the cuspidal divisor C =S − S. We work over R 0 , and denote by C (n) the n-th infinitesimal neighborhood of C inS. The closed immersion i : C ֒→ S admits a canonical left inverse r : S → C, a retraction satisfying r • i = Id C . This is not automatic, but rather a consequence of the rigidity of tori, as explained in [Bel], Proposition II.2.4.2. As a corollary, the universal semi-abelian scheme A /C (n) is the pull-back of A /C via r. The same therefore holds for P and L, namely there are natural isomorphisms r * (P| C ) ≃ P| C (n) and r * (L| C ) ≃ L| C (n) . As a consequence, the filtration (2.42) 0 → P 0 → P → P µ → 0 extends canonically to C (n) . Since L, P 0 and P µ are trivial on C, they are trivial over C (n) as well.
2.4.2. Arithmetic Fourier-Jacobi expansions. We fix an arbitrary noetherian R 0algebra R and consider all our schemes over R, without a change in notation.
As usual, we let O S = lim ← O C (n) (a sheaf in the Zariski topology on C). Via r * , this is a sheaf of O C -modules. Choose a global nowhere vanishing section s ∈ H 0 (C, L) trivializing L. Such a section is unique up to a unit of R on each connected component of C. This s determines an isomorphism To understand the structure of H 0 (C, O S ) let I ⊂ OS be the sheaf of ideals defining C, so that C (n) is defined by I n . The conormal sheaf N = I/I 2 is the restriction i * OS(−C) of OS(−C) to C. It is an ample invertible sheaf on C, since (over R N ) its degree on each component E j is −E 2 j > 0. Now r * supplies, for every n ≥ 2, a canonical splitting of (2.45) 0 → I/I n → OS/I n → OS/I → 0.
Inductively, we get a direct sum decomposition 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 H 0 (C, N m ).

2.4.3.
Fourier-Jacobi expansions over C. Working over C, we shall now relate the infinitesimal retraction r to the geodesic retraction, and the powers of the conormal bundle N to theta functions. Recall the analytic compactification of X Γ described in Proposition 1.7. Let E be the connected component ofX Γ −X Γ corresponding to the standrad cusp c ∞ . As before, we denote by E (n) its nth infinitesimal neighborhood. The line bundle T | E is just the analytic normal bundle to E, hence we have an isomorphism (2.49) N an ≃ T ∨ between the analytification of N = I/I 2 and the dual of T .
Lemma 2.12. The infinitesimal retraction r : E (n) → 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 (2.50) r an : E (n) an → E an where E an is the analytic space associated to E with its sheaf of analytic functions O hol E , and E (n) an is the same topological space with the sheaf O hol S /I n an . The geodesic retraction (sending (z, u) to u mod Λ) is an analytic map r geo : E an (ε) → E an , where E an (ε) is our notation for some tubular neighborhood of E an inS an . On the other hand, there is a canonical map can of ringed spaces from E (n) an to E an (ε). We claim that these three maps satisfy r geo • can = r an .
To prove the lemma, note that the infinitesimal retraction r : E (n) → E is uniquely characterized by the fact that the O K -semi-abelian variety A x at a point x of E (n) , obtained as a specialization of the universal family A, is the pull-back of the universal semi-abelian variety at r(x). See [Bel], II.2.4.2. The computations of section 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 C, f ∈ M k (N, C). Using the trivialization of L an over the symmetric space X given by 2πi · dζ 3 as discussed in Section 2.1.2, we identify f with a collection of functions f j on X, transforming under Γ j according to the automorphy factor j(γ; z, u) k . As usual we look at Γ = Γ 1 only, and at the expansion of f = f 1 at the standard cusp c ∞ , 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 C corresponding to g 1 = 1. It also means that as the section s used to trivialize L along E, we must use a section that, analytically, coincides with 2πi · dζ 3 .
Pulling back the sheaf N an from E = C/Λ to C, it is clear that q = q(z) = e 2πiz/M maps, at each u ∈ C, to a generator of T ∨ = N an = I an /I 2 an , and we denote by q m the corresponding generator of N m an = I m an /I m+1 an . If 2.5. The Gauss-Manin connection in a neighborhood of a cusp.
2.5.1. A computation of ∇ in the complex model. We shall now compute the Gauss-Manin connection in the complex ball model near the standard cusp c ∞ . Recall that we use the coordinates (z, u, ζ 1 , ζ 2 , ζ 3 ) as in Section 1.2.4. Here dζ 1 and dζ 2 form a basis for P and dζ 3 for L. The same coordinates served to define also the semiabelian variety G u (denoted also A u ) over the cuspidal component E at c ∞ , cf Section 1.6. As explained there (1.69), the projection to the abelian part is given by the coordinate ζ 1 (modulo O K ), so dζ 1 is a basis for the sub-line-bundle of ω A/E coming from the abelian part, which was denoted P 0 . In section 2.4.1 above it was explained how to extend the filtration P 0 ⊂ P canonically to the formal neighborhood 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 A (z,u) along a geodesic, it becomes clear that P We shall now pull back these vector bundles to the ball X, and compute the Gauss Manin connection ∇ complex analytically on ω A/X . We write P 0 = O X · dζ 1 for P 0,an etc. dropping the decoration an. Let These 6 vectors span L ′ (z,u) over Z. Let β 1 , . . . , β ′ 3 be the dual basis to {α 1 , . . . , α ′ 3 } in H 1 dR (A/O X ), i.e. α1 β 1 = 1 etc. As the periods of the β i 's along the integral homology are constant, the β-basis is horizontal for the Gauss-Manin connection. The first coordinate of the α i and α ′ i gives us (2.54) dζ 1 = 0 · β 1 + 1 · β 2 + u · β 3 + 0 · β ′ 1 + ω K · β ′ 2 + ω K u · β ′ 3 , and we find that

2.5.2.
A computation of KS in the complex model. We go on to compute the Kodaira-Spencer map on P, i.e. the map denoted KS(Σ). For that we have to take ∇(dζ 1 ) and ∇(dζ 2 ) and project them to R 1 π * O A (Σ) ⊗ Ω 1 X . We then pair the result, using the polarization form , λ on H 1 dR (A) (reflecting the isomorphism (2.57) To perform the computation we need two lemmas Lemma 2.13. The Riemann form on L ′ x , associated to the polarization λ, is given in the basis α 1 , α 2 , α 3 , α ′ 1 , α ′ 2 , α ′ 3 by the matrix Proof. This is an easy computation using the transition map T between L and L ′ x and the fact that on L the Riemann form is the alternating form , = Im δ (, ).
For the formulation of the next lemma recall that if A is a complex abelian variety, a polarization λ : A → A t induces an alternating form , λ on H 1 dR (A) as well as a Riemann form on the integral homology H 1 (A, Z). We compare the two.
Lemma 2.14. Let (A, λ) be a principally polarized complex abelian variety. If α 1 , . . . , α 2g is a symplectic basis for H 1 (A, Z) in which the associated Riemann form is given by a matrix J, and β 1 , . . . , β 2g is the dual basis of H 1 dR (A), then the matrix of the bilinear form , λ on H 1 dR (A) in the basis β 1 , . . . , β 2g is (2πi) −1 J. Proof. These are essentially Riemann's bilinear relations. For example, if A is the Jacobian of a curve C and the basis α 1 , . . . , α 2g has the standard intersection matrix (2.59) J = 0 I −I 0 then the lemma follows from the well-known formula for the cup product (ξ, η being differentials of the second kind on C) Using the two lemmas we get We summarize.
Proposition 2.15. Let z, u, ζ 1 , ζ 2 , ζ 3 be the standard coordinates in a neighborhood of the cusp c ∞ . Then, complex analytically, the Kodaira-Spencer isomorphism (2.63) KS(Σ) : P ⊗ L ≃ Ω 1 X is given by the formulae Corollary 2.16. The Kodaira-Spencer isomorphism P ⊗ L ≃ Ω 1 S extends meromorphically overS. Moreover, in a formal neighborhood S of C, its restriction to the line sub-bundle P 0 ⊗ L is holomorphic, and on any direct complement of P 0 ⊗ L in P ⊗ L it has a simple pole along C.
Proof. As we have seen, dζ 1 ⊗ dζ 3 and dζ 2 ⊗ dζ 3 define a basis of P ⊗ L at the boundary, with dζ 1 ⊗ dζ 3 spanning the line sub-bundle P 0 ⊗ L. On the other hand du is holomorphic there, while dz has a simple pole along the boundary.
Proof. As we have seen, dζ 1 is a basis for P 0 .
Proof. The isomorphism det P ≃ L carries dζ 1 ∧ dζ 2 to a constant multiple of dζ 3 , so the corollary follows from (2.64).

2.5.3.
Transferring the results to the algebraic category. The computations in the analytic category over X of course descend (still in the analytic category) to S C , because they are local in nature. They then hold a fortiori in the formal completion S C along the cuspidal component E. But the Gauss-Manin and Kodaira-Spencer maps are defined algebraically on S, and both Ω 1 S and ω A/ S are flat over R 0 , so from the validity of the formulae over C we deduce their validity in S over R 0 , provided we identify the differential forms figuring in them (suitably normalized) with elements of Ω 1 S and ω A/ S defined over R 0 . In particular, they hold in the characteristic p fiber as well.
From the relation we deduce that the map κ 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.
2.6.1. Rationality of local sections of P and L. We have compared the arithmetic surface S with the complex analytic surfaces Γ j \X (1 ≤ j ≤ m), and the compactifications of these two models. We have also compared the universal semi-abelian scheme A and the automorphic vector bundles P and 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. To avoid issues of class numbers, we shall work rationally and not integrally.
We shall need to look at local parameters at the cusps, and as the cusps are defined only over K N , we shall work with S KN instead of S K . With a little more care, working with Galois orbits of cusps, we could probably prove rationality over K, but for our purpose K N is good enough.
If ξ and η belong to a K N -module, we write ξ ∼ η to mean that η = cξ for some c ∈ K × N . We begin with the vector bundles P and L. Over C they yield analytic vector bundles P an and L an on each X Γj (1 ≤ j ≤ m). Assume for the rest of this section that j = 1 and write Γ = Γ 1 . Similar results will hold for every j. The vector bundles P and L are trivialized over the unit ball X by means of the nowhere vanishing sections dζ 3 ∈ H 0 (X, L an ) and dζ 1 , dζ 2 ∈ H 0 (X, P an ). These sections do not descend to X Γ , but (2.67) σ an = (dζ 1 ∧ dζ 2 ) ⊗ dζ −1 3 ∈ H 0 (X Γ , det P ⊗ L −1 ) does, as the factors of automorphy of dζ 1 ∧ dζ 2 and dζ 3 are the same (cf Section 2.1.2). Furthermore, this factor of automorphy (i.e. j(γ; z, u)) is trivial on Γ cusp , the stabilizer of c ∞ in Γ, so dζ 1 ∧ dζ 2 and dζ 3 define sections of det P and L on S C , the formal completion ofS C along the cuspidal divisor E c = p −1 (c ∞ ) ⊂S C . We have noted already that along E c , P has a canonical filtration (2.68) 0 → P 0 → P → P µ → 0 and that dζ 1 is a generator of P 0 . (Compare (1.63) and (1.71) and note that the projection to C/O K = B(C) is via the coordinate ζ 1 , so dζ 1 is a generator of P 0 | Ec = ω B ). As we have shown in Section 2.4.1, this filtration extends to the formal neighborhood S C of E c . The vector bundles P and L, as well as the filtration on P over S C , are defined over K N . It makes sense therefore to ask if certain sections are K N -rational. Recall that the cusp c ∞ is of type (O K , O K ).
(iii) Let B be the elliptic curve over K N associated with the cusp c ∞ as in Section 1.5.1. Let Ω B ∈ C × be a period of a basis ω of ω B = H 0 (B, Ω 1 B/KN ) (i.e. the lattice of periods of ω is Ω B · O K ). This Ω B is well-defined up to an element of K × N . Then Proof. Let E be the component of C KN which over C becomes E c . Let G be the universal semi-abelian scheme over E. Then G is a semi-abelian scheme which is an extension of B × KN E by the torus (O K ⊗G m,KN )× KN E. At any point u ∈ E(C) we have the analytic model G u (1.69) for the fiber of G at u, but the abelian part and the toric part are constant. Over E the line bundle P 0 is (by definition) ω B×E/E . As the lattice of periods of a suitable K N -rational differential is Ω B · O K , while the lattice of periods of dζ 1 is O K , part (iii) follows. For parts (i) and (ii) observe that the toric part of G is in fact defined over K, and that e * OK maps the cotangent space of O K ⊗ G m,K isomorphically to the K-span of 2πidζ 2 and 2πidζ 3 .
Corollary 2.20. Ω B · σ an is a nowhere vanishing global section of det P ⊗ L −1 over S Γ , rational over K N .
Proof. Recall that we denote by S Γ the connected component of S KN whose associated analytic space is the complex manifold X Γ . We have seen that as an analytic section Ω B · σ an descends to X Γ and extends to the smooth compactificationX Γ . By GAGA, it is algebraic. SinceX Γ 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 ∞ ) is defined over K N .
The complex periods Ω B (and their powers) appear as the transcendental parts of special values of L-functions associated with Grossencharacters of K. They are therefore instrumental in the construction of p-adic L functions on K. We expect them to appear in the p-adic interpolation of holomorphic Eisenstein series on the group G, much as powers of 2πi (values of ζ(2k)) appear in the p-adic interpolation of Eisenstein series on GL 2 (Q).
2.6.2. Rationality of local parameters at the cusps. We keep the assumptions and the notation of the previous section. Analytically, neighborhoods of E c∞ were described in Section 1.4.1 with the aid of the parameters (z, u). Let S denote the formal completion ofS KN along E. Let r : S → E be the infinitesimal retraction discussed in Section 2.4.1. If i : E ֒→ S is the closed embedding then r • i = Id E . If I is the sheaf of definition of E, then N = I/I 2 is the conormal bundle to E, hence its analytification is the dual of the line bundle T , Consider r * N on S. The retraction allows us to split the exact sequence The map i • r : S → S induces a sheaf homomorphism r * i * Ω 1 S → Ω 1 S , 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 (2.71) Ω 1 S = r * i * Ω 1 S = r * N × r * Ω 1 E . Let x ∈ E and represent it by u ∈ C (modulo Λ). Then q = e 2πiz/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 dq, the image of q in I an /I 2 an , is a basis of N an (on U x ∩ E). But (2.72) 2πi · dz = M dq q is independent of u (see (1.55)), so represents a global meromorphic section of r * N an , with a simple pole along E ⊂ S C . By GAGA, this section is (meromorphic) algebraic.
Proposition 2.21. (i) The section 2πi · dz is K N -rational, i.e. it is the analytification of a section of r * N . (ii) The section Ω B · du is K N -rational, i.e. belongs to H 0 (E, Ω 1 E/KN ).
Proof. The proof relies on the Kodaira-Spencer isomorphism KS(Σ) (2.64), which is a K N -rational (even K-rational) algebraic isomorphism between P ⊗L and Ω 1 S . As we have shown, it extends to a meromorphic homomorphism from P ⊗ L to Ω 1 S over S. Over S it induces an isomorphism of P 0 ⊗ L onto r * Ω 1 E ⊂ Ω 1 S carrying the K Nrational section Ω B dζ 1 ⊗ 2πidζ 3 to −Ω B δ · du, proving part (ii) of the proposition. It also carries 2πidζ 2 ⊗ 2πidζ 3 to 2πidz, but the latter is only meromorphic. We may summarize the situation over S by the following commutative diagram with exact rows: Let h be a K N -rational local equation of E, i.e. a K N -rational section of I in some Zariski open U intersecting E non-trivially, vanishing to first order along E ∩ U.
The differential η = h · (2πidz) is regular on U, and to prove that it is K N -rational we may restrict it to S and check rationality there. But in S we have a K Nrational product decomposition Ω 1 S = r * N × r * Ω 1 E and the projection of η to the second factor is 0, so it is enough to prove rationality of its projection to r * N . This projection is the image, under KS(Σ), of h · (2πidζ 2 ⊗ 2πidζ 3 mod P 0 ⊗ L), so our assertion follows from parts (i) and (ii) of the previous proposition. This proves that η, hence h −1 η = 2πidz is a K N -rational differential. An alternative proof of part (ii) is to note that E is isogenous over K N to B, so up to a K N -multiple has the same period.
Remark 2.1. Unlike 2πidz = M dq/q, 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 ∈ Λ) then q is multiplied by the factor e 2πiδs(u+s/2)M , so although O hol S C ,x ⊂ OS C ,x and analytic parameters may be considered as formal parameters, the question whether q itself is K N -rational is not well-defined (in sharp contrast to the case of modular curves!).
2.6.3. Normalizing the isomorphism det P ≃ L. Let us fix a nowhere vanishing section This section is determined up to K × . From now on we shall use this section to identify det P with L whenever such an identification is needed. From Corollary 2.20 we deduce that when we base change to C, on each connected component X Γ (2.75) σ ∼ Ω B · σ an .
3.1.1. The three strata. Fix a prime p > 2 which is inert in K and relatively prime to N . Then R 0 /pR 0 = F p 2 and we consider the characteristic p fiber of the smooth compactificationS of S, From now on we shall writeS to denote this scheme, rather than the original one over R 0 . We let A, as before, stand for the universal semi-abelian variety overS. The structure ofS has been worked out by Vollaard [V]. We record her main results.
Recall that an abelian variety over a field of characteristic p is called supersingular if the Newton polygon of its p-divisible group is of constant slope 1/2. It is called superspecial if it is isomorphic to a product of supersingular elliptic curves.
Theorem 3.1. (i) There exists a closed reduced 1-dimensional subscheme S ss ⊂S (the supersingular locus), disjoint from the cuspidal divisor (i.e. contained in S), which is uniquely characterized by the fact that for every geometric point x of S, the abelian variety A x is supersingular if and only if x ∈ S ss . (ii) Let S ssp be the singular locus on S ss . Then x lies in S ssp if and only if A x is superspecial. If s 0 ∈ S ssp then s 0 is rational over F p 2 and (iii) Assume that N is large enough (depending on p). Then the irreducible components of S ss are rational over F p 2 , nonsingular, and in fact are all isomorphic to the Fermat curve (3.3) x p+1 + y p+1 + z p+1 = 0.
There are p 3 + 1 points of S 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.
We callS µ =S − S ss (or S µ =S µ ∩ S) the µ-ordinary or generic locus, S gss = S ss − S ssp the general supersingular locus, and S ssp the superspecial locus. Then S =S µ ∪ S gss ∪ S ssp is a stratification. The three strata are of dimensions 2,1, and 0 respectively, the closure of each stratum contains the lower dimensional ones, and each of the three is open in its closure. -We] and Vollaard describe the p-divisible group A x (p) of the abelian variety A x for x in the various strata. Let x : Spec(k) → S µ be a geometric point (k an algebraically closed field of characteristic p) and A = A x (an abelian variety over k). Then the p-divisible group A(p) has a three-step O K -invariant filtration

The p-divisible groups. Bültel and Wedhorn [Bu
such that gr 2 = F il 2 is a group of multiplicative type (connected, withétale dual), gr 1 = F il 1 /F il 2 is connected with a connected dual, and gr 0 = F il 0 /F il 1 iś etale. Each of the three graded pieces is a p-divisible group of height 2 (O K -height 1). As every p-divisible group over an algebraically closed field of characteristic p splits canonically into a product of a group of multiplicative type, a group which is connected with a connected dual, and anétale group, the above filtration splits.
We have where a and c are ideals of O K and G is the p-divisible group of a supersingular elliptic curve overF p (the group denoted by G 1,1 in the Manin-Dieudonné classification [Dem]). Although, up to isomorphism, it is possible to substitute O K for a or c, it is sometimes more natural to allow this extra freedom in notation.
Geometric points of S ss classify abelian varieties A = A x for which A(p) is isogenous to G 3 , and such a point x is superspecial if and only if A(p) is isomorphic to G 3 .
The a-number of A is the dimension of the k-linear space Hom(α p , A[p]). It is 1 if x ∈ S µ ∪ S gss and 3 if x ∈ S ssp. The stratification which we have described coincides, in our simple case, with the Ekedahl-Oort stratification [Oo], [Mo], [We].

New relations between automorphic vector bundles in characteristic p.
3.2.1. The line bundles P 0 and P µ overS µ . Consider the universal semi-abelian variety A over the Zariski open setS µ . As we have seen in Section 2.3.1, over the cuspidal divisor C, P = ω A (Σ) admits a canonical filtration (3.6) 0 → P 0 → P → P µ → 0 where P 0 is the cotangent space to the abelian part of A, and P µ the Σ-component of the cotangent space to the toric part of A. This filtration exists already in characteristic 0, but when we reduce modulo p it extends, as we now show, to the whole ofS µ . Let A[p] 0 be the connected part of the subgroup scheme A[p]. Then A[p] 0 is finite flat overS µ of rank p 4 . (It is clearly flat and quasi-finite, and the fiber rank can be computed separately on C and on S µ . Since the rank is constant, the morphism tō S µ is actually finite, cf. [De-Ra], Lemme 1.19.) Let be the maximal subgroup-scheme of multiplicative type. Since at every geometric point ofS µ , A[p] µ is of rank p 2 , this subgroup is also finite flat overS µ . It is also O K -invariant. Over the cuspidal divisor C, A[p] µ is the p-torsion in the toric part of A, and over S µ As ω A is killed by p, we have ω is then a line bundle P 0 of type (1, 0) and we get the short exact sequence (3.9) 0 → P 0 → ω A → ω µ A → 0 over the whole ofS µ . Decomposing according to types and setting P µ = ω µ A (Σ), we get the desired filtration.

Frobenius and Verschiebung.
Write G for A[p] 0 (a finite flat O K -group scheme overS) and G (p) for the base-change of G with respect to the absolute Frobenius morphism of degree p. In other words, if we denote by φ the homomorphism x → x p (of any F p -algebra), and by Φ :S →S the corresponding map of schemes, then (3.10) The relative Frobenius is an OS-linear homomorphism F rob G : G → G (p) , characterized by the fact that pr 2 • F rob G is the absolute Frobenius morphism of G.
The relative Verschiebung is an O S -linear homomorphism V er G : A , and taking Σ-components we get (3.11) The right vertical arrow is 0 since V kills P 0 , as G is of local-local type. The left vertical map is an isomorphism, since V er is an isomorphism on p-divisible groups of multiplicative type. We conclude that (3.13) P 0 = ker(V : P → L (p) ).

3.2.3.
Relations between P 0 , P µ and L overS µ . We first recall a general lemma. (3.14) a ⊗ m → a · m ⊗ · · · ⊗ m is an isomorphism of line bundles over S.
Since L (p) ≃ L p by the lemma, we have (3.15) L p ≃ P/P 0 = P µ .
Finally, from P 0 ⊗ P µ ≃ det P ≃ L we get We have proved: Proposition 3.3. OverS µ , P µ ≃ L p and P 0 ≃ L 1−p .
In the same vein we get a commutative diagram for theΣ parts Thus overS µ , P has a canonical filtration by P 0 , but the induced filtration on P (p) already splits as a direct sum.
Remark 3.1. Working overF p , and restricting attention to a connected component E of C, P| E is a non-split extension of P µ by P 0 . However, both P µ and P 0 are trivial on E, so the extension is described by a non-zero class ξ ∈ H 1 (E, O E ). The extension P (p) | E is then described by ξ (p) . The semilinear map ξ → ξ (p) is the Cartier-Manin operator, and since E is a supersingular elliptic curve, ξ (p) = 0 and P (p) | E splits. Thus at least over C, the splitting of P (p) is consistent with what we know so far.
Since V induces an isomorphism of L onto P (p) /P (p) 0 ≃ (P/P 0 ) p and P/P 0 ≃ L p we conclude that overS µ , L ≃ L p 2 . In the next section we realize this isomorphism via the Hasse invariant. Combining what was proved so far we easily get the following.

3.2.4.
Extending the filtration on P over S gss . In order to determine to what extent the filtration on P and the relation between L and the two graded pieces of the filtration extend into the supersingular locus, we have to employ Dieudonné theory.
Proof. Everything is a formal consequence of the fact that V maps P onto L (p) , and the relation det P ≃ L. OverS µ the proposition was verified in the previous subsection, so it is enough to prove that V (P) = L (p) in the fiber of any geometric point x ∈ S gss (k) (k algebraically closed). We use the description of H 1 dR (A x /k) given in Lemma 4.9 below, due to Bültel and Wedhorn. In the notation of that lemma, P x is spanned over k by e 1 and e 2 and L x by f 3 , while V (e 1 ) = 0, V (e 2 ) = f (p) 3 . This concludes the proof. Proposition 3.7. Over the whole ofS − S ssp , V maps L injectively onto a subline-bundle of P (p) .
Proof. Once again, we know it already overS µ , and it remains to check the assertion fiber-wise on S gss . We refer again to Lemma 4.9, and find that V (f 3 ) = e (p) 1 , which proves our claim.
The emerging picture is this: Outside the superspecial points, V maps L injectively onto a sub-line-bundle of P (p) , and V (p) maps P (p) surjectively onto L (p 2 ) . However, the line V (L) coincides with the line P (p) 0 = ker (V (p) ) only on the general supersingular locus, while on its complementS µ the two lines make up a frame for P (p) (3.18). One can be a little more precise. The equation is the defining equation of S ss in the sense that when expressed in local coordinates it defines S ss with its reduced subscheme structure. See Proposition 3.9 below. For a superspecial x, A x is isomorphic to a product of three supersingular elliptic curves, so V vanishes on the whole of ω Ax . The analysis of the last paragraph breaks up. To complete the picture, we shall now prove that there does not exist any way to extend the filtration (3.19) across such an x.
Proposition 3.8. It is impossible to extend the filtration 0 → P 0 → P → P µ → 0 along S ss in a neighborhood of a superspecial point x.
Proof. To simplify the computations we change the base field toF p . At any superspecial point there are p + 1 branches of S ss meeting transversally. We shall prove the proposition by showing that along any one of these branches (labelled by ζ, a p + 1-st root of −1) P 0 approaches a line P x [ζ] ⊂ P x , but these p + 1 lines are distinct. In other words, on the normalization of S ss we can extend the filtration uniquely, but the extension does not come from S ss .
Before we go into the proof a word of explanation is needed. In Section 4.3 below we study the deformation of the action of V on ω A near a general supersingular point x ∈ S gss . For that purpose the first infinitesimal neighborhood of x suffices, and we end up using Grothendieck's crystalline deformation theory, even though we cast it in the language of de Rham cohomology. At a point x ∈ S ssp , in contrast, we need to work in the full formal neighborhood of x in S, or at least in an Artinian neighborhood which no longer admits a divided power structure. The reason is that the singularity of S ss at x is formally of the type Spec (F p [[u, v]]/(u p+1 + v p+1 )), as we shall see in (3.27). Crystalline deformation theory is inadequate, and we need to use Zink's "displays". As the theory of displays is covariant, we start with the covariant Cartier module of A = A x rather than the contravariant Dieudonné module, and look for its universal deformation.
Let us review the (confusing) functoriality of these two modules. For the moment, let A be any abelian variety overF p . If D is the (contravariant) Dieudonné module of A and M is its (covariant) Cartier module, then D/pD = H 1 dR (A) and M/pM = H 1 dR (A t ) are set in duality. The dual of (3.21) V : D/pD → (D/pD) (p) F rob A t being the Frobenius isogeny from A t to A t(p) ). As usual, sinceF p is perfect, we may view V as a φ −1 -linear map of D/pD, and F as a φ-linear map of M/pM. Replacing A by A t we then also have a map F on D/pD and V on M/pM. The Hodge filtration This reminder tells us that when we pass from the contravariant theory to the covariant one, instead of looking for the deformation of V on ω A we should look for the deformation of With an appropriate choice of the basis, the Frobenius F on M/pM is the φ-linear map whose matrix with respect to the basis f 3 , e 1 , e 2 , e 3 , f 1 , f 2 is All this can be deduced from [Bu -We], 3.2.
To construct the universal display we follow the method of [Go-O]. See also [An-Go]. With local coordinates u and v we write S = SpfF p [[u, v]] for the formal completion of S at x. We study the deformation of F to We use a basis f 3 , e 1 , . . . , f 2 satisfying the same assumptions as above with respect to the O K -type and the polarization pairing. Then one can choose u and v and the basis of H 1 dR (A t / S) so that the universal Frobenius is given by the matrix Since the first three vectors project onto a basis of Lie(A), the matrix of F : Lie(A) (p) → Lie(A) is the 3 × 3 upper left corner, and the matrix of F 2 (= F • F (p) ) is (note the semilinearity) This matrix is sometimes called the Hasse-Witt matrix. Thus on Lie(A)(Σ) (p 2 ) = L (p 2 )∨ the action of F 2 is given by multiplication by u p+1 + v p+1 . As the supersingular locus is the locus where the action of F on the Lie algebra is nilpotent, it follows that the local (formal) equation of S ss at x is Note that this equation guarantees also that the lower 2 × 2 block, representing the action of F 2 on the Σ-part of the Lie algebras is (semi-linearly) nilpotent, i.e. (3.28) We write S ss = Spf (F p [[u, v]]/(u p+1 + v p+1 )) for the formal completion of S ss at x. Letting ζ run over the p + 1 roots of −1 we recover the p + 1 formal branches through x as the "lines" (3.29) u = ζv.
We write S ss [ζ] = Spf (F p [[u, v]]/(u − ζv)) for this branch. When we restrict (pull back) the vector bundle Lie(A) to S ss [ζ], ker(F ) ∩ Lie(A) (p) (the dual of P µ ) becomes When v = 0 (i.e. outside the point x) this is the line As these lines are distinct, the filtration of P can not be extended across x.

The Hasse invariant hΣ.
3.3.1. Definition of hΣ. The construction and main properties of the Hasse invariant that we are about to describe, have been given (for any unitary Shimura variety) by Goldring and Nicole in [Go-Ni]. Let R be an R 0 /pR 0 = F p 2 -algebra. Let A be an abelian scheme over R and where φ is the p-power map. If α is an endomorphism of A then α (p) = α × 1 is an endomorphism of A (p) and (3.33) If, via ι, A has CM by O K and type (2, 1), then A (p) will have, via ι (p) , type (1, 2). The isogenies F rob A/R and V er A/R induce R-linear maps (F A/R ) * and (V A/R ) * on Lie algebras. Since must have a kernel in Lie(A (p) )(Σ) which is at least one-dimensional. Consider This means that the Hasse invariant is a modular form of weight p 2 − 1 in characteristic p.
Theorem 3.9. The Hasse invariant is invertible on S µ and vanishes on S ss = S − S µ to order one. More precisely, S µ is an open subscheme whose complement S ss is a divisor, and when we endow this divisor with its induced reduced subscheme structure, it becomes the Cartier divisor div(hΣ).
Proof. Dualizing the definition, the Hasse invariant vanishes precisely where V * A (L) is contained in ker (V * A (p) : P (p) → L (p 2 ) ). We have already seen that over S µ the latter is the line bundle P (p) 0 and that V * sends L isomorphically onto a direct complement of P (p) 0 , cf (3.18). Thus the Hasse invariant does not vanish on S µ . To prove that hΣ vanishes on S ss to order 1 we must study the Dieudonné module at an infinitesimal neighborhood of a point x ∈ S gss and compute V * (p) • V * using local coordinates there. This can be extracted from [Bu -We], but since later in our work we explain it in deatil (for the purpose of studying the theta operator), we shall now refer to Section 4.3. In Lemma 4.9 we describe the (contravariant) Dieudonné module at x. In subsection 4.3.3 we describe its infinitesimal deformation. Using the local coordinates u and v, and the notation used there, f 3 − uf 1 − vf 2 becomes a basis for L over the first infinitesimal neighborhood of x. We then compute Apart from working in the cotangent space instead of the tangent space, V * (p) • V * is the Hasse invariant (use Hom(M, N ) = Hom(N ∨ , M ∨ )). It follows that after L has been locally trivialized, the equation hΣ = 0 becomes u = 0, which is the local equation for S gss .
Proposition 3.10. The Hasse invariant extends to a holomorphic section of L p 2 −1 overS, which is nowhere vanishing onS µ . If we trivialize L| C then the restriction of hΣ to the cuspidal divisor C becomes a nowhere vanishing locally constant function.
Proof. Extendibility holds by the Koecher principle for any modular form. One can even deduce non-vanishing at the cusps, but here we may argue directly. The same definition as the one given over S, with the abelian variety A replaced by G = A[p] 0 (A signifying now the semi-abelian variety overS), defines hΣ over the complete Picard surface: The same argument as above shows that hΣ is nowhere vanishing on the whole of S µ . Since L| C is trivial, the last statement is obvious.
Corollary 3.11. The scheme S * µ = S * − S ss is affine and overF p , the intersection of S ss with every connected component of S is connected.
Proof. The line bundle L is ample on S, even over R 0 . 4 Hence for large enough m, which we can take to be a multiple of p 2 − 1, L m is very ample, and by [La1] the Baily-Borel compactification S * is the closure of S in the projective embedding supplied by the linear system H 0 (S, L m ). It follows that L m has an extension to a line bundle on S * which we denote O S * (1), since it comes from the restriction of the O(1) of the projective space to S * . Moreover, Larsen proves that on the smooth compactificationS, L m = π * O S * (1) where π :S → S * . 5 Replacing hΣ by its power h m/(p 2 −1) Σ , this power becomes a global section of L m , hence its zero locus S ss a hyperplane section of S * in the projective embeding supplied by H 0 (S, L m ). Its complement is therefore affine. The last claim follows from the fact [Hart], III, 7.9, that a positive dimensional hyperplane section of a smooth (or more generally, normal) projective variety is connected.
The schemeS µ is of course far from affine, as it contains the complete curves E as cuspidal divisors.

A secondary Hasse invariant on the supersingular locus.
In his forthcoming Ph.D. thesis [Bo], Boxer develops a general theory of secondary Hasse invariants defined on lower strata of Shimura varieties of Hodge type. See also [Kos]. In this section we provide an independent approach, in the case of Picard modular surfaces, affording a detailed study of its properties. As an application we relate the number of irreducible components of S ss to the Euler number of S C , and through it to the value of the function L(s, ( DK · )) at s = 3. 3.4.1. Definition of h ssp . As we have seen, along the general supersingular locus S gss , Verschiebung induces isomorphisms (3.43) V L : L ≃ P (p) 0 , V P : P µ ≃ L (p) . (The first is unique to S gss , the second holds also on the µ-ordinary stratum.) Consider the isomorphism . Its source is the line bundle det P (p) which is identified with L (p) ≃ L p . We therefore get a nowhere vanishing section (3.45)h ssp ∈ H 0 (S gss , L p 2 −p+1 ).
4 One way to see it is to use the ampleness of the Hodge bundle det ω A ≃ L 2 (pull back from Siegel space, where it is known to be ample by [Fa-Ch]). 5 It is not clear that L itself has an extension to a line bundle on S * , or that π * L, which is a coherent sheaf extending L| S , is a line bundle (the problem lying of course only at the cusps). In other words, it is not clear that we can extract an mth root of O S * (1) as a line bundle.
We shall show that h ssp extends to a holomorphic section on S ss , and vanishes at the superspecial points (to a high order).

3.4.2.
Computations at the superspecial points. We refer again to the computations of Proposition 3.8, and work overF p . Dualizing the display described there, and using the letters e i , f j to denote the dual basis to the basis used there we get the following.
Lemma 3.12. Let x ∈ S ssp be a superspecial point. There exist formal coordinates u and v so that the formal completion of S at x is S = Spf (F p [[u, v]]), and D = H 1 dR (A/ S) has a basis f 3 , e 1 , e 2 , e 3 , f 1 , f 2 overF p [[u, v]] with the following properties: (i) f 3 , e 1 , e 2 is a basis for ω A (ii) The basis is symplectic, i.e. the polarization form is (iii) O K acts on the e i via Σ and on the f j viaΣ.
Using the lemma, we compute along S ss [ζ], where u = ζv (ζ p+1 = −1). Denote by P[ζ], P 0 [ζ] and L[ζ] the pull-backs of the corresponding vector bundles to the branch S ss [ζ]. The map V L is given by Use e 1 ∧ e 2 = e 1 ∧ (ζ p e 1 + e 2 ) as a basis for det P[ζ]. Since V Lemma 3.13. There does not exist a function g ∈F p [[u, v]]/(u p+1 + v p+1 ) on S ss whose restriction to the branch S ss [ζ] is ζu p−1 .
Proof. Had there been such a function, represented by a power series G ∈F p [[u, v]], then we would get vg = u p on S ss [ζ] for every ζ, hence But any power series in the ideal (u p+1 +v p+1 ) contains only terms of degree ≥ p+1, while in vG − u p we can not cancel the term u p .
The lemma means thath ssp can not be extended over S ss to a section of Hom(det P (p) , L p 2 +1 ) ≃ L p 2 −p+1 . However, when we raise it to a p + 1 power the dependence on ζ disappears. It then extends to a section h ssp of L p 3 +1 over S ss , given over S by the equation where ε ∈F p [[u, v]] × depends on the isomorphism between det P and L.
Theorem 3.14. The secondary Hasse invariant h ssp belongs to H 0 (S ss , L p 3 +1 ). It vanishes precisely at the points of S ssp . The subscheme "h ssp = 0" of S ss is not reduced. At x ∈ S ssp , with u and v as above, it is the spectrum of From now on we assume that N is large enough (depending on p) so that Theorem 3.1(iii) holds. Each irreducible component of S ss is non-singular, and h ssp has a zero of order p 2 − 1 at each superspecial point on such a component. Each component contains p 3 + 1 superspecial points. It follows that if Z is such a component, We get the following corollary.
Proof. Computing intersection numbers, Denote by KS a canonical divisor on the given connected component ofS. From the adjunction formula, As we have seen in Proposition 2.11, O(KS + C) ≃ L 3 where C is the cuspidal divisor. Hence by the last Corollary. We get Plugging this into the expression for (Z.Z) we get (3.60) (Z.Z) = n(p 2 − 1) 2 .
On the other hand, since Z is the divisor of the Hasse invariant onS, div(hΣ) = Z, we get O(Z) = L p 2 −1 so (3.61) n = c 1 (L) 2 .
3.4.4. The classes of Ekedahl-Oort strata in the Chow ring ofS. Let us denote by CH = CH(S) the Chow ring with Q coefficients ofS overF p , and let CH L the Qsubalgebra generated by c 1 (L) (note that we now use c i to denote Chern classes of vector bundles in CH and not in cohomology). Recall that the Ekedahl-Oort strata ofS consist ofS itself, Z = S ss in codimension 1, and W = S ssp in codimension 2. Proof. Without loss of generality we may assume that N is large enough, so that Theorem 3.1(iii) applies. As the divisor of the Hasse invariant is Z, [Z] = (p 2 − 1)c 1 (L). It remains to deal with [W ]. Let f :Z →S be the map from the normalization of Z toS with image Z. Then the projection formula implies

WriteZ =
n i=1Z i and note that (p 3 + 1)c 1 (f * L) is the class of div(h ssp ) onZ, which is represented by the 0-cycle consisting of the superspecial points on eachZ i with multiplicities p 2 − 1. It follows that 3.5.1. The Igusa scheme. Let N ≥ 3 as always, and let M be the moduli problem of Section 1.3.1. Let n ≥ 1 and consider the following moduli problem on R 0 /pR 0algebras: is a finite flat O K -subgroup scheme of rank p 2n of multiplicative type, and It is clear that if (A, A[p n ] µ , ε) ∈ M Ig(p n ) (R) then A is fiber-by-fiber µ-ordinary and therefore A ∈ M(R) defines an R-point of S µ . It is also clear that the functor R M Ig(p n ) (R) is relatively representable over M, and therefore as N ≥ 3 and M is representable, this functor is also representable by a scheme Ig µ (p n ) which maps to S µ . See [Ka-Ma] for the notion of relative representability. We call Ig µ (p n ) the Igusa scheme of level p n .
Proposition 3.18. The morphism τ : Ig µ (p n ) → S µ is finite andétale, with the Galois group ∆(p n ) = (O K /p n O K ) × acting as a group of deck transformations.
Proof. Every µ-ordinary abelian variety has a unique finite flat O K -subgroup scheme of multiplicative type A[p n ] µ of rank p 2n . Such a subgroup scheme is, locally in thé etale topology, isomorphic to δ −1 K O K ⊗ µ p n , and any two isomorphisms differ by a unique automorphism of δ −1 ∆(p n ) becomes a group of deck transformation and the proof is complete.
3.5.2. A compactification over the cusps. The proof of the following proposition mimics the construction ofS. We omit it.
Proposition 3.19. Let Ig µ (p n ) be the normalization ofS µ =S − S ss in Ig µ (p n ). Then Ig µ (p n ) →S µ is finiteétale and the action of ∆(p n ) extends to it. The boundary Ig µ (p n ) − Ig µ (p n ) is non-canonically identified with ∆(p n ) × C.
We define similarly Ig * µ , and note that it is finiteétale over S * µ . Proposition 3.20. Let A denote the pull-back of the universal semi-abelian variety fromS µ to Ig µ (p n ). Then A is equipped with a canonical Igusa level structure Over C and after base change to R N /pR N the toric part of A is locally Zariski of the form a ⊗ G m and ε is then an O K -linear isomorphism between δ −1 K O K ⊗ µ p n and a ⊗ µ p n .
3.5.3. A trivialization of L over the Igusa surface. From now on we focus on Ig µ = Ig µ (p) although similar results hold when n > 1, and would be instrumental in the study of p-adic modular forms. The vector bundle ω A pulls back to a similar vector bundle over Ig µ . But there is a rank 2 quotient bundle stable under O K (of type (1, 1)), and the isomorphism ε induces an isomorphism Proposition 3.21. The line bundles L, P 0 and P µ are trivial over Ig µ .
Proof. Use ε * as an isomorphism between vector bundles and note that L = ω µ A (Σ) and P µ = ω µ A (Σ). The relation P 0 ⊗ P µ = det P ≃ L implies the triviality of P 0 as well.
Note that the trivialization of L and P µ is canonical, because it uses only the tautological map ε which exists over the Igusa scheme. The trivialization of P 0 on the other hand depends on how we realize the isomorphism det P ≃ L.
We can now give an alternative proof to the fact that L p 2 −1 and P p 2 −1 µ are trivial onS µ . Since Ig µ is anétale cover ofS µ of order p 2 −1, det τ * (τ * L) ≃ L p 2 −1 . As τ * L is already trivial, so is L p 2 −1 on the base. The same argument works for P µ and for P 0 . The fact that P p+1 0 is already trivial could be deduced by a similar argument had we worked out an analogue of Ig(p) classifying symplectic isomorphisms of G[p] with gr 1 A[p]. The role of ∆(p) for such a moduli space would be assumed by (3.73) ∆ 1 (p) = ker(N : , which is a group of order p + 1. We do not go any further in this direction here.
3.6. Compactification of the Igusa surface along the supersingular locus.
(ii) The section a(1) is a p 2 − 1 root of the Hasse invariant over Ig µ , i.e.
Proof. (i) This part is a restatement of the action of ∆(p). At two points of Ig µ (R) lying over the same point of S µ (R) and differing by the action of γ ∈ ∆(p), the canonical embeddings 3.68). The induced trivializations of Lie(A)(Σ) differ byΣ(γ) and by duality we get (i).
(ii) Since over any F p -base, V er Gm = 1, we have a commutative diagram (3.78) .
Using the isomorphism Lie(A)(Σ) (p 2 ) ≃ Lie(A)(Σ) p 2 to which we alluded before, we get the commutative diagram (3.79) , from which we deduce that hΣ = a(p 2 − 1). Note that we can interprete a(1) as an arrow as above rather than as a section of L = Lie(A)(Σ) ∨ because of the identity Hom(M, N ) = Hom(N ∨ , M ∨ ).
3.6.2. The compactification Ig of Ig µ . Quite generally, let L → X be a line bundle associated with an invertible sheaf L on a scheme X. Write L n for the line bundle L ⊗n over X. Let s : X → L n be a section. Consider the fiber product where the two maps to L n are λ → λ n and s. Let p : Y pr2 → X be the projection which can also factor as Y pr1 → L → L n → X (since X s → L n → X is the identity). Consider This line bundle on Y has a tautological section t : Y → p * L, Here s(x) = λ n and (3.83) t n (y) = (λ n , y) = (s(x), y) = p * s(y) 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 → X, and t 1 ∈ Γ(Y 1 , p * 1 L) is such that t n 1 = p * 1 s, then there exists a unique morphism h : Y 1 → Y covering the two maps to X such that t 1 = h * t.
The map L → L n is finite flat of degree n and if n is invertible on the base, finité etale away from the zero section. Indeed, locally on X it is the map A 1 ×X → A 1 ×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 → X : this map is finite flat of degree n andétale 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, Ig = Ig(p) as where the map S → L p 2 −1 is hΣ. From the universal property and part (ii) of Proposition 3.22 we get a map ofS-schemes This map is an isomorphism overS µ because both schemes areétale torsors for ∆(p) = (O K /pO K ) × and the map respects the action of this group. We summarize the discussion in the following theorem (for the last point, consult [Mu2], Proposition 2, p.198).
Theorem 3.23. The morphism τ : Ig →S satisfies the following properties: (i) It is finite flat of degree p 2 − 1,étale overS µ , totally ramified over S ss .
(ii) ∆ acts on Ig as a group of deck transformations and the quotient isS.
(iii) Let s 0 ∈ S gss (F p ). Then there exist local parameters u, v at s 0 such that O S,s0 =F p [[u, v]], S gss ⊂ S is formally defined by u = 0, and ifs 0 ∈ Ig maps to s 0 under τ, then O Ig,s0 =F p [[w, v]] where w p 2 −1 = u. In particular, Ig is regular in codimension 1.
(iv) Let s 0 ∈ S ssp (F p ). Then there exist local parameters u, v at s 0 such that O S,s0 =F p [[u, v]], S gss ⊂ S is formally defined by u p+1 + v p+1 = 0, and ifs 0 ∈ Ig maps to s 0 under τ, then In particular,s 0 is a normal singularity of Ig.
3.6.3. Irreducibility of Ig. So far we have avoided the delicate question of whether Ig is "relatively irreducible", i.e. whether τ −1 (T ) is irreducible if T ⊂S is an irreducible (equivalently, connected) component. Using an idea of Katz, and following the approach taken by Ribet in [Ri], the irreducibility of τ −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 γ ∈ (O K /p n O K ) × there exists a µ-ordinary abelian variety with PEL structure A ∈ S µ (F q r ) such that the image of Gal(F q /F q r ) in Instead, we shall give a different argument valid for the case n = 1.
Proposition 3.24. The morphism τ : Ig →S induces a bijection on irreducible components.
Proof. Since Ig is a normal surface, connected components and irreducible components are the same. Let T be a connected component ofS and T ss = T ∩ S ss . Let τ −1 (T ) = Y i be the decomposition into connected components. As τ is finite and flat, each τ (Y i ) = T. Since τ is totally ramified over T ss , there is only one Y i .
Lemma 4.1. Fix 0 ≤ k < p 2 − 1. Then we have a surjective homomorphism Proof. Take f ∈ H 0 (Ig µ , O) (k) , so that f · a(k) ∈ H 0 (Ig µ , L k ) (0) , hence descends to g ∈ H 0 (S µ , L k ). This g may have poles along S ss , but some h n Σ g will extend holomorphically to S, 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 n Σ a(k).
Proposition 4.2. The resulting ring homomorphism obtained by dividing a modular form of weight k by a(k) is surjective, respects the Z/(p 2 − 1)Z-grading on both sides, and its kernel is the ideal generated by (hΣ − 1).
Proof. We only have to prove that anything in ker(r) is a multiple of hΣ − 1, the rest being clear. Since r respects the grading, we may assume that for some k ≥ 0 we have f j ∈ M k+j(p 2 −1) (S, R 0 /pR 0 ) and As a result we get that

4.1.2.
Fourier-Jacobi expansions modulo p. The arithmetic Fourier-Jacobi expansion (2.48) depended on a choice of a nowhere vanishing section s of L along the boundary C =S − S ofS. As the boundaryC = Ig µ − Ig µ is (non-canonically) identified with ∆(p) × C, we may "compute" the Fourier-Jacobi expansion on the Igusa surface rather than on S. But on the Igusa surface, a(1) is a canonical choice for such an s. We may therefore associate a canonical Fourier-Jacobi expansion along the boundary of Ig, for every (R an R 0 /pR 0 -algebra). The following proposition becomes almost a tautology. Proof. The first statement is tautologically true. For the second, note that for where Let ω(f ) = k − (p 2 − 1)n. Then n is the order of vanishing of f along S ss . Equivalently, k − ω(f ) is the order of vanishing of the pull-back of f to Ig along Ig ss . In addition, ω(f m ) = mω(f ).

4.2.1.
Definition of Θ(f ). From now on we work overF p . We first recall some notation. Let S be the (open) Picard surface overF p and Ig = Ig(p) the Igusa surface of level p over S (completed along the supersingular locus as explained above). We denote by Z = S ss = S −S µ the supersingular locus of S, byZ = Ig ss = Ig −Ig µ its pre-image under the covering map τ : Ig → S, by Z ′ = S gss = S ss −S ssp the smooth part of Z, and byZ ′ = Ig gss = Ig ss − Ig ssp the pre-image of Z ′ under τ. When we need to compactify these schemes at the cusps, we letS and Ig stand for the smooth compactifications. By the Koecher principle, the space of mod p modular forms of weight k ≥ 0 and level N can be regarded as sections of L k over either S orS: We recall (Theorem 3.23) that τ : Ig →S is finite Galois of degree p 2 − 1, that it isétale outside Z, and fully ramified over Z. The Galois group of the covering may be canonically identified with (4.12) If M is anF p [∆] module, and χ : ∆ →F × p a character, we let M χ be the submodule on which ∆ acts via χ. We continue to denote by A the universal abelian scheme over Ig (or its extension to a semi-abelian scheme over Ig) and by P and L the Σ-andΣ-parts of ω A/Ig . These vector bundles are just the base-change by τ * of their counterparts overS. We let a(1) be the canonical section of L over Ig, trivializing L over Ig µ = Ig −Z, and vanishing to first order alongZ. The Galois group ∆ acts on a(1) viaΣ −1 . Let . This function has a pole of order k alongZ, and the Galois group acts on it viaΣ k . Let dg ∈ H 0 (Ig µ , Ω 1 be the map obtained from KS(Σ) −1 when we apply V to P. Over S − S ssp this is the same as dividing out by the line sub-bundle P 0 = ker V P , since (4.15) P/P 0 = P µ ≃ V (P) = L (p) .
A priori, this extends only to a meromorphic modular form of weight k + p + 1, as it may have poles along Z. Let P be a prime of R N above p at which we reduce the Picard surface. The standard component of C over R N /P R N is the reduction modulo P of the standard component of C over R N . Finally, Ig maps toS (over R N /P R N ) and the cuspidal components mapping to a given component E of C are classified by the embedding of δ −1 K O K ⊗µ p in the toric part of A. Since the toric part of the universal semi-abelian variety over the standard component is O K ⊗ G m , we may define the standard cuspidal component of Ig to be the component where the map Here we use the fact that (4.20) (ii) The effect of Θ on Fourier-Jacobi expansions is a "Tate twist". More precisely, let  Here M (equal to N or 2 −1 N ) is the width of the cusp.
Corollary 4.6. The operator Θ extends to a derivation of the graded ring of modular forms, and for any f, Θ(f ) is a cusp form.
Parts (iii) and (iv) of the theorem are clear from the construction. The proof of (i), that Θ(f ) is in fact holomorphic along S ss , will be given in the next section. 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 ′ = N Q then the conormal bundle becomes the Q-th power of the conormal bundle of level N ′ , i.e. N = N ′Q (see Section 1.4.3). It follows that what was the m-th FJ coefficient at level N becomes the Qm-th coefficient at level N ′ . The operator Θ commutes with level-change, but the factor M −1 , which changes to (QM ) −1 , takes care of this. 4.2.3. The effect of Θ on FJ expansions. Let E be the standard cuspidal component ofS (over the ring R N ). As the reader probably noticed, we have trivialized the line bundle L along E on two different occasions in two seemingly different ways, that we now have to compare. On the one hand, after reducing modulo P and pulling L back to the Igusa surface, we got a canonical nowhere vanishing section a(1) trivializing L over Ig µ , and in particular along any of the p 2 − 1 cuspidal components lying over E in Ig µ . On the other hand, extending scalars from R N to C, shifting to the analytic category, restricting to the connected componentX Γ on which E lies, and then pulling back to the unit ball X, we have trivialized L| E by means of the section 2πidζ 3 , which we showed to be K N -rational.
Lemma 4.7. The sections a(1) and 2πidζ 3 "coincide" in the sense that they come from the same section in H 0 (E, L).
Proof. Let A be the universal semi-abelian variety over E. Its toric part is O K ⊗G m , hence, takingΣ-component of the cotangent space at the origin (4.24) L|Ẽ = ω A/Ẽ (Σ) = (δ −1 K O K ⊗ ω Gm )(Σ) admits the canonical section eΣ · (1 ⊗ dT /T ). Tracing back the definitions and using (1.69), this section becomes, under the base change R N ֒→ C, just 2πidζ 3 . On the other hand, when we reduce it modulo P and use the Igusa level structure ε at the standard cusp, it pulls back to the section "with the same name" eΣ · (1 ⊗ dT /T ), because alongẼ (4.19) induces the identity on cotangent spaces. The lemma follows from the fact that, by definition, ε * a(1) = eΣ · (1 ⊗ dT /T ) too.
Lemma 4.8. The sections a(1) p+1 and 2πidζ 2 ⊗ 2πidζ 3 "coincide" in the sense that they come from the same section in H 0 (E, P µ ⊗ L).
Proof. Let σ 2 (resp. σ 3 ) be the K N -rational section of P µ (resp. L) along E, which over C becomes the section 2πidζ 2 (resp. 2πidζ 3 ). We have just seen that modulo P, when we identifyẼ with E (via the covering map τ : Ig →S), σ 3 reduces to a(1). To conclude, we must show that the map (4.25) V : P/P 0 = P µ ≃ L (p) carries σ 2 to σ (p) 3 . This will map, under L (p) ≃ L p , to a(1) p . Along E the line bundles P µ and L are just the Σ-andΣ-parts of the cotangent space at the origin of the torus O K ⊗ G m , and σ 2 and σ 3 are the sections (4.26) σ 2 = e Σ · (1 ⊗ dT /T ), σ 3 = eΣ · (1 ⊗ dT /T ).
To prove part (ii) we argue as follows. Let g = f /a(1) k be the function on Ig µ obtained by trivializing the line bundle L. We have to study the FJ expansion along E of κ(dg)/a(1) p+1 , where κ is the map defined in (4.14). For that purpose we may restrict to a formal neighborhood ofẼ. This formal neighborhood is isomorphic, under the covering map τ : Ig µ →S µ , to the formal neighborhood S of E in S. We may therefore regard dg as an element of Ω 1 S . Now (4.27) κ : Ω 1 S → P µ ⊗ L is a homomorphism of O S -modules defined over R N so, having restricted to S, we may study the effect of κ on FJ expansions by embedding S C in a tubular neighborhoodS(ε) 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
(4.28) where q = e 2πiz/M and θ m is a theta function, so that θ m (u)q m is a section of N m along E (now over C). Then According to Corollary 2.17, κ(du) = 0, and κ(dz) = 2πidζ 2 ⊗ dζ 3 . It follows that Recalling that in characteristic p, 2πidζ 2 ⊗ 2πidζ 3 reduced to a(1) 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: (4.31) We next turn to part (i).

4.3.
A study of the theta operator along the supersingular locus.  Both F and V are everywhere of rank 3, which implies that their kernel and image are locally free direct summands. Moreover, ImF = ker V and ImV = ker F = ω A (p) /R . The maps F and V preserve the types Σ,Σ, but note that D (p) (Σ) = D(Σ) (p) etc. The principal polarization on A induces one on A (p) , and these polarizations induce symplectic forms (4.35) , where the second form is just the base-change of the first. For x ∈ D (p) , y ∈ D we have (4.36) F x, y = x, V y (p) .
As V F = F V = 0, the first relation implies that ImF and ImV are isotropic subspaces. So is ω A/R . The Gauss-Manin connection is an integrable connection It is a priori defined (e.g. in [Ka-O]) when R is smooth overF p , but we can define it by base change also when R is a nilpotent thickening of a point of S (see [Kob], where R is a local Artinian ring). Note however that if R = O S,s0 and R m = O S,s0 /m m+1 S,s0 (m ≥ 0), and if we extend scalars from R to R m we get by base change the connection (4.39) Rm is not an isomorphism. In fact, Ω 1 Rm is not R m -free. As a result the Kodaira-Spencer map (4.40) KS(Σ) : P Rm ⊗ Rm L Rm → Ω 1 Rm will not be an isomorphism over R m , as it is over R. It is nevertheless true that KS(Σ) induces an isomorphism (4.41) KS(Σ) : P Rm−1 ⊗ Rm−1 L Rm−1 ≃ Ω 1 Rm ⊗ Rm R m−1 . This follows from the fact that Ω 1 Rm , the only dependencies between these differentials occuring for i + j = m − 1). We shall need to deal only with the first infinitesimal neighborhood of a point, R = O S,s0 /m 2 S,s0 . In this case, D has a basis of horizontal sections. Indeed, R = F p [u, v]/(u 2 , uv, v 2 ) where u and v are local parameters at s 0 , and Ω 1 R = (Rdu + Rdv)/ udu, vdv, udv + vdu (p is odd). If x ∈ D and But if ∇x 1 = du ⊗ x 11 + dv ⊗ x 12 and ∇x 2 = du ⊗ x 21 + dv ⊗ x 22 then This means thatx is a horizontal section having the same specialization as x in the special fiber, so the horizontal sections span D over R by Nakayama's lemma.
Let e 1 , e 2 , f 3 , f 1 , f 2 , e 3 be any six horizontal sections over R, specializing to a basis of H 1 dR (A s0 /F p ). Let D 0 be theirF p -span. As we have just seen, (4.46) R ⊗F p D 0 = D and ∇ = d ⊗ 1. Since R d=0 =F p , it follows that D 0 = D ∇ , i.e. there are no more horizontal sections besides D 0 . Thus every x ∈ H 1 dR (A s0 /F p ) has a unique extension to a horizontal section x ∈ H 1 dR (A/R). There is a similar connection on D (p) . The isogenies F rob and V er, like any isogeny, take horizontal sections with respect to the Gauss-Manin connection to horizontal sections, e.g. if x ∈ D and ∇x = 0 then V x ∈ D (p) satisfies ∇(V x) = 0.
Remark 4.1. In the theory of Dieudonné modules one works over a perfect base. It is then customary to identify D with D (p) via x ↔ 1 ⊗ x. This identification is only σ-linear where σ = φ, now viewed as an automorphism of R. The operator F becomes σ-linear, V becomes σ −1 -linear and (4.36) reads F x, y = x, V y σ . With this convention F and V switch types, rather than preserve them.

4.3.2.
The Dieudonné module at a gss point. Assume from now on that s 0 ∈ Z ′ = S gss is a closed point of the general supersingular locus. We write D 0 for H 1 dR (A s0 /F p ). Lemma 4.9. 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 (iii) The vectors e 1 , e 2 , f 3 form a basis for the cotangent space ω A0/Fp . Hence e 1 and e 2 span P and f 3 spans L.
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 byB(3), cf [Bu -We] (3.2). The classification in loc. cit. Proposition 3.6 shows that the Dieudonné module of a µ-ordinary abelian variety is isomorphic toB(2) ⊕S, that of a gss abelian variety is isomorphic toB(3) and in the superspecial case we getB(1) ⊕ S 2 .

Infinitesimal deformations.
Let O S,s0 be the local ring of S at s 0 , m its maximal ideal, and R = O S,s0 /m 2 . This R is a truncated polynomial ring in two variables, isomorphic toF p [u, v]/(u 2 , uv, v 2 ).
As remarked above, the de Rham cohomology D = H 1 dR (A/R) has a basis of horizontal sections, and since D 0 =F p ⊗ R D, we may identify D ∇ canonically with D 0 . Grothendieck tells us that A/R is completely determined by A 0 and by the Hodge filtration ω A/R ⊂ D = R ⊗F p D 0 . Since A is the universal infinitesimal deformation of A 0 , we may choose the coordinates u and v so that (4.50) P = Span R {e 1 + ue 3 , e 2 + ve 3 }.
The fact that ω A/R is isotropic implies then that Consider the abelian scheme A (p) . It is not the universal deformation of A (p) 0 over R. In fact, the map φ : R → R factors as and therefore A (p) , unlike A, is constant: 0 , ∇ = d ⊗ 1, but this time the basis of horizontal sections can be obtained also from the trivalization of A (p) , and 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 4.9 continue to hold. Indeed, the matrix of V in the basis at s 0 prescribed by that lemma, continues to represent V over Spec(R) by "horizontal continuation". The matrix of F is then derived from the relation (4.36).
The condition V (L) = P (p) 0 , which is the "equation" of the closed subscheme Z ′ ∩ Spec(R) (see Proposition 3.9) means Lemma 4.10. Let s 0 ∈ S gss as above. Then the closed subscheme S gss ∩ Spec(R) is given by the equation u = 0.
Note that when we divide by ω A/R and project H 1 dR (A/R) to H 1 (A, O), e 1 + ue 3 dies, and the image e 3 of e 3 becomes a basis for the line bundle that we called L ∨ (ρ) = H 1 (A, O)(Σ). Recall the definition of κ given in (4.14), but note that this definition only makes sense over Spec(O S,s0 ) or its completion, where KS(Σ) is an isomorphism, and can be inverted.
Proposition 4.11. Let s 0 ∈ Z ′ = S gss . Choose local parameters u and v at s 0 so that in O S,s0 the local equation of Z ′ becomes u = 0. Then at s 0 , κ(du) has a zero along Z ′ .
Proof. Let i : Z ′ ֒→ 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 ′ , i * κ(du) = 0. It is enough to show that the image of κ(du) in the fiber at every point s of Z ′ near s 0 , vanishes. All points being alike, it is enough to do it at s 0 . In other words, we denote by κ 0 the map (4.56) κ 0 : Ω 1 S,s0 → P µ ⊗ L| s0 ≃ L p+1 | s0 . and show that κ 0 (du) = 0. We may now work over Spec(R), where R = O S,s0 /m 2 . It is enough to show that in the diagram (4.57) KS(Σ) maps the line sub-bundle P 0,R ⊗ L R onto Rdu. Once we have passed to the infinitesimal neighborhood Spec(R) we can replace the local parameters u, v by any two formal parameters for which u = 0 defines Z ′ ∩ Spec(R). We may therefore assume, in view of Lemma 4.10, that u and v have been chosen as in section 4.3.3. But then equation (4.55) shows that the restriction of KS(Σ) to Z ′ , i.e. the homomorphism i * KS(Σ), maps i * P 0 onto i * R · du ⊗ e 3 . This concludes the proof.

4.3.5.
A computation of poles along the supersingular locus. We are now ready to prove the following.
Proposition 4.12. Let k ≥ 0, and let f ∈ H 0 (S, L k ) be a modular form of weight k in characteristic p. Then Θ(f ) ∈ H 0 (S, L k+p+1 ).

Proof.
A priori, the definition that we have given for Θ(f ) produces a meromorphic section of L k+p+1 which is holomorphic on the µ-ordinary part S µ but may have a pole along Z = S ss . Since S is a non-singular surface, it is enough to show that Θ(f ) does not have a pole along Z ′ = S gss , the non-singular part of the divisor Z. Consider the degree p 2 − 1 covering τ : Ig → S, which is finite,étale over S µ and totally ramified along Z. Let s 0 ∈ Z ′ and lets 0 ∈ Ig be the closed point above it. Let u, v be formal parameters at s 0 for which Z ′ is given by u = 0, as in Theorem 3.23. As explained there we may choose formal parameters w, v ats 0 where w p 2 −1 = u (and v is the same function v pulled back from S to Ig). It follows that in Ω 1 Ig we have (4.58) du = −w p 2 −2 dw.
We now follow the steps of our construction. Dividing f by a(1) k we get a function g = f /a(1) k on Ig with a pole of order k alongZ, the supersingular divisor on Ig, whose local equation is w = 0. In O Ig,s0 we may write Applying the map κ (extended O Ig -linearly from S to Ig), and noting that κ(du) has a zero along Z ′ , hence a zero of order p 2 − 1 alongZ ′ , we conclude that κ(dg) has a pole of order k (at most) alongZ ′ . Finally Θ(f ) = a(1) k · κ(dg) becomes holomorphic alongZ ′ , and also descends to S. It is therefore a holomorphic section of P µ ⊗ L k+1 ≃ L k+p+1 .
It is amusing to compare the reasons for the increase by p+1 in the weight in Θ(f ) for modular curves and for Picard modular suraces. 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, hence a total increase by p + 1 = 2 + (p − 1) in the weight. In our case, the Kodaira-Spencer isomorphism is responsible for a shift by p + 1 (the p coming from P µ ≃ L p ), but the section turns out to be holomorphic along the supersingular locus. See Section 4.5.

4.4.
Relation to the filtration and theta cycles. In part (ii) of the main theorem we have described the way Θ acts on Fourier-Jacobi expansions at the standard cusp. A similar formula inevitably exists at any other cusp. We may deduce from it that modular forms in the image of Θ have vanishing FJ coefficients in degrees divisible by p. Moreover, for such a form f ∈ Im(Θ), Θ 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(Θ(f 1 )) = r(Θ(f 2 )). We may therefore define unambiguously (4.61) Θ(r(f )) = r(Θ(f )).
For any other i in this range (4.65) ω(Θ i+1 (f )) = ω(Θ i (f )) + p + 1. This is reminiscent of the "theta cycles" for classical (i.e. elliptic) modular forms modulo p, see [Se], [Ka2] and [Joc]. Recall that if f is a mod p modular form of weight k on Γ 0 (N ) with q-expansion a n q n (a n ∈F p ), then Θ(f ) is a mod p modular form of weight k + p + 1 with q expansion na n q n (Katz denotes Θ(f ) by Aθ(f )). One has ω(Θ(f )) < ω(f ) + p + 1 if and only if ω(f ) ≡ 0 mod p. In such a case we say that the filtration "drops" and we have (4.66) ω(Θ(f )) = ω(f ) + p + 1 − a(p − 1) for some a > 0. As a corollary, ω(f ) can never equal 1 mod p for an f ∈ Im(Θ).
Assume now that f ∈ Im(Θ) is a "low point" in its "theta cycle", namely, ω(f ) is minimal among all ω(Θ i (f )). Then ω(Θ i+1 (f )) < ω(Θ i (f )) + p + 1 for one or two values of i ∈ [0, p − 2], which are completely determined by ω(f ) mod p [Joc]. This is not true anymore for Picard modular forms. Not only the drop in the theta cycle is 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 Θ(f ) can not be determined by ω(f ) modulo p alone. Let f and k be as in the Proposition.
(3) If k < p + 1 but f / ∈ Im(Θ) then starting with Θ(f ) instead of f, one sees that the drop in the theta cycle of Θ(f ) occurs either in passing from Θ p−2 (f ) to Θ p−1 (f ), or in passing from Θ p−1 (f ) to Θ p (f ).

4.5.
Compatability between theta operators for elliptic and Picard modular forms.
4.5.1. 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, cf. [Se], but its geometric construction was given by Katz in [Ka1] and [Ka2], where it is denoted Aθ. Katz' construction relied on a canonical splitting of the Hodge filtration over the ordinary locus, but it coincides with a slightly modified construction, similar to the one we have been using over the Picard modular surface. This construction was suggested in [Gr], Proposition 5.8, see also [An-Go] in the Hilbert modular case.
Let X be the modular curve X(N ) overF p (N ≥ 3, p ∤ N ) and I ord the Igusa curve of level p lying over X ord = X −X ss , the ordinary part of X. LetX andĪ ord be the curves obtained by adjoing the cusps to X and I ord respectively. Let L = ω 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 L k over X. Let a(1) be the tautological nowehere vanishing section of L overĪ ord . Given a modular form f of weight k, we consider r(f ) = τ * f /a(1) k where τ :Ī ord →X is the covering map, and apply the inverse of the Kodaira-Spencer isomorphism KS : L 2 → Ω 1 I ord to get a section κ(dr(f )) of L 2 overĪ ord . When multiplied by a(1) k it descends toX ord , and when this is multiplied further by h = a(1) p−1 , the Hasse invariant for elliptic modular forms, it extends holomorphically over X ss to an element (4.67) θ(f ) = a(1) k+p−1 κ(dr(f )) ∈ Γ(X, L k+p+1 ).
4.5.2. An embedding of a modular curve inS. To illustrate our idea, and to simplify the computations, we assume that N = 1 and d K ≡ 1 mod 4, so that D = D K = d K . We shall treat only one special embedding of the modular curveX = X 0 (D) intoS (there are many more).
This embedding induces an embedding of symmetric spaces H ֒→ X, z → t (z, 0). One can easily compute that the intersection of Γ, the stabilizer of the lattice L 0 in G ′ ∞ , with SL 2 (R), is the subgroup of SL 2 (Z) given by Let E 0 = C/O K , endowed with the canonical principal polarization and CM type Σ. For z ∈ H let Λ z = Z + Zz and E z = C/Λ z . Let M z be the cyclic subgroup of order D of E z generated by D −1 z mod Λ z . Using the model (1.27) of the abelian variety A z associated to the point t (z, 0) ∈ X, we compute that with the obvious O K -structure. 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 O K ⊗ E z /δ K ⊗ M z obtained by descending the polarization (4.71) λ can : O K ⊗ E z → δ −1 K ⊗ E z = (O K ⊗ E z ) t of degree D 2 , modulo the maximal isotropic subgroup δ K ⊗ M z of ker(λ can ).
It is now clear that over any R 0 -algebra R we have the same moduli theoretic construction, sending a pair (E, M ) where M is a cyclic subgroup of degree D to A(E, M ), with O K structure and polarization given by the same formulae. This gives a modular embedding X → 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 . 4.5.3. Comparison of the two theta operators. From now on we work overF p . The modular interpretation of the embedding j :X →S allows us to complete it to a diagram Lemma 4.14. The pull-back j * ω A/S decomposes as a product ω E0 × (O K ⊗ ω E/X ). Under this isomorphism j * L = (O K ⊗ ω E/X )(Σ) (4.73) j * P 0 = ω E0 j * P µ ≃ (O K ⊗ ω E/X )(Σ).
The line bundle j * P 0 is constant, and P µ , originally a quotient bundle of P, becomes a direct summand when restricted toX.
Proof. This is straightforward from the construction of j, and the fact that E 0 is supersingular, while E is ordinary overX ord . Note that O K ⊗E/δ K ⊗M and O K ⊗E have the same cotangent space.
Proof. We abbreviate I ord by I and Ig µ by Ig. The pull-back via j of the tautological section a(1) of L over Ig is the tautological section a(1) of j * L = ω E/X . We therefore have (4.75) j * (dr(f )) = dr(j * (f )) (r(f ) = τ * f /a(1) k is the function on Ig denoted earlier also by g). It remains to check the commutativity of the following diagram (4.76) Here j * 0 is the map j * Ω 1 Ig → Ω 1 I 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, P/P 0 = P µ pulls back under j to L(ρ) (the line bundle L with the O K action conjugated). But, KS(Σ)(P 0 ⊗ L) maps under j * to the conormal bundle, so we obtain a commutative diagram [Bo] G. Boxer, Ph.D. Thesis, Harvard, to appear.