Siegel Varieties and p-Adic Siegel Modular Forms

To John Coates for his sixtieth birthday

J. Tilouine

Received: November 30, 2005 Revised: August 10, 2006

Abstract. In this paper, we present a conjecture concerning the classicality of a genus two overconvergent Siegel cusp eigenform whose associated Galois representation happens to be geometric, and more precisely, given by the Tate module of an abelian surface. This conjecture is inspired by the Fontaine-Mazur conjecture. It generalizes known results in the genus one case, due to Kisin, Buzzard-Taylor and Buzzard. The main difference in the genus two case is the complexity of the arithmetic geometry involved. This is why most of the paper consists in recalling (mostly with proofs) old and new results on the bad reduction of parahoric type Siegel varieties, with some consequences on their rigid geometry. Our conjecture would imply, in certain cases, a conjecture posed by H. Yoshida in 1980 on the modularity of abelian surfaces defined over the rationals.

2000 Mathematics Subject Classification:

11F33, 11F46, 11F80, 11G18 Keywords and Phrases:

Arithmetic Siegel varieties, q-expansion, Bad reduction of Siegel varieties of parahoric level, Overconvergent Siegel modular forms, Galois representations

Siegel Varieties and p-Adic Siegel Modular Forms 783

In a previous paper, we showed under certain assumptions (Theorem 4 of [26]) that a degree four symplectic Galois representation ρ with singular Hodge-Tate weights which is congruent to a cohomological modular Galois representation (we say then that ρ is residually cohomologically modular) is p-adically modular. The precise definitions of the expressions above can be found in [26] Sect.2 and 4. As a corollary, we obtain that certain abelian surfaces A_{/ℚ} do correspond, if they are residually cohomologically modular, to overconvergent Siegel cusp forms of weight (2, 2) (see Theorem 8 of [26]), in the sense that their Galois representations coincide. This result fits a Generalized Shimura-Taniyama Conjecture due to H. Yoshida ([30], Section 8.2) according to which for any irreducible abelian surface A defined over ℚ, there should exist a genus two holomorphic Siegel cusp eigenform g of weight (2, 2) such that L(h^{1}(A), s) = L_{spin}(g, s), where L(h^{1}(A), s) is the Grothendieck L function associated to the motive h^{1}(A) and L_{spin}(g, s) is the degree four automorphic L function associated to g (with Euler factors defined via Hecke parameters rather than Langlands parameters, for rationality purposes). One should notice that this conjecture presents a new feature compared to the genus one analogue. Namely, contrary to the genus one case, the weight (2, 2) occuring here is not cohomological; in other words, the Hecke eigensystem of g does not occur in the singular cohomology of the Siegel threefold (it occurs however in the coherent cohomology of this threefold). In particular, the only way to define the Galois representation ρ_{g,p} associated to such a form g, either classical or overconvergent, is to use a p-adic limit process, instead of cutting a piece in the étale cohomology with coefficients of a Siegel threefold. This can be achieved in our case because g fits into a two-variable Hida family of p-nearly ordinary cusp eigenforms. Note that, more generally, for a classical cusp eigenform g of weight (2, 2) with (finite) positive slopes for its Hecke eigenvalues at p, one believes that two-variable Coleman families of cusp eigenforms passing through g in weight (2, 2) could also be constructed, and this would allow a similar construction of ρ_{g,p}. For our p-nearly ordinary overconvergent g, Theorem 8 of [26] states that the associated Galois representation ρ_{g,p} does coincide with the p-adic realization of a motive h^{1}(A). Therefore, ρ_{g,p} is geometric; several results in the analogue situation for genus 1 (see [18], [6] and [7]) lead us to conjecture that this g is actually classical. The goal of the present paper is to generalize slightly and state precisely this conjecture (Sect.4.2). We also take this opportunity to gather geometric facts about Siegel threefolds with parahoric level p, which seem necessary for the study of the analytic continuation of such overconvergent cusp eigenforms to the whole (compactified) Siegel threefold; the rigid GAGA principle would then imply the classicity of such g. We are still far from fulfilling this program. However, we feel that the geometric tools presented here, although some of them can actually be found in the literature, may be useful for various arithmetic applications besides this one, for instance to establish the compatibility between global and local Langlands correspondence for cusp forms of parahoric level for GSp(4, ℚ). As a final remark, we should point out that there exist other Generalized Shimura-Taniyama Conjectures for submotives of rank 3 resp.4 of the motive h^{1}(A) for certain abelian threefolds resp. fourfolds A (see [3]). For those, Theorem 8 of [26] seems transposable; the question of classicity for the resulting overconvergent cusp eigenforms for unitary groups U(2, 1) resp. U(2, 2) could then be posed in a similar way. It would then require a similar study of the (rigid) geometry of Shimura varieties of parahoric type for the corresponding groups. Part of this paper has been written during visits at NCTS (Taiwan) and CRM (Montreal). The excellent working conditions in these institutions were appreciated. The author wishes to express his thanks to Professors Jing Yu and A. Iovita for their invitations, as well as the Clay Institute which financed part of the stay in Montreal. Discussions with H. Hida, A. Iovita, C.-F. Yu and especially A. Genestier were very useful to remove several falsities and add truths to an earlier draft (but the author alone is responsible for the remaining errors).

Contents

1 Notations 784

2 Integral models and local models 785

2.1 The case ∗ = B . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

2.1.1 The case ∗ = U_{B} . . . . . . . . . . . . . . . . . . . . . . 793 2.2 The case ∗ = P . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 2.3 The case ∗ = Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 2.4 Rigid geometry of Siegel varieties . . . . . . . . . . . . . . . . . 802

3 Siegel modular forms 804 3.1 Arithmetic Siegel modular forms and q-expansion . . . . . . . . 804 3.2 p-adic Siegel modular forms and q-expansion . . . . . . . . . . 808 3.3 Overconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . 810

4 Galois representations of low weight and overconvergent modular forms 812 4.1 Eichler-Shimura maps . . . . . . . . . . . . . . . . . . . . . . . 812 4.2 Galois representation associated to a cohomological cusp eigenform812 4.3 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

1 Notations

Let

G = GSp(4) = {X ∈ GL_{4};^{t}XJX = ν · J}

be the split reductive group scheme over ℤ of symplectic simitudes for the anti-

symmetric matrix J, given by its 2 × 2 block decomposition: J = ( \begin{matrix}0 & −s \\ s & 0 \end{matrix} )

where s is the 2 × 2 antidiagonal matrix whose non zero entries are 1. This group comes with a canonical character ν : X ↦ ν(X) ∈ 𝔾_{m} called the similitude factor. The center of G is denoted by Z, the standard (diagonal) maximal torus by T and the standard (upper triangular) Borel by B; U_{B} denotes its unipotent radical, so that B = T U_{B}. Let γ_{P} = t_{1}/t_{2} resp. γ_{Q} = ν^{−1}t_{2}^{2} be the short, resp. the long simple root associated to the triple (G, B, T). The standard maximal parabolic P = MU, associated to γ_{P}, is called the Klingen parabolic, while the standard maximal parabolic Q = M′U′, associated to γ_{Q}, is the Siegel parabolic. The Weyl group of G is denoted W_{G}. It is generated

by the two reflexions s_{P} and s_{Q} induced by conjugation on T by ( \begin{matrix}s & 0 \\ 0 & s \end{matrix} )

resp.( \begin{matrix}1 \\ & s \\ & & 1 \end{matrix} ) . Let us fix a pair of integers (a, b) ∈ ℤ^{2}, a ≥ b ≥ 0; we

identify it with a dominant weight for (G, B, T), namely the character

T ∋ t = diag(t_{1}, t_{2}, ν^{−1}t_{2}, ν^{−1}t_{1}) ↦ t_{1}^{a}t_{2}^{b}

Let V_{a,b} be a generically irreducible algebraic representation of G associated to (a, b) over ℤ. Let 𝔸 = 𝔸_{f} × ℚ_{∞} be the ring of rational adeles. Fix a compact open subgroup K of G_{f} = G(𝔸_{f}); let N ≥ 1 be an integer such that K = K^{N} × K_{N} with K^{N} = G(ℤ^{N}) maximal compact and K_{N} = ∏_{ℓ|N} K_{ℓ} for local components K_{ℓ} to be specified later. Let H^{N} be the unramified Hecke algebra outside N (that is, the tensor product algebra of the unramified local Hecke algebras at all prime-to-N rational primes); for each rational prime ℓ prime to N, one defines the abstract Hecke polynomial P_{ℓ} ∈ H^{N}[X] as the monic degree four polynomial which is the minimal polynomial of the Hecke Frobenius at ℓ (see Remarks following 3.1.5 in [12]). Let C_{∞} be the subgroup of G_{∞} = G(ℚ_{∞}) generated by the standard maximal compact connected subgroup K_{∞} and by the center Z_{∞}. For any neat compact open subgroup L of G(𝔸_{f}), the adelic Siegel variety of level L is defined as: S_{L} = G(ℚ)\G(𝔸)/LC_{∞}; it is a smooth quasi-projective complex 3-fold. If L ⊂ L′ are neat compact open subgroups of G_{f}, we have a finite etale transition morphism ϕ_{L,L′} : S_{L} → S_{L′}.

2 Integral models and local models

Let K be a compact open subgroup of G(ℤ̂) such that K(N) ⊂ K. For any integer M ≥ 1, we write K_{M} resp. K^{M} for the product of the local components of K at places dividing M, resp. prime to M. Let p be a prime not dividing N we denote by I, Π_{P} resp. Π_{Q} the Iwahori subgroup, Klingen parahoric, resp. Siegel parahoric subgroup of G(ℤ_{p}). We consider K_{B}(p) = K∩I ×K^{p}, K_{P} (p) = K∩Π_{P} ×K^{p} and K_{Q}(p) = K∩Π_{Q}×K^{p} and the corresponding Shimura varieties S_{B}(p), S_{P} (p) resp. S_{Q}(p). Let us consider the moduli problems

F_{∅} : ℤ[\frac{1}{N}]−Sch → Sets, S ↦ \{A, λ, \overline{η})_{/S}\}/ ∼,

F_{B} : ℤ[\frac{1}{N}]−Sch → Sets, S ↦ \{A, λ, \overline{η}, H_{1} ⊂ H_{2} ⊂ A[p])_{/S}\}/ ∼,

F_{P} : ℤ[\frac{1}{N}]−Sch → Sets, S ↦ \{A, λ, \overline{η}, H_{1} ⊂ A[p])_{/S}\}/ ∼

and

F_{Q} : ℤ[\frac{1}{N}]−Sch → Sets, S ↦ \{A, λ, \overline{η}, H_{2} ⊂ A[p])_{/S}\}/ ∼

where A/S is an abelian scheme, λ is a principal polarisation on A, \overline{η} is a Klevel structure (see end of Sect.6.1.1 of [12]), H_{i} is a rank p^{i} finite flat subgroup scheme of A[p] with H_{2} totally isotropic for the λ-Weil pairing. As in Th.6.2.1 of [12] or [16] Prop.1.2, one shows Theorem 1 If K is neat, the functors above are representable by quasiprojective ℤ[\frac{1}{N}]-schemes X_{∅}, X_{B}(p), X_{P} (p) and X_{Q}(p). The first one is smooth over ℤ[\frac{1}{N}] while the others are smooth away from p; the functors of forgetfulness of the level p structure provide proper morphisms π_{B,∅} : X_{B}(p) → X_{∅}, π_{P,∅} : X_{P} (p) → X_{∅}, and π_{Q,∅} : X_{Q}(p) → X_{∅} which are finite etale in generic fiber.

We’ll see that these morphisms are not necessarily finite hence not necessarily flat. We’ll also consider a moduli problem of level Γ_{1}(p). Let U_{B} be the unipotent radical of the Borel B of G. Let F_{U_{B}} be the functor on ℚ−Sch sending S to \{A, λ, \overline{η}, P_{1}, P_{2})_{/S}\}/ ∼ where P_{1} is a generator of a rank p finite flat subgroup scheme H_{1} of A[p] while P_{2} is a generator of the rank p finite flat group scheme H_{2}/H_{1} for H_{2} a lagrangian of A[p]. Over ℚ, it is not difficult to show that it is representable by a scheme X_{U_{B}}(p)_{ℚ}. Following [14] and [12] Sect.6.2.2, we define the ℤ[\frac{1}{N}]-scheme X_{U_{B}}(p) as the normalisation of X_{B}(p) in X_{U_{B}}(p)_{ℚ}; it comes therefore with a morphism π_{U_{B},B} : X_{U_{B}}(p) → X_{B}(p) which is generically finite Galois of group T (ℤ/pℤ).

Remark: All schemes above have geometrically connected generic fibers if and only if ν(K) = ℤ̂^{×}. However, in general, the morphisms π_{∗,∅} induce bijections between the sets of geometric connected components of X_{∗}(p) and X_{∅}; therefore the descriptions of irreducible components of the special fiber at p given below should be interpreted as relative to an arbitrary given connected component of the special fiber at p of X_{∅}. We still denote by X_{∗}(p) the base change to ℤ_{p} of X_{∗}(p)_{/ℤ| \frac{1}{N}]} (∗ = ∅, B, P, Q). The results that we will explain below are essentially due to de Jong [16], Genestier [11], Ngô-Genestier [22], Chai-Norman [9], C.-F. Yu [29]. As most of these authors, we make first use of the theory of local models [23], which allows to determine the local structure of X_{∗}(p); then, one globalizes using the surjectivity of the monodromy action due to [10]. This argument is sketched in [16] for g = 2 and developed for any genus and for any parahoric level structure in [29]. The determination of the local model and of its singularities has been done in case ∗ = B by de Jong [16], in case ∗ = P in [12] Sect.6.3 (inspired by [14]) and in case ∗ = Q in [12] Appendix. Let us recall the results.

2.1 The case ∗ = B

We first recall the definition of the local model M_{B} of X_{B}(p) over ℤ_{p}. Let St_{0} = ℤ_{p}^{4}, with its canonical basis (e_{0}, e_{1}, e_{2}, e_{3}), endowed with the standard unimodular symplectic form ψ: ψ(x, y) = ^{t}xJy. We consider the standard

diagram St_{2} →^{α_{2}} St_{1} →^{α_{1}} St_{0} where α_{i+1} sends e_{i} to pe_{i} and e_{j} to e_{j} (j ≠ i).

We endow St_{2} resp. St_{0} with the unimodular standard symplectic form ψ, which we prefer to denote ψ_{2} resp. ψ_{0}. Let α^{2} = α_{1} ∘ α_{2}; then we have ψ_{0}(α^{2}(x), α^{2}(y)) = pψ_{2}(x, y).

Then, M_{B} is the scheme representing the functor from ℤ_{p}−Sch to Sets sending a scheme S to the set of triples (ω_{i})_{i=0,1,2}, where ω_{i} is a direct factor of St_{i} ⊗O_{S}, ω_{0} and ω_{2} are totally isotropic, and α_{i+1}(ω_{i+1}) ⊂ ω_{i} for i = 0, 1. It is a closed subscheme of the flag variety over ℤ_{p} G(St_{2}, 2) × G(St_{1}, 2) × G(St_{0}, 2). Let \overline{ξ}_{0} = (\overline{ω}_{2}, \overline{ω}_{1}, \overline{ω}_{0}) ∈ M_{B}(𝔽_{p}) be the point given by \overline{ω}_{2} = ⟨e_{0}, e_{1}⟩, \overline{ω}_{1} = ⟨e_{0}, e_{3}⟩ and \overline{ω}_{0} = ⟨e_{2}, e_{3}⟩. Consider the affine neighborhood U of \overline{ξ}_{0} in M_{B} given by ω_{2} = ⟨e_{0} + c_{11}e_{2} + c_{12}e_{3}, e_{1} + c_{21}e_{2} + c_{22}e_{3}⟩, ω_{1} = ⟨e_{0} + b_{11}e_{1} + b_{12}e_{2}, e_{3} + b_{21}e_{1} + b_{22}e_{2}⟩ and ω_{0} = ⟨e_{2} + a_{11}e_{0} + a_{12}e_{1}, e_{3} + a_{21}e_{0} + a_{22}e_{1}⟩. We’ll see below that it is enough to study the geometry of U because this open set is “saturating” in M_{B} (i.e. its saturation G_{B}U for the action of the group G_{B} of automorphisms of M_{B} is M_{B}). Let us first study the geometry of U. The equations of U are c_{11} = c_{22}, a_{11} = a_{22}, pe_{1} + c_{21}e_{2} + c_{22}e_{3} = c_{22}(e_{3} + b_{21}e_{1} + b_{22}e_{2}), e_{0} + c_{11}e_{2} + c_{12}e_{3} = e_{0} + b_{11}e_{1} + b_{12}e_{2} + c_{12}(e_{3} + b_{21}e_{1} + b_{22}e_{2}), and similarly pe_{0} + b_{11}e_{1} + b_{12}e_{2} = b_{12}(e_{2} + a_{11}e_{0} + a_{12}e_{1}), e_{3} + b_{21}e_{1} + b_{22}e_{2} = e_{3} + a_{21}e_{0} + a_{22}e_{1} + b_{22}(e_{2} + a_{11}e_{0} + a_{12}e_{1}). Equating the coordinates of the two members, one gets the set of equations (2) of [16] Sect.5. Putting x = a_{11}, y = b_{12}, a = c_{12}, b = a_{12} and c = b_{22}, an easy calculation shows that U = spec ℤ_{p}[x, y, a, b, c]/(xy − p, ax + by + abc). The special fiber U_{0} ⊂ M_{B} ⊗ 𝔽_{p} of U is an affine threefold given by the equations xy = 0 and ax + by + abc = 0; it is the union of its four smooth irreducible components Z_{00} = V (x, b), Z_{01} = V (x, y + ac), Z_{10} = V (y, a) and Z_{11} = V (y, x + bc). Let R = ℤ_{p}^{ur}[x, y, a, b, c]/(xy − p, ax + by + abc); then \overline{ξ}_{0} has coordinates (0, 0, 0, 0, 0) in U_{0}(\overline{𝔽}_{p}). Let \overline{ζ}_{0} = (\overline{x}_{0}, \overline{y}_{0}, \overline{a}_{0}, \overline{b}_{0}, \overline{c}_{0}) be an arbitrary point of U_{0}(\overline{𝔽}_{p}). Note that \overline{x}_{0}\overline{y}_{0} = 0 and \overline{a}_{0}\overline{x}_{0}+\overline{b}_{0}(\overline{y}_{0}+\overline{a}_{0}\overline{c}_{0}) = \overline{b}_{0}\overline{y}_{0}+\overline{a}_{0}(\overline{x}_{0}+\overline{b}_{0}\overline{c}_{0}) = 0. Let 𝔪_{0} be the maximal ideal of R corresponding to \overline{ζ}_{0}. The completion of R at 𝔪_{0} is given by the following easy lemma ([16] Section 5).

Lemma 2.1 • If \overline{x}_{0} + \overline{b}_{0}\overline{c}_{0} ≠ 0, then if \overline{y}_{0} ≠ 0, R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[u, β, γ]],

• If \overline{x}_{0} + \overline{b}_{0}\overline{c}_{0} ≠ 0 and \overline{y}_{0} = 0, then R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[x, y, b, c]]/(xy − p),

• If \overline{a}_{0} ≠ 0, if \overline{y}_{0} = \overline{b}_{0} = 0 then R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[y, b, t, c]]/(ybt − p), and if \overline{y}_{0} ≠ 0 or \overline{b}_{0} ≠ 0, if \overline{y}_{0}\overline{b}_{0} = 0 then R̂_{𝔪_{0}} is ℤ_{p}^{ur}[[y, b, t, c]]/(yt − p), or it is smooth if \overline{y}_{0}\overline{b}_{0} ≠ 0,

• If \overline{c}_{0} ≠ 0 and \overline{x}_{0} = \overline{b}_{0} = \overline{a}_{0} = \overline{y}_{0} = 0, if moreover \overline{c}_{0} ≠ 0, then R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[x, y, u, v, w]]/(xy − p, uv − p),

• If \overline{x}_{0} = \overline{b}_{0} = \overline{a}_{0} = \overline{y}_{0} = \overline{c}_{0} = 0, that is, if s_{0} = x_{0} (defined above), then

R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[x, y, a, b, c]]/(xy − p, ax + by + abc), The other cases are brought back to those by permuting the variables x and y resp. a and b.

Proof: If \overline{x}_{0} + \overline{b}_{0}\overline{c}_{0} ≠ 0, and \overline{y}_{0} ≠ 0, we choose liftings x_{0}, a_{0}, b_{0}, c_{0} ∈ ℤ_{p}^{ur} and y_{0} ∈ ℤ_{p}^{ur}^{×} and introduce new variables u, α, β, γ by putting y = y_{0} + u and a = a_{0} + α, b = b_{0} + β, c = c_{0} + γ (in case \overline{b}_{0} = 0 for instance, we choose b_{0} = 0 so that β = b, and similarly for γ). Then, the relation ax + by + abc = 0 in R̂_{𝔪_{0}} reads a(x + bc) + by = 0, so that the image of the variable α can be expressed as a series of the images of the variables u, β, γ; similarly, the relation xy = p allows to express x as a series of u; in conclusion, we have R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[u, β, γ]]. If \overline{x}_{0} ≠ 0 = \overline{y}_{0} = 0, this reasoning shows that R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[x, y, β, γ]]/(xy − p). If \overline{a}_{0} ≠ 0, let us omit the centering at 0 of variables as above (needed for instance if \overline{b}_{0} ≠ 0 or \overline{y}_{0} ≠ 0). Let us write the relation ax + by + abc = 0 as x = −a^{−1}by − bc = b(−a^{−1}y − c). We introduce a new variable t = −a^{−1}y − c. Then we have p = xy = bty so that R̂_{𝔪_{0}} ≅ ℤ_{p}^{ur}[[y, b, t, c]]/(ybt − p) unless, as mentioned, \overline{b}_{0} ≠ 0 or \overline{y}_{0} ≠ 0 where things become simpler. If \overline{x}_{0} = \overline{b}_{0} = \overline{a}_{0} = \overline{y}_{0} = 0 but \overline{c}_{0} ≠ 0, then (x+bc)(y+ac) = p+c(ax+by+abc) = p; hence, putting u = x + bc and v = y + ac, one defines a change of variables from the set of variables (x, y, a, b, c) to (x, y, u, v, c) (actually, as above, one should use γ = c − c_{0} instead of c) and the conclusion follows. The last case is clear.QED.

By the theory of local models, we have a diagram

W_{B}
π ↙  ↘ f
X_{B}(p)  M_{B}

where W_{B} classifies quintuples (A, λ, H_{1}, H_{2}; ϕ : St_{·} ⊗ O_{S} ≅ D(A_{·})) over a scheme S (see Sect.3 of [16], especially Prop.3.6, for the definition of ϕ). One sees easily that it is representable by a X_{B}(p)-scheme π : W_{I} → X_{B}(p). The morphism f consists in transporting the Hodge filtration from the Dieudonné modules to St_{·} by ϕ and π consists in forgetting ϕ. Recall that those morphisms are smooth and surjective. Given a point z = (A_{0} → A_{1} → A_{2}, λ_{0}, λ_{2}; ϕ) of W_{B}(\overline{𝔽}_{p}), the degree p isogenies A_{0} → A_{1} → A_{2} (defined by quotienting A = A_{0} by H_{1} and H_{2}) give rise to morphisms of filtered Dieudonné modules (writing M_{i} for D(A_{i})_{S}): M_{2} → M_{1} → M_{0}, sending ω_{i+1} into ω_{i}. Let us consider the rank p finite flat group schemes G_{0} = H_{1} = Ker (A_{0} → A_{1}) and G_{1} = H_{2}/H_{1} : Ker (A_{1} → A_{2}). Then, we have a canonical isomorphism

1) ω_{i}/α(ω_{i+1}) ≅ ω_{G_{i}}. Recall that ω_{A_{i}^{∨}} = ω_{i}^{∨} = M_{i}/ω_{i}, hence by Th.1, Sect.15 of [20]), if G_{i}^{∨} denotes the Cartier dual of G_{i}, we have

2) ω_{G_{i}^{∨}} = M_{i}/(ω_{i} + α(M_{i+1})). For z ∈ W_{B}(\overline{𝔽}_{p}) as above, let x = π(z) = (A_{0} → A_{1} → A_{2}, λ_{0}, λ_{2}) and s = f(z) = (ω_{2}, ω_{1}ω_{0}). We define σ_{i}(s) = dim ω_{i}/α(ω_{i+1}) and τ_{i}(s) = dim M_{i}/(ω_{i} + α(M_{i+1})). Then, • if G_{i} is μ_{p}, σ_{i}(s) = 1 and τ_{i}(s) = 0

• if G_{i} is ℤ/pℤ, σ_{i}(s) = 0 and τ_{i}(s) = 1

• if G_{i} is α_{p}, σ_{i}(s) = 1 and τ_{i}(s) = 1

We define M_{B}(\overline{𝔽}_{p})^{ord} as the set of points s such that (σ_{i}(s), τ_{i}(s)) ∈ {(1, 0), (0, 1)} for i = 1, 2. One determines its four connected components and we check their Zariski closures are the irreducible components of M_{B}(\overline{𝔽}_{p}) as follows. The calculations of the lemma above show that M_{B}(\overline{𝔽}_{p}) ∩ U is the union of the loci

• (1) x = b = 0,

• (2) x = y + ac = 0,

• (3) y = a = 0,

• (4) y = x + bc = 0,

Then, let us check that the component x = b = 0 is the Zariski closure of the locus (m, m) where H_{1} and H_{2}/H_{1} are multiplicative. This component consists in triples (ω_{2}, ω_{1}, ω_{0}) such that the generators of ω_{0} satisfy a_{11} = a_{12} = 0, that is, by equations (1) of U_{0} in Sect.6 of [16], such that ω_{0} = ⟨e_{2}, e_{3}⟩. Then one sees that α(ω_{1}) = ⟨b_{12}e_{2}, e_{3} + b_{22}e_{2}⟩ has codimension 1 in ω_{0} if b_{12} = 0, and codimension 0 otherwise, while α(ω_{2}) = ⟨e_{0} + c_{11}e_{2} + c_{12}e_{3}, c_{21}e_{2} + c_{22}e_{3}⟩ has codimension 1 if c_{11} = 0 and 0 otherwise. On the other hand, α(M_{1}) is generated by (e_{1}, e_{2}, e_{3}) so M_{0}/α(M_{1}) is generated by the image of e_{0}; since ω_{0} = ⟨e_{2}, e_{3}⟩, we see that τ_{0}(s) = 1 for any s ∈ Z_{00}, while α(M_{2}) is generated by (e_{0}, e_{2}, e_{3}) so that M_{1}/α(M_{2}) is generated by the image of e_{1}; since ω_{1} = ⟨e_{0}+b_{12}e_{2}, e_{3}+b_{22}e_{2}⟩, we see that τ_{1}(s) = 1 also on Z_{00}. Hence the open dense locus defined by b_{12} ≠ 0 and c_{11} ≠ 0 is the ordinary locus of this component (that is, the set of points s such that (σ_{i}(s), τ_{i}(s)) = (0, 1) (i = 1, 2). One can do similar calculations for the other components; to obtain the table at bottom of page 20 of [16] (note however that our labeling of the components is different). This calculation proves the density of the ordinary locus in each irreducible component in U_{0} and provides at the same time the irreducible components of the non-ordinary locus and of the supersingular locus. We find

Lemma 2.2 The open subset U_{0} of M_{B} ⊗ 𝔽_{p} is an affine scheme with four irreducible components

• (1) x = b = 0, Zariski closure of the locus (m, m) where H_{1} and H_{2}/H_{1} are multiplicative

• (2) x = y + ac = 0, Zariski closure of the locus (m, e) where H_{1} is multiplicative and H_{2}/H_{1} is étale • (3) y = a = 0, Zariski closure of the locus (e, e) where H_{1} and H_{2}/H_{1}

are étale

• (4) y = x + bc = 0, Zariski closure of the locus (e, m) where H_{1} is étale and H_{2}/H_{1} is multiplicative.

The singular locus U_{0}^{sing} can be viewed as the union of two loci: “H_{1} biconnected”,whose equation is x = y = 0, and “H_{2}/H_{1} biconnected”, whose equation is y + ac = x + bc = 0. The intersection of those two is the supersingular locus U_{0}^{ssing}. The locus “H_{1} biconnected” is the union of U_{0}^{ssing} and two 2-dimensional irreducible components

• (14) the locus x = b = y = 0,equation of the Zariski closure of the locus where H_{1} is biconnected and H_{2}/H_{1} is multiplicative,

• (23) the locus y = x = a = 0, equation of the Zariski closure of the locus where H_{1} is biconnected and H_{2}/H_{1} is étale,

where the label (ij) denotes the irreducible 2-dimensional intersection of (i) and (j). The supersingular locus U_{0}^{ssing} coincides with the intersection (2) ∩ (4) which is the union of one 2-dimensional component x = y = c = 0, which we denote by (24) and one 1-dimensional component a = b = x = y = 0. The locus “H_{2}/H_{1} biconnected” is the union of U_{0}^{ssing} and of two irreducible components

• (12) x = b = y +ac = 0, equation of the Zariski closure of the locus where H_{1} is multiplicative and H_{2}/H_{1} is biconnected,

• (34) y = a = x+bc = 0, equation of the Zariski closure of the locus where H_{1} is étale and H_{2}/H_{1} is biconnected .

with the same convention (ij) = (i) ∩ (j) (here, those are irreducible 2dimensional components); Finally, the three irreducible components of the one-dimensional stratum associated to the four irreducible components of U_{0}^{sing} are

• x = y = a = b = 0,

• x = y = a = c = 0,

• x = y = b = c = 0,

They are all contained in U_{0}^{ssing}. More precisely, the second and third are contained in (24), and U_{0}^{ssing} is the union of the first and of (24). Thus, the supersingular locus of M_{B} is not equidimensional, it is union of a twodimensional irreducible component, namely the Zariski closure of the locus (24), and a one-dimensional irreducible component, closure of x = y = a = b = 0.

Let us consider the Iwahori group scheme G_{B}; it is a smooth group scheme over ℤ_{p} representing the functor S ↦ Aut_{S}(St_{·} ⊗ O_{S}). Its generic fiber is the symplectic group G while its special fiber is extension of the upper triangular Borel B by the opposite unipotent radical. The complete list of the G_{B}-orbits in M_{B} ⊗ 𝔽_{p} follows from the analysis above. There are thirteen such orbits. There are four 3-dimensional orbits (whose Zariski closures are the irreducible components), five 2-dimensional orbits, three 1-dimensional orbits, and one 0-dimensional orbit, intersection of all the closures of the other orbits. These orbits can be detected from the irreducible components as complement in an irreducible component of the union of the other components of smaller dimension. In [13] p.594, they are described in terms of thirteen alcoves in an apartment of the Bruhat-Tits building. Let us explain now the property of saturation of U: G_{B} · U = M_{B}. To prove this, we note that U_{0} meets all the orbits of G_{B} because it contains the smallest orbit, namely the point \overline{ξ}_{0} defined above and that this point is in the closure of all the other orbits. (cf. the remark of [11] above Lemma 3.1.1). This observation, together with the previous lemma implies [16], [22]

Proposition 2.3 The scheme M_{B} is flat, locally complete intersection over ℤ_{p}. Its special fiber is the union of four smooth irreducible components. Its ordinary locus coincides with the regular locus and is dense; the singular locus has 5 2-dimensional irreducible components, all smooth, and two one-dimensional irreducible components, also smooth; the p-rank zero locus has 3 irreducible components, all smooth; one is 2-dimensional and two are 1-dimensional.

The local and global geometry of X_{B}(p) is mostly contained in the following:

Theorem 2 The scheme X_{B}(p) is flat, locally complete intersection over ℤ_{p}. The ordinary locus in the special fiber coincides with the regular locus; it is therefore dense in the special fiber X_{B}(p) ⊗ 𝔽_{p}; this scheme is the union of four smooth irreducible components X^{mm}, X^{me}, X^{em}, X^{ee}. They are the Zariski closures of their ordinary loci, which are given respectively by the following conditions on the filtration 0 ⊂ H_{1} ⊂ H_{2} ⊂ A[p]: H_{2} is multiplicative, H_{1} is multiplicative and H_{2}/H_{1} étale, H_{1} is étale and H_{2}/H_{1} is multiplicative, H_{2} is étale. The singular locus of X_{B}(p) ⊗ 𝔽_{p} is therefore the locus where either H_{1} or H_{2}/H_{1} is étale-locally isomorphic to α_{p}. There exists a semistable model X̃_{B}(p) of X_{B}(p) over ℤ_{p} with a proper morphism h : X̃_{B}(p) → X_{B}(p) whose generic fiber h ⊗ ℚ_{p} is an isomorphism and whose special fiber h ⊗ 𝔽_{p} is an isomorphism over the ordinary locus.

Remark: The stratification of the special fiber of M_{B} by the G_{B}-orbits (called the Kottwitz-Rapoport stratification) defines also a stratification of the special fiber of X_{B}(p); the stratum X_{S} associated to the (irreducible) stratum S of M_{B} is defined as π(f^{−1}(S)). The four orbits corresponding to the irreducible components are connected because of the monodromy theorem of [10] (due to C.-F. Yu [29]). It has been pointed out to the author by A.Genestier that for the 2-dimensional orbits, no such connexity result is available yet by a p-adic monodromy argument. However, C.F. Yu explained to us how to prove that the p-rank one stratum does consist of four 2-dimensional irrreducible components as listed above for M_{B}^{sing}. Indeed, for any p-rank one geometric closed point x of X_{B}(p) ⊗ 𝔽_{p}, we have A_{x}[p] = G_{1,1}[p] × μ_{p} × ℤ/pℤ where G_{1,1} denotes the p-divisible group of a supersingular elliptic curve; hence the possibilities for the pairs (H_{1}, H_{2}/H_{1}) are (α_{p}, μ_{p}), (α_{p}, ℤ/pℤ), (μ_{p}, α_{p}), (ℤ/pℤ, α_{p}). This shows that the p-rank one stratum has exactly four connected components, so that the components of each type are irreducible. For the supersingular locus X_{B}(p)^{ss}, it is known by Li-Oort that the number of irreducible components is in general strictly greater than 3 (which is the number of irreducible components of M_{B}^{ss}). Proof: By [16] Sect.4, the morphisms π : W_{B} → X_{B}(p) and f : W_{I} → M_{B} are smooth and surjective and for any geometric point x of X_{B}(p), there exists a geometric point s ∈ f(π^{−1}({x}) of M_{B} and a local ring isomorphism

Ô_{X_{B}(p),x} ≅ Ô_{M_{B},s}

The description of the strictly henselian local rings Ô_{X_{B}(p),x} is therefore given by the list of Lemma 2.2. They are flat, complete intersection over ℤ_{p}^{ur}. The ordinary subcheme X_{B}(p)^{ord} of the special fiber 𝕏_{B}(p) ⊗ 𝔽_{p} is the locus where the connected component of A[p] is of multiplicative type. By total isotropy of H_{2} it follows easily that X_{B}(p)^{ord}(\overline{𝔽}_{p}) = π(f^{−1}(M_{B}^{ord})). Therefore, X_{B}(p)^{ord} is the disjoint union of four open subsets X^{mm,ord}, X^{me,ord}, X^{em,ord}, X^{ee,ord}, defined by the conditions: “the type of the pair (H_{1}, H_{2}/H_{1}) is (m, m) resp. (m, e), resp. (e, m), resp. (e, e), where m means multiplicative and e means étale”. Let us denote by X^{mm}, X^{me}, X^{em}, X^{mm} their Zariski closures in X_{B}(p) ⊗ 𝔽_{p}. By density of the ordinary locus, one has X_{B}(p) ⊗ 𝔽_{p} = X^{mm} ∪ X^{me} ∪ X^{em} ∪ X^{mm}. Let us show that these four subschemes are smooth irreducible. For i, j ∈ {0, 1}, let M_{B}^{αβ} ( α and β in {m, e}) be the irreducible components of M_{B} ⊗𝔽_{p} such that M_{B}^{αβ} ∩U_{0} is the component (α, β) in Lemma 2.2; then we have π(f^{−1}(M_{B}^{αβ})) = X^{αβ}. Thus, the smoothness of the components M_{B}^{αβ} of M_{B} ⊗ 𝔽_{p} yields the smoothness of X^{αβ} ∩ U_{0} for all α and β in {m, e}. The connectedness of X^{αβ} follows from a simple argument due to C.-F. Yu [29] which we repeat briefly, with a small correction (of the wrong statement (2.2) p.2595). let A → X_{∅} be the universal abelian variety; let X_{∅}^{o} be the ordinary locus of X_{∅}⊗𝔽_{p}; then for any closed geometric point \overline{x}, by Sect.V.7 of [10] the monodromy representation π_{1}(X_{∅}^{o}, \overline{x}) → GL_{g}(ℤ_{p}) is surjective; this is equivalent to saying that the finite étale X_{∅}^{o}-cover Ig(p) = Isom_{X_{∅}^{o}}(μ_{p}^{2}, A[p]^{o}) is connected. Consider the scheme Ig_{b}(p) = Isom_{X_{∅}^{o}}((μ_{p}^{2} ×(ℤ/pℤ)^{2}, A[p]) where the second member consists in symplectic isometries between the standard symplectic space (for the pairing given by the matrix J) and A[p] endowed with the Weil pairing.

By extension of isomorphisms between lagrangians to symplectic isometries, we see that Ig_{b}(p) is a purely inseparable torsor above Ig(p) under the group scheme μ_{p} ⊗ U(ℤ/pℤ) where U denotes the unipotent radical of the Siegel parabolic. Hence Ig_{b}(p) is connected. Now, for each connected component X^{αβ,ord} of X_{B}(p)^{ord}, one can define a finite surjective morphism Ig_{b}(p) → X^{o,αβ}. For instance for X^{me,ord}, we define a filtration inside μ_{p}^{2} × (ℤ/pℤ)^{2} by H_{1}^{me} = μ_{p} × 1 × 0 × 0 ⊂ H_{2}^{me} = μ_{p} × 1 × ℤ/pℤ × 0, and we define f^{me} as sending (A, λ, ξ) ∈ Ig(p) to (A, λ, 0 ⊂ ξ(H_{1}^{me}) ⊂ ξ(H_{2}^{me}) ⊂ A[p]) ∈ X^{me,ord}. This shows the connectedness of X^{me,ord}. A similar argument applies to the other components. The construction of the G_{B}-equivariant semistable model M̃_{B} of M_{B} has been done first by de Jong [16] by blowing-up M_{B} along either of the irreducible components (m, m) or (e, e), while Genestier constructs a semistable scheme L̃ by three consecutive blowing-ups of the lagrangian grassmannian L in such a way that the resulting scheme has an action of G_{B}; then he shows that the isomorphism from the generic fiber of L̃ to that of M_{B} extends to a proper morphism L̃ → M_{B}. He also shows [11] Construction 2.4.1 that the two constructions coincide: M̃_{B} = L̃. Then, both authors define X̃_{B}(p) as (W_{B} ×_{M_{B}} M̃_{B})/G_{B} (for its diagonal action). QED Remark: The previous calculations show also that the proper morphism π_{B,∅} is not finite over the supersingular locus C of X_{∅}, for instance the inverse image π_{B,∅}^{−1}(CSS) of the (zero dimensional) superspecial locus CSS ⊂ C coincides with the locus where the lagrangian H_{2} coincides with the lagrangian α_{p} × α_{p} of G_{1,1}[p] × G_{1,1}[p], and H_{1} ⊂ H_{2}; thus by [20] Sect.15, Th.2, the fiber of π_{B,Q} at each superspecial point of X_{Q}(p) is a projective line. On the other hand, the morphism π_{Q,∅} : X_{Q}(p) → X_{∅} is finite.

2.1.1 The case ∗ = U_{B}

Recall that U_{B} denotes the unipotent radical of B. The study of X_{U_{B}}(p) can be deduced from that of X_{B}(p) following the lines of [14] Sect.6, using Oort-Tate theory. More precisely, let W be the G_{B}-torsor considered above and W_{U} = f^{−1}(U) the inverse image of the affine open subset U of M_{B} (see beginning of 2.1). The locus where H_{1} and H_{2}/H_{1} are connected has equation x = b = 0. This locus can also be described by oort-Tate theory as follows. There exist two line bundles L_{1}, L_{2} on X_{B}(p) and two global sections u_{i} ∈ H^{0}(X_{B}(p), L_{i}^{⊗(p−1)}, i = 1, 2, together with scheme isomorphisms H_{1} ≅ Spec (O_{X_{B}(p)}[T]/(T^{p} − u_{1}T)), resp. H_{2}/H_{1} ≅ Spec (O_{X_{B}(p)}[T]/(T^{p} − u_{2}T)) such that the neutral sections correspond to T = 0; then the locus where H_{1} and H_{2}/H_{1} are connected is given by u_{1} = u_{2} = 0 in X_{B}(p). Moreover, the (ramified) covering X_{U_{B}}(p) → X_{B}(p) is defined by p − 1st roots t_{i} of u_{i}. More precisely, when L is a line bundle on a scheme X and u is a global section of L, one defines the scheme X[u^{1/n}] as the closed subscheme of \underline{Spec}_{X}(Symm^{•}L) given by the (well-defined) equation t^{n} = u; it is finite flat over X.

Hereafter, we pull back the line bundles and sections u_{i} to W_{U}. The divisor x = 0 has two irreducible components: x = b = 0 and x = y + ac = 0 along which u_{1} has a simple zero. Moreover, u_{1}/x is well defined and does not vanish on W_{U}. Similarly, u_{2}/(x + bc) is defined everywhere and does not vanish on W_{U}. By extracting p−1st roots of these nowhere vanishing sections, one defines an etale covering Z → W_{U}. Define Z_{U_{B}} = X_{U_{B}} ×_{X_{B}(p)} Z. On this scheme, the functions x and x + bc admit p − 1st roots. Moreover, one has a diagram analogue to the local model theory:

X_{U_{B}}(p) ← Z_{U_{B}} → U′ = U[f_{1}, f_{2}]/(f_{1}^{p−1} − x, f_{2}^{p−1} − (x + bc))

Lemma 2.4 The two morphisms of the diagram above are smooth and surjective. The scheme U′ is a local model of X_{U_{B}}(p).

Proof: The morphism Z → X_{B}(p) is smooth since it is the composition of an étale and a smooth morphism; the same holds therefore for its base change Z_{U_{B}} → X_{U_{B}}(p). The smoothness of the other morphism is proved in a similar way, noticing that one also has Z_{U_{B}} = Z ×_{U} U′. The surjectivity of W_{U} → X_{B}(p) (hence of Z_{U_{B}} → X_{U_{B}}(p)) follows because U is G_{B}-saturating. The surjectivity of Z_{U_{B}} → U′ comes from the surjectivity of W → M_{B}.

Corollary 2.5 The singular locus of the reduced irreducible components of X_{U_{B}}(p) is either empty or zero-dimensional.

Let T′ be the diagonal torus of the derived group G′ of G.

Proposition 2.6 The morphism π_{U_{B},B} : X_{U_{B}}(p) → X_{B}(p) is finite flat, generically étale of Galois group T′(ℤ/pℤ). The special fiber X_{U_{B}}(p) ⊗ 𝔽_{p} of X_{U_{B}}(p) has four irreducible components mapped by π_{U_{B},B} onto the respective irreducible components of X_{B}(p)⊗𝔽_{p}; each irreducible component of X_{U_{B}}(p)⊗𝔽_{p} has prime to p multiplicities and the singular locus of the underlying reduced subscheme of each component is at most zero dimensional.

One can also describe a local model of the quasisemistable scheme X̃_{U_{B}}(p) = X_{U_{B}}(p) ×_{X_{B}(p)} X̃_{B}(p). Namely, recall that the map M̃_{B} → M_{B} restricted to the affine subscheme U ⊂ M_{B} as before, is described (in de Jong’s approach) as the blowing-up of U along x = b = 0. It is the union of two charts V : (b, [x/b]) and V′ : (x, [b/x]); the first is more interesting as it is G_{B}-saturating in the blowing-up. In V, one has y = −([x/b] + c), hence after eliminating y, one finds a single equation for V in the affine space of a, b, c, [x/b], namely: p = −ab[x/b]([x/b] + c). Therefore the inverse image V_{U_{B}} of V in X̃_{U_{B}}(p) has equations

p = −ab[x/b]([x/b] + c), f_{1}^{p−1} = b.[x/b], f_{2}^{p−1} = b · ([x/b] + c)

This scheme is not regular, but has toric, hence mild, singularities. The restriction of Z̃_{U_{B}} above V provides again a diagram

X̃_{U_{B}}(p) ← Z̃_{U_{B},V} → V_{U_{B}}

with smooth and surjective arrows (for the left one, the surjectivity comes from the G_{B}-saturating character of V). Therefore, V_{U_{B}} is a local model of X̃_{U_{B}}(p).

2.2 The case ∗ = P

We follow the same method (see [12] Sect.6 for a slightly different proof). We keep the same notations (so p is prime to the level N of the neat group K). In order to study X_{P} (p) over ℤ_{p}, we consider the diagram of morphisms

W_{P}
π ↙  ↘ f
X_{P} (p)  M_{P}

W_{P} is the ℤ_{p}-scheme which classifies isomorphism classes of (A, λ, \overline{η}, H_{1}, ϕ) where ϕ : St_{·} ⊗ O_{S} → M_{·}(A) is an isomorphism between two diagrams. The first is St_{·} ⊗ O_{S}, ψ_{0} where St_{i} = ℤ_{p}^{4} (i = 0, 1) and the diagram St_{·} consists in the inclusion α_{1} : St_{1} → St_{0}, α_{1}(e_{0}) = pe_{0} and α_{1}(e_{i}) = e_{i} (i ≠ 0), and as before, ψ_{0} is the standard unimodular symplectic pairing on St_{0} given by J. The second is given by the inclusion of Dieudonné modules D(A_{1}) → D(A_{0}) associated to the p-isogeny A_{0} → A_{1} where A_{0} = A and A_{1} = A/H_{1}. Let G_{P} be the group scheme representing the functor S ↦ Aut_{S} (St_{·} ⊗ O_{S}); is is a smooth group scheme of dimension 11 over ℤ_{p} whose generic fiber is G and the special fiber is an extension of the Klingen parahoric P by the opposite unipotent radical. Then π : cW_{P} → X_{P} (p) is a G_{P} -torsor . The local model M_{P} is the projective ℤ_{p}-scheme classifying isomorphism classes of pairs (ω_{1}, ω_{0}) of rank 2 direct factors ω_{i} ⊂ St_{i} (i = 0, 1) such that α_{1}(ω_{1}) ⊂ ω_{0} and ω_{0} is totally isotropic for ψ_{0}. The map f send a point of W_{P} to the pair obtained by transporting the Hodge filtrations to St_{·} ⊗O_{S} via the isomorphism ϕ We introduce again an open neighborhood U of the point ξ_{0} = (\overline{ω}_{1}, \overline{ω}_{0}) in M_{P} with \overline{ω}_{1} = ⟨e_{0}, e_{3}⟩ and \overline{ω}_{0} = ⟨e_{2}, e_{3}⟩. Its importance, as in the Iwahori case, stems from the fact that it is G_{P} -saturating G_{P} U = M_{P} (same proof as above). It consists in the points (ω_{1}, ω_{0}) where ω_{1} = ⟨e_{0}+b_{11}e_{1}+b_{12}e_{2}, e_{3}+b_{21}e_{1}+b_{22}e_{2}⟩ and ω_{0} = ⟨e_{2} + a_{11}e_{0} + a_{12}e_{1}, e_{3} + a_{21}e_{0} + a_{22}e_{1}⟩. The condition α_{1}(ω_{1}) ⊂ ω_{0} yields the relations p = b_{12}a_{11}, b_{11} = b_{12}a_{12}, 0 = a_{21} + b_{22}a_{11} and b_{21} = a_{22} + b_{22}a_{12}. The isotropy relation yields a_{11} = a_{22}. By putting x = a_{11}, y = b_{12}, z = a_{12}, t = b_{22}, we find that U = spec R where R = ℤ_{p}[x, y, z, t]/(xy − p), so that for any maximal ideal 𝔪_{0} corresponding to (\overline{x}_{0}, \overline{y}_{0}, \overline{z}_{0}, \overline{t}_{0}) of U(\overline{𝔽}_{p}), the completion R̂_{𝔪_{0}} is ℤ_{p}^{ur}[[x, y, z, t]]/(xy − p), if \overline{x}_{0}\overline{y}_{0} = 0, and smooth otherwise. In any case, the local rings are ℤ_{p}-regular.

Via transitive action of G_{P} we conclude that M_{P} is semistable, with special fiber a union of two smooth irreducible components Z_{0} (locally: x = 0) and Z_{1} (locally: y = 0). In this situation, it is natural to consider only the maps

σ_{0} : s ↦ dim ω_{0}(s)/α_{1}(ω_{1}(s)) and τ_{0} : s ↦ dim M_{0}/ω_{0}(s) + α_{1}(M_{1})

as above; the regular locus M_{P}^{r} of M_{P} ⊗ 𝔽_{p} coincides with the locus where (σ_{0}(s), τ_{0}(s)) ∈ {(0, 1), (1, 0)}. As for ∗ = B, we conclude that

Theorem 3 The scheme X_{P} (p) is flat, semistable over ℤ_{p}. The ordinary locus in the special fiber is dense, strictly contained in the regular locus. The special fiber X_{B}(p) ⊗ 𝔽_{p} is the union of two smooth irreducible components X^{m} and X^{e} where X^{m} −X^{e} is the locus where H_{1} is multiplicative, and X^{e} −X^{m} is the locus where H_{1} is étale. The singular locus of X_{P} (p) ⊗ 𝔽_{p} is a smooth surface; it is the locus where H_{1} is étale-locally isomorphic to α_{p}.

The proof of the density of the ordinary locus is as follows. The forgetful morphism X_{B}(p) → X_{P} (p) sends the ordinary locus of X_{B}(p) onto the one of X_{P} (p); hence the density of the first implies that of of the second. The singular locus is the intersection of the two components; it is the locus where H_{1} is étale-locally isomorphic to α_{p}. Remark: We give an ad hoc proof of the density of the ordinary locus of X_{P} (p) ⊗ 𝔽_{p} in[12] Prop.6.4.2.

2.3 The case ∗ = Q

Again, the same method applies; however, in order to study X_{Q}(p) over ℤ_{p} and find a semistable model X̃_{Q}(p) → X_{Q}(p), we’ll first perform calculations in the flavor of de Jong’s method [16], as a motivation for Genestier’s approach ([11] Sect.3.3.0 and 3.3.3 and [12] Appendix) which we will follow and further a little. We consider the diagram of morphisms

W_{Q}
π ↙  ↘ f
X_{Q}(p)  M_{Q}

where π_{Q} : W_{Q} → X_{Q}(p) is the X_{Q}(p)-scheme classifying isomorphism classes of (A, λ, \overline{η}, H_{2}, ϕ) where ϕ : St_{·} ⊗ O_{S} → M_{·}(A) is a symplectic isomorphism between two diagrams. The first is St_{·} ⊗ O_{S}, ψ_{0}, ψ_{2} where St_{i} = ℤ_{p}^{4} (i = 0, 2) and the diagram St_{·} consists in the inclusion α^{2} : St_{2} → St_{0}, α^{2}(e_{i}) = pe_{i} (i = 0, 1) and α_{1}(e_{i}) = e_{i} (i > 1), and as before, ψ_{0} and ψ_{2} both denote the standard unimodular symplectic pairing on ℤ_{p}^{4} given by J. Note that α^{2} is a symplectic similitude of similitude factor p: ψ_{2}(α^{2}(x), α^{2}(y)) = p · ψ_{0}(x, y).

Let G_{Q} be the ℤ_{p}-group scheme of automorphisms of M_{Q}. It acts on W_{Q} as well and π_{Q} is a G_{Q}-torsor. Let L be the grassmannian of lagrangian direct factors in St_{0} over ℤ_{p}. Following [11] and [12] Appendix, we shall construct a G_{Q}-equivariant birational proper morphism L^{(2)} → L over ℤ_{p}, composition of two blowing-up morphisms along closed subschemes of the special fiber such that L^{(2)} is semistable and is endowed with a canonical G_{Q}-equivariant proper morphism h : L^{(2)} → M_{Q} (an isomorphism in generic fiber). We shall call h the Genestier morphism for (GSp_{4}, Q). For the easiest case (GSp_{2g}, P), see Prop.6.3.4. of [12]. As a motivation for the detailed construction below by two blowing-ups, we introduce the open subset U of M_{Q} consisting of pairs (ω_{2}, ω_{0}) ∈ M_{Q} where ω_{0} is spanned by e_{3} + a_{21}e_{0} + a_{11}e_{1} and e_{2} + a_{22}e_{0} + a_{12}e_{1} (with a_{12} = a_{21}) and ω_{2} = ⟨e_{1} + c_{21}e_{2} + c_{11}e_{3}, αe_{0} + c_{22}e_{2} + c_{12}e_{3}⟩ (with c_{12} = c_{21}), such that α^{2}(ω_{2}) ⊂ ω_{0}; it is therefore isomorphic to the affine set of A_{ℤ_{p}}^{6} consisting of pairs (A, C) of 2 × 2 symmetric matrices such that AC = p1_{2} by the map

(A, C) ↦ ( \begin{matrix}s \\ sC \end{matrix} ) , ( \begin{matrix}sA \\ s \end{matrix} )

Its special fiber has three irreducible components, given by A = 0, B = 0 and the Zariski closure of the locally closed set: rk A = rk B = 1. One then defines Ũ in M̃_{Q} as the quotient by 𝔾_{m} of the affine open set of triples (λ, A′, μ) such that A′ ≠ 0 is symmetric and λμdet A′ = p, the action of 𝔾_{m} being given by t · (λ, A′, μ) = ((tλ, t^{−1}A′, tμ). The map (λ, A′, μ) ↦ (A, C) given by A = λA′, C = μ^{t}com(A′) is the blowing-up of U along the component A = 0.

Remark: One checks easily that Ũ is also the blowing-up of U along C = 0. Hence the projection is invariant under the symmetry (A, C) ↦ (C, A). This allows the definition of an involution W on Ũ. This involution will extend to M̃_{B}. See after Prop. below. Note however that the following construction is dyssymmetrical, and does not make explicit use of the open set U defined above.

The first blowing-up L^{(1)} of the lagrangian grassmannian L over ℤ_{p} along the closure of Q · \overline{ω}_{23} where \overline{ω}_{23} is the 𝔽_{p}-lagrangian spanned by e_{2} and e_{3}. Note that by functoriality of the blowing-up, L^{(1)} is endowed with a natural action of G_{Q} (which acts on L through the canonical morphism G_{Q} → G and leaves the center of blowing-up stable). Namely, let us consider the affine open subset Ω_{0} of L consisting of the lagrangian planes ω_{0} = ⟨e_{3} + a_{11}e_{0} + a_{12}e_{1}, e_{2} + a_{21}e_{0} + a_{22}e_{1}⟩ (with a_{12} = a_{21}), the blowing-up L^{(1)}|Ω_{0} is the closed ℤ_{p}-subscheme of 𝔸^{3} × ℙ^{3} of points (a_{11}, a_{12}, a_{22}; [A_{11}, A_{12}, A_{22}, S]) such that

a_{11}A_{12} − a_{12}A_{11} = 0, a_{11}A_{22} − a_{22}A_{11} = 0, a_{12}A_{22} − a_{22}A_{12,} = 0

and

pA_{11} = a_{11}S, pA_{12} = a_{12}S, pA_{22} = a_{22}S.

The scheme L^{(1)}|Ω_{0} can be described as the quotient by 𝔾_{m} of the locally closed ℤ_{p}-subscheme T_{1} of the affine space 𝔸^{5} defined in terms of the coordinates (λ_{0}, P_{0}, A_{11}, A_{12}, A_{22}) as the intersection of the closed subscheme λ_{0}P_{0} = p with the complement of the closed subscheme P_{0} = A_{11} = A_{12} = A_{22} = 0. The action of 𝔾_{m} is given by multiplication by λ^{−1} on the first variable and by λ on the rest. Indeed, the quotient map T_{1} → L^{(1)}|Ω_{0} is

(λ_{0}, P_{0}, A_{11}, A_{12}, A_{22}) ↦ (a_{11}, a_{12}, a_{22}; [A_{11}, A_{12}, A_{22}, S])

where a_{11} = λ_{0}A_{11}, a_{12} = λ_{0}A_{12}, a_{22} = λ_{0}A_{22}, S = P_{0}. To take care of equation (1), following [11] Theorem, one forms the blowup L^{(2)} of L^{(1)} along the strict transform Z_{02}^{c,(1)} of the Zariski closure Z_{02}^{c} of Z_{02} = Q · \overline{ω}_{02} where \overline{ω}_{02} is the lagrangian spanned by e_{0} and e_{2}. The equations of L^{(2)}|Ω_{0} can be determined as follows. First, one notes that Z_{02}^{c,(1)}|Ω_{0} is given as a ℤ_{p}-subscheme of L^{(1)}|Ω_{0} by the equations A_{11}A_{22}−A_{12}^{2} = P_{0} = 0. Its inverse image in T_{1} is given by the same equations (this time, viewed in an affine space). Let δ = A_{11}A_{22} − A_{12}^{2}. Then, the blowing-up T^{(2)} of T_{1} along this inverse image is the subscheme of T_{1} × ℙ^{1} with coordinates (λ_{0}, P_{0}, A_{11}, A_{12}, A_{22}, [P_{1}, δ_{1}]) given by the equation δP_{1} = δ_{1}P_{0} (with (P_{1}, δ_{1}) ≠ (0, 0)). Introducing λ_{1} such that P_{0} = λ_{1}P_{1}, and δ = λ_{1}δ_{1}, one can rewrite T^{(2)} as the quotient by 𝔾_{m} of the affine locally closed subscheme T_{2} of 𝔸^{7} with affine coordinates (λ_{0}, λ_{1}, P_{1}, A_{11}, A_{12}, A_{22}, δ_{1}) and equations λ_{0}λ_{1}P_{1} = p and λ_{1}δ_{1} = A_{11}A_{22} − A_{12}^{2} in the open subset of 𝔸^{7} intersection of the locus (λ_{1}P_{1}, A_{11}, A_{12}, A_{22}) ≠ (0, 0, 0, 0) with (δ_{1}, P_{1}) ≠ (0, 0); the action of μ ∈ 𝔾_{m} being the trivial one on λ_{0} and A_{ij}, the multiplication by μ^{−1} on λ_{1} and the multiplication by μ on P_{1} and δ_{1}. The quotient map is

(λ_{0}, λ_{1}, P_{1}, A_{11}, A_{12}, A_{22}, δ_{1}) ↦ (λ_{0}, P_{0}, A_{11}, A_{12}, A_{22}, [P_{1}, δ_{1}])

with P_{0} = λ_{1}P_{1}. We can thus write L^{(2)}|Ω_{0} as a quotient T_{2}/𝔾_{m}^{2}, for the action of (λ, μ) ∈ 𝔾_{m}^{2} on (λ_{0}, λ_{1}, P_{1}, A_{11}, A_{12}, A_{22}, δ_{1}) ∈ T_{2} by multiplication by λ^{−1} on λ_{0}, μ^{−1} on λ_{1}, by λμ on P_{1}, by λ on A_{ij} and λ^{2}μ on δ_{1}. The ℤ_{p}-scheme T_{2} is clearly semistable. It implies by Lemme 3.2.1 of [11] that L^{(2)}|Ω_{0} is also semistable. Since G_{Q} · L^{(2)}|Ω_{0} = L^{(2)}, the same holds for L^{(2)}. Let us consider the forgetful morphism π_{0} : M_{Q} → L, (ω_{2}, ω_{0}) ↦ ω_{0}; the open subset U′′ = π^{−1}(Ω_{0}) ⊂ M_{Q}. This open set is not affine, it is dyssymmetrical, it contains the affine open set U defined above. We can now define the Genestier morphism h on L^{(2)}|Ω_{0}. It is given by the 𝔾_{m}^{2}-invariant map

T_{2} → U′′, (λ_{0}, λ_{1}, P_{1}, A_{11}, A_{12}, A_{22}, δ_{1}) ↦ (ω_{2}, ω_{0})

where ω_{0} is given by a_{ij} = λ_{0}A_{ij} and ω_{2} is given in terms of its Plücker coordinates on the basis (e_{0} ∧ e_{1}, e_{0} ∧ e_{2}, e_{0} ∧ e_{3}, e_{1} ∧ e_{2}, e_{1} ∧ e_{3}, e_{2} ∧ e_{3}): [δ_{1}, P_{1}A_{11}, −P_{1}A_{12}, P_{1}A_{12}, P_{1}A_{22}, λ_{1}P_{1}^{2}]. This point of ℙ^{5} is well defined because if δ_{1} = 0, we have P_{1} ≠ 0 and if λ_{1} = 0, one of the A_{ij} ≠ 0. It is invariant by the action of 𝔾_{m}^{2} hence factors through L^{(2)}|Ω_{0}. Moreover it corresponds to an isotropic plane because the third and fourth coordinates are opposite. By [11] Sect.3 before Lemme3.1.1, the saturation of Ω_{0} under G_{Q} is L, hence by G_{Q} equivariance, it is defined everywhere on L^{(2)}. One sees easily the surjectivity of h restricted to L^{(2)}|Ω_{0} onto U′′ (which consists of points in ℙ^{5} [u_{0}, u_{1}, u_{2}, −u_{2}, u_{3}, u_{4}] such that u_{0}u_{4} = u_{1}u_{3} −u_{2}^{2}), hence by G_{Q}-equivariance, to the whole of M_{Q}.

Definition 2.7 We put M̃_{Q} = L^{(2)}, it is a semistable ℤ_{p}-scheme; its special fiber has three smooth irreducible components. We define X̃_{Q}(p) = (W_{Q} × M̃_{Q})/G_{Q}; it is a semistable model of X_{Q}(p) over ℤ_{p} with smooth irreducible components; their number is at least three. It comes with a proper birational morphism h_{X} : X̃_{Q}(p) → X_{Q}(p) which we call the Genestier morphism which is an isomorphism on the generic fiber.

What precedes is a developed version of [12] Appendix, which may be useful to non expert algebraic geometers. We give now some new information on h and h_{X}. For any geometric point s = (ω_{2}, ω_{0}) of the special fiber of M_{Q}, let k = k(s) be the residue field; we define

σ(s) = dim ω_{0}/α^{2}(ω_{2}), τ(s) = dim M_{0}/(α^{2}(M_{2}) + ω_{0})

Let x = (A, λ, H_{2}) be a geometric point of X_{Q}(p) corresponding to s. Note that σ(s) is the p-rank of the connected component H_{2}^{0} of the group scheme H_{2}, while τ(s) is the p-rank of the connected component of the Cartier dual H_{2}^{t} of H_{2}. It can be identified by the Weil pairing to A[p]/H_{2}. From this it is easy to verify that the condition

(Ord) (σ(s), τ(s)) ∈ {(0, 2), (2, 0), (1, 1)}

is equivalent to the ordinarity of the point x. Let M_{Q}^{ord} be the locus where (Ord) is satisfied. Then the ordinary locus of X_{Q}(p)^{ord} of X_{Q}(p) × 𝔽_{p} is equal to π(f^{−1}(M_{Q}^{ord}). We have a partition M_{Q}^{ord} = M_{Q}^{ee,ord}⊔M_{Q}^{mm,ord}⊔M_{Q}^{em,ord}. corresponding to the conditions (σ(s), τ(s)) ∈ {(0, 2), resp. (σ(s), τ(s)) ∈ {(2, 0), resp. (σ(s), τ(s)) ∈ {(1, 1). Similarly, by taking the inverse images in M̃_{Q} by h, we can define a similar partition of \overline{M}_{Q}^{ord}:

M̃_{Q}^{ord} = M̃_{Q}^{ee,ord} ⊔ M̃_{Q}^{mm,ord} ⊔ M̃_{Q}^{em,ord}.

Let M̃_{Q}^{reg} resp. M̃_{Q}^{sing} be the regular locus resp. singular locus of the special fiber of M̃_{Q}. Let us determine the locus M_{Q}^{ee,ord} ∩ U′′ where (σ(s), τ(s)) = (0, 2) in U′′, together with its inverse image M̃_{Q}^{ee,ord}|Ω_{0} by h. The condition τ(s) = 2 translates as a_{11} = a_{12} = a_{22} = 0; this implies λ_{0} = 0. On the other hand, σ(s) = 0 implies, using Plücker coordinates, that λ_{1}P_{1} ≠ 0. One checks easily that actually (σ(s), τ(s)) = (0, 2) if and only if λ_{0} = 0 and λ_{1}P_{1} ≠ 0. In particular,M̃_{Q}^{ee,ord}|Ω_{0} coincides with the (smooth) irreducible component λ_{0} = 0 deprived from M̃_{Q}^{sing}; moreover, h induces an isomorphism between M̃_{Q}^{ee,ord}|Ω_{0} and M_{Q}^{ee,ord} ∩ U′′. Similarly for the locus M_{Q}^{mm} ∩ U′′ where (σ(s), τ(s)) = (2, 0) in U′′; the condition τ(s) = 0 is given by the equation a_{11}a_{22} − a_{12}^{2} ≠ 0, that is, λ_{0}^{2}λ_{1}δ_{1} ≠ 0; while σ(s) = 2 implies P_{1} = 0. Conversely, one sees easily that (σ(s), τ(s)) = (2, 0) if and only if P_{1} = 0 and λ_{0}λ_{1} ≠ 0. Therefore,M̃_{Q}^{mm,ord}|Ω_{0} coincides with the smooth irreducible component P_{1} = 0 minus M̃_{Q}^{sing}. Finally, we consider the locus M_{Q}^{em}∩U′′ where (σ(s), τ(s)) ∈ {(1, 1)} in U′′. We see that τ(s) = 1 is equivalent to a_{11}a_{22} −a_{12}^{2} = 0 and (a_{11}, a_{12}, a_{22}) ≠ (0, 0, 0), that is, λ_{0}^{2}λ_{1}δ_{1} = 0 and (λ_{0}A_{ij} ≠ (0, 0, 0). While σ(s) = 1 implies λ_{1}P_{1}^{2} = 0. Conversely, one sees easily that (σ(s), τ(s)) ∈ {(1, 1)} if and only if λ_{1} = 0 and λ_{0}P_{1} ≠ 0. In other words M̃_{Q}^{em,ord}|Ω_{0} coincides with the smooth irreducible component λ_{1} = 0 minus M̃_{Q}^{sing}. In the three cases, one deduces also from the previous calculations that h induces an isomorphism between M̃_{Q}^{αβ,ord}|Ω_{0} and M_{Q}^{αβ,ord} ∩ U′′.

We define then the Zariski closures M_{Q}^{αβ} of M_{Q}^{αβ,ord} and M̃_{Q}^{αβ} of M̃_{Q}^{αβ,ord}. Using G_{Q}-equivariance, we define X̃^{αβ} as (W_{Q}×M_{Q}^{αβ})/G_{Q} for all α, β ∈ {e, m} (with the convention that em = me ) We can then conclude

Theorem 4 The scheme X_{Q}(p) is flat, locally complete intersection over ℤ_{p}. The ordinary locus in the special fiber is dense in every irreducible component; it is contained in the regular locus. The special fiber X_{Q}(p) ⊗ 𝔽_{p} is the union of three irreducible components X^{mm} and X^{me} and X^{em} which are the Zariski closures respectively of the locus where H_{2} is of multiplicative type, the locus where, locally for the étale topology, H_{2} = μ_{p} × ℤ/pℤ and the locus where H_{2} is étale. The singular locus of X_{Q}(p) ⊗ 𝔽_{p} is the locus where H_{2} étale-locally contains α_{p}. There is a semistable model together with a blowing-up morphism h_{X} : X̃_{Q}(p) → X_{Q}(p) whose center is in the special fiber; the special fiber of X̃_{Q}(p) consists of three smooth irreducible components X̃^{mm}, X̃^{me} and X̃^{em} crossing transversally. The ordinary locus X̃_{Q}(p)^{ord} coincides with the regular locus X̃_{Q}(p)^{reg}.The restriction of h_{X} induces proper surjective morphisms X̃^{αβ} → X^{αβ} (α, β ∈ {e, m}) which are isomorphisms between the respective ordinary loci.

The irreducibility of the components X^{αβ} follows from [29] as explained above. This implies the irreducibility of the three components X̃^{αβ} because h_{X} is an isomorphism between the two dense open subsets X̃^{αβ,ord} and X^{αβ,ord}, the latter being irreducible. Remark: Note that we have thus recovered part of the results of [9]; however, this paper contains extra informations: the singular locus of X_{Q}(p) ⊗ 𝔽_{p} coincides with the finite set of superspecial abelian surfaces (that is, the cartesian products of supersingular elliptic curves); these isolated singularities are Cohen-Macaulay. The description of the intersections two by two and of the three components is given in Sect.6.2 there. Finally, we introduce an involution W of the ℤ_{p}-schemes X_{Q}(p) and \overline{X}_{Q}(p) compatible with h_{X}. The automorphism of the functor F_{Q} given by (A, λ, η, H_{2}) ↦ (\overline{A}, \overline{λ}, \overline{η}, \overline{H}_{2}) where \overline{A} = A/H_{2}, \overline{λ}, resp. \overline{η} is the quotient polarization resp. Γ-level structure on \overline{A} deduced from λ resp. η and \overline{H}_{2} = A[p]/H_{2}, induces an involution of the ℤ[1/N]-scheme X_{Q}(p), hence of its pull-back to ℤ_{p}. If one writes the test objects as (α : A_{0} → A_{2}, η_{0}, η_{2}) where A_{i}’s are principally polarized abelian varieties, α is an isogeny with lagrangian kernel in A[p] respecting the polarizations and the Γ-level structures η_{i} on A_{i}, we see that the involution W can be written as the duality α ↦ ^{t}α followed by the identifications of the dual abelian varieties ^{t}A_{i} to A_{i}; hence W maps (α : A_{0} → A_{2}, η_{0}, η_{2}) to (^{t}α : A_{2} → A_{0}, η_{2}, η_{0}). This involution W therefore extends to the torsor W_{Q} by replacing the diagram M_{·}(A) = (M(α) : M(A_{2}) → M(A_{0})) by its dual M(^{t}A) = (M(^{t}α) : M(A_{0}) → M(A_{2})) and by interchanging the two isomorphisms ϕ_{0} and ϕ_{2} in the isomorphism of diagrams ϕ : St_{·} ⊗ O_{S} → M_{·}(A) to obtain ϕ′ : St_{·} ⊗ O_{S} → M_{·}(A). The involution W on W_{Q} is compatible with the forgetful morphism W_{Q} → M_{Q} where W on M_{Q} is given by taking the dual of α^{2} : St_{2} → St_{0} with respect to the standard symplectic pairings ψ_{0} and ψ_{2}, and exchanging ω_{0} and ω_{2}. Hence, the involution acts on the diagram X_{Q}(p) ← W_{Q} → M_{Q}. Remark: By taking symplectic bases, its matricial interpretation is

( \begin{matrix}0 & −s \\ p · s & 0 \end{matrix} ); note that this matrix normalizes the automorphism group

G_{Q} of the diagram St_{·}. The involution W exchanges the two extreme irreducible components X^{ee} and X^{mm} of X_{Q}(p) ⊗ 𝔽_{p} and it leaves the intermediate component X^{em} stable. Remarks:

1) There is another construction of the morphism h : M̃_{Q} → M_{Q} by noticing that the restriction of h above the open subset U introduced at the beginning of the present section coincides with the map Ũ → U defined above and is G_{Q}-equivariant. Since U, Ũ and h|Ũ is symmetric under (A, C) ↦ (C, A); W extends by G_{Q}-action to an involution of M̃_{Q}-still denoted W, compatible to h. We thus obtain an involution W of the ℤ_{p}-scheme X̃_{Q}(p) = (W_{Q} × M̃_{Q})/G_{Q} compatible to h_{X} : X̃_{Q}(p) → X_{Q}(p); it exchanges the irreducible components X̃^{ee} and X̃^{mm} of X̃_{Q}(p) ⊗ 𝔽_{p} and leaves X̃^{em} stable.

2) Genestier’s construction [11] of the semistable model M̃_{B} of the local model M_{B} of X_{B}(p) in a way similar to that of M̃_{Q} implies that the forgetful morphism M_{B} → M_{Q}, (ω_{2}, ω_{1}, ω_{0}) ↦ (ω_{2}, ω_{0}) extends to the semistable models M̃_{B} → M̃_{Q}; an easy argument provides then a canonical morphism between the Genestier models X̃_{B}(p) → X̃_{Q}(p). However, it should be noted that the morphism M_{B} → M_{Q} is NOT a local model of the morphism X_{B}(p) → X_{Q}(p). This is already false for the case of the classical modular curve X_{0}(p) and the classical modular curve X of level prime to p. Finally, note that as explained in the case ∗ = Q, there is a Fricke-Weil involution W on X_{B}(p); it extends to the semistable models and the forgetful morphism π̃_{B,Q} is compatible with W.

2.4 Rigid geometry of Siegel varieties

We gather here some informations concerning the rigid geometry of the Siegel varieties X = X_{∅} and X_{Q}(p). Some (Prop.2.6, 2) are used in the formulation of the conjecture of Sect.4.3. We hope to develop them in another paper for studying analytic continuation of overconvergent Siegel cusp eigenforms. Let X^{rig}, X^{∗,rig} resp. \overline{X}^{rig} be the rigid analytic space associated to the p-adic completion of its corresponding ℤ_{p}-scheme (for the toroidal compactification, we assume throughout this section that we fixed a fine Γ-admissible polyedral cone decomposition Σ). Choosing a Γ_{Q}(p)-admissible refinement Σ′ of Σ, one can define a smooth toroidal compactification \overline{X}_{Q}(p)_{/ℚ_{p}} of the ℚ_{p}-scheme X_{Q}(p) ⊗ ℚ_{p} (actually, by [10], it exists as a proper smooth scheme over ℤ[\frac{1}{Np}]). Because of the compatibility of Σ and Σ′, we see that the forgetful morphism π = π_{Q∅} : X_{Q}(p) → X extends uniquely as a morphism \overline{π} : \overline{X}_{Q}(p) → \overline{X}. Let \overline{X}_{Q}(p)^{rig} be the rigid space over ℚ_{p} corresponding to the scheme \overline{X}_{Q}(p)_{/ℚ_{p}} (cf. Chapter 9, Ex.2 of [5]). Let \overline{X} be the formal completion of \overline{X} along the special fiber. The ordinary locus \overline{X}^{ord} is an open formal subscheme of \overline{X}; let \overline{X}^{rig,ord} be the corresponding admissible rigid open subset of \overline{X}^{rig}. Let \overline{X}_{Q}(p)^{rig,ord} be the inverse image of \overline{X}^{rig,ord} by \overline{π}^{rig}. We want to describe the connected components of this admissible rigid open set and strict neighborhoods thereof, in terms of a suitable model of \overline{X}_{Q}(p)^{rig}. For this purpose, we write simply X_{G} for the semistable model X̃_{Q}(p) of X_{Q}(p) over ℤ_{p}. We briefly explain the construction of a “toroidal compactifcation of X_{G}” associated to Σ′, by which we mean a proper regular ℤ_{p}-scheme \overline{X}_{G} together with a toroidal open immersion X_{G} ↪ \overline{X}_{G} such that \overline{X}_{G} ⊗ ℚ_{p} is the (smooth) toroidal compactification \overline{X}_{Q}(p)_{/ℚ_{p}} associated to Σ′ mentioned above. Details on this construction, specific to the genus 2 case, should appear in the thesis of a student of A. Genestier. The model of \overline{X}_{Q}(p)^{rig} that we are looking for is then defined as the formal completion \overline{X}_{G} of \overline{X}_{G} along the special fiber. The construction is as follows. One first takes the normalization of the ℤ_{p}- toroidal compactification \overline{X} associated to Σ, in the finite étale morphism X_{Q}(p)_{/ℚ_{p}} → X_{/ℚ_{p}}. Let \overline{X}_{Q}(p)^{Σ} be this normalization. The morphism X_{G} → X_{Q}(p) is an isomorphism outside the supersingular locus X_{Q}(p)^{ss} and this locus is proper (because we are in genus 2). We can therefore glue the schemes \overline{X}_{Q}(p)^{Σ} and X_{G} along their common open subscheme X_{Q}(p)\X_{Q}(p)^{ss}. We obtain a ℤ_{p}-scheme denoted \overline{X}_{G}^{Σ}. Let Z(Σ′)_{/ℚ_{p}} be the closed subscheme of \overline{X}_{Q}(p)_{/ℚ_{p}}^{Σ} which is the center of the blowing-up morphism

\overline{X}_{Q}(p)_{/ℚ_{p}} = \overline{X}_{Q}(p)_{/ℚ_{p}}^{Σ′} → \overline{X}_{Q}(p)_{/ℚ_{p}}^{Σ}

We consider the Zariski closure Z(Σ′) of Z(Σ′)_{/ℚ_{p}} in the ℤ_{p}-scheme \overline{X}_{G}^{Σ}. The

blowing-up of \overline{X}_{G}^{Σ} along Z(Σ′) is the desired scheme. It is denoted \overline{X}_{G}; by restricting the construction to the local charts of Faltings-Chai, it can be proven that \overline{X}_{G} is regular over ℤ_{p} and that X_{G} ↪ \overline{X}_{G} is toroidal, although the divisor at infinity doesn’t have good reduction.

Remark: For the sake of completion, let us mention another abstract construction. Let X_{G} be the formal completion of X_{G} along the special fiber. One can apply the notion of normalization studied in[4] to define the “normalization” \overline{X} _{G}^{(U_{i})} of X_{G} along \overline{X}_{Q}(p)^{rig} associated to an admissible affinoid cover of \overline{X}_{Q}(p)^{rig} (we denote by U_{i} the formal scheme associated to the affinoid U_{i}). The ℤ_{p}-formal scheme \overline{X} _{G}^{(U_{i})} is endowed with an open immersion of formal schemes X_{G} → \overline{X} _{G}^{(U_{i})}. However, this construction does depend on the choice of the covering. This is why the specific construction described above is better suited for our purpose.

We still denote by \overline{π} the morphism \overline{X}_{G} → \overline{X} as well as its p-adic completion \overline{X}_{G} → \overline{X}. We define the ordinary locus \overline{X} _{G}^{ord} as the inverse image in \overline{X}_{G} of the ordinary locus \overline{X}^{ord} of \overline{X}. We observe that \overline{X} _{G}^{ord} is smooth. Its underlying 𝔽_{p}-scheme is denoted by \overline{X}_{G}^{ord}. Let X̃^{αβ,ord} (α, β ∈ {e, m}) be the three connected components of X_{G}⊗𝔽_{p}. We denote by \overline{X}_{G}^{αβ,ord} the Zariski closure of X̃^{αβ,ord} in \overline{X}_{G}^{ord}. We have a partition into three smooth open subschemes

\overline{X}_{G}^{ord} = \overline{X}_{G}^{mm,ord} ⊔ \overline{X}_{G}^{me,ord} ⊔ \overline{X}_{G}^{ee,ord}

Therefore, by taking the inverse image by the specialization map associated to the model \overline{X}_{G}, we obtain three connected components of the open admissible subset \overline{X}_{Q}(p)^{rig,ord}:

\overline{X}_{Q}(p)^{rig,ord} =]\overline{X}_{G}^{mm,ord}[⊔]\overline{X}_{G}^{me,ord}[⊔]\overline{X}_{G}^{ee,ord}[

We need to extend this to admissible quasi-compact neighborhoods of \overline{X}_{Q}(p)^{rig,ord}. First we fix a lifting E of the Hasse invariant (see [15] Sect.3, or see next section below). Let G^{rig} → \overline{X}^{rig} be the rigid analytification of the semi-abelian scheme G → \overline{X} (as in Chap.9, ex.2 of [5]). By a Theorem of Abbès and Mokrane [1] Prop.8.2.3 (and [2] for an improved radius of convergence), the open subdomain \overline{X}^{rig}(p^{−a}) of \overline{X}^{rig} defined as the locus where the lifting E of the Hasse invariant satisfies |E|_{p} > p^{−a} (a = \frac{1}{p(p−1)} for [1], and a = \frac{p−1}{2p−1} for [2]) is endowed with a finite flat group scheme C_{can} of rank p^{2} whose restriction to the ordinary locus is canonically isomorphic to G[p]^{0}. For each r ∈]p^{−a}, 1[∩p^{ℚ}, we define

\overline{X}\{r\} = \{x ∈ \overline{X}^{rig}(L); |E|_{p} ≥ r\}

These domains are admissible, quasi-compact relatively compact neighborhoods of \overline{X}_{Q}(p)^{rig,ord} (cf.[19] Sect.3.1.6). Let \overline{X}_{Q}(p)\{r\} be the inverse image of \overline{X}\{r\} by \overline{π}^{rig}.

Proposition 2.8 1) For any r sufficiently close to 1, the neighborhood \overline{X}_{Q}(p)\{r\} has still three connected components denoted \overline{X}_{G}^{αβ}\{r\} (α, β ∈ \{e, m\}); \overline{X}_{G}^{αβ}\{r\} is defined as the largest connected subset of \overline{X}_{Q}(p)\{r\} containing ]\overline{X}_{G}^{αβ,ord}[.

2) For any r ∈]p^{−a}, 1[, the isomorphism ]\overline{X}_{G}^{mm,ord}[≅ \overline{X}^{rig,ord} induced by the forgetful morphism extends to an isomorphism \overline{X}_{G}^{mm,rig}\{r\} ≅ \overline{X}\{r\} (the inverse morphism being given by the canonical subgroup).

Proof: Since we won’t need the first part of the proposition, we won’t prove it in this paper. For the second statement, which is crucial to our conjecture, we notice that by definition, the morphism \overline{π} sends \overline{X}_{G}^{mm}\{r\} into \overline{X}\{r\} while the inverse map is provided by the canonical subgroup as in [1] Prop.8.2.3. Finally, we note that the involution W extends to the toroidal compactifications hence defines an involution of \overline{X}_{Q}(p)^{rig} which exchanges ]\overline{X}_{G}^{mm,ord}[ and ]\overline{X}_{G}^{ee,ord}[ resp. \overline{X}_{G}^{mm}\{r\} and \overline{X}_{G}^{ee}\{r\} and leaves stable the middle component ]\overline{X}_{G}^{em,ord}[ resp. \overline{X}_{G}^{em}\{r\}.

Finally, we can consider in a similar way the extension to compatible toroidal compactifications \overline{X}_{U_{B}}(p) and \overline{X}_{B}(p) of the morphisms π_{U_{B},B} and π_{B,Q}. We shall consider the inverse image by

π_{B,Q} ∘ π_{U_{B},B} : \overline{X}_{U_{B}}(p)^{rig} → \overline{X}_{Q}(p)^{rig}

of \overline{X}_{G}^{mm}\{r\}.

3 Siegel modular forms

3.1 Arithmetic Siegel modular forms and q-expansion

In [26], care has been taken to define the arithmetic Siegel varieties and modular forms adelically. However, here for simplicity, we restrict our attention to one connected component X corresponding to a discrete subgroup Γ ⊂ Sp_{4}(ℤ). We assume that X has a geometrically connected model over ℤ[1/N]. We also assume that Γ is neat, so that the problem of classifying principally polarized abelian surfaces with Γ-level structure is a fine moduli problem (if it is not the case, see [26] Section 3 where X is only a coarse moduli problem). Let f : A → X be the universal principally polarized abelian surface with Γlevel structure η over ℤ[1/N]. We put \underline{ω} = e^{∗}Ω_{A/X}, where e denotes the unit section. For any pair of integers κ = (k, ℓ) (k ≥ ℓ), we consider the rational representation of GL(2): W_{κ}(ℚ) = Sym^{k−ℓ} ⊗ det^{ℓ} St_{2}. Here, St_{2} denotes the standard two-dimensional representation of GL(2); the standard Levi M of the Siegel parabolic of Sp_{4} is identified to GL(2) by (4.1.1) U ↦ diag(U, s^{t}U^{−1}s) The twist by s occurs because our choice of the symplectic matrix J defining G involves the matrix s instead of 1_{2}. We use (4.1.1) to identify M to GL(2). Let B_{M} = T N_{M} be the Levi decomposition of the standard Borel of M (corresponding to the group of upper triangular matrices in GL(2)). In order to define integral structures on the space of Siegel modular forms, it will be useful to consider an integral structure of W_{κ}(ℚ). Since there is in general an ambiguity for such an integral structure, we need to make our choice explicit: following [15] Sect.3, we take it to be the induced ℤ-module W_{κ} = Ind_{B_{M}}^{M} κ. For any ring R, we put W_{κ}(R) = W_{κ} ⊗ R. Let T = Isom_{X}(O_{X}^{2}, \underline{ω}) be the right GL(2)-torsor over X of isomorphisms ϕ : O_{X}^{2} → \underline{ω}. By putting ω_{1} = ϕ((1, 0)) and ω_{1} = ϕ((0, 1)), it can also be viewed as the moduli scheme classifying quintuples (A, λ, η, ω_{1}, ω_{2}) where A, λ is a principally polarized abelian varieties with a Γ level structure η over a base S, endowed with a basis (ω_{1}, ω_{2}) of \underline{ω}_{A/S}. One writes π : T → X for the structural map. Note that π_{∗}O_{T} carries a left action (by right translation) of GL(2). Then, for any κ = (k, ℓ) ∈ ℤ^{2}, one defines the locally free sheaf \underline{ω}^{κ} over X as (π_{∗}O_{T})^{N_{M}} [κ^{−1}]. Its sections are functions on T such that for any ϕ ∈ Isom_{X}(O_{X}^{2}, \underline{ω}), for any t ∈ T and any n ∈ N_{M}, f(A, λ, η, ϕ ∘ tn) = κ(t)^{−1}f(A, λ, η, ϕ). One sees easily that π^{∗}\underline{ω}^{κ} = W_{κ}(O_{T}), so that \underline{ω}^{κ} is a locally free sheaf which is non zero if and only if k ≥ ℓ.

We briefly recall some notations concerning toroidal compactifications, canonical extensions of sheaves and q-expansions. It will allow us in particular to define the cuspidal subsheaf \underline{ω}_{κ} of the canonical extension of \underline{ω}^{κ}. For any ring R, let S_{2}(R) be the module of symmetric 2 × 2-matrices with entries in R. Recall that the bilinear form T r : S_{2}(ℝ) × S_{2}(ℝ) → ℝ identifies

the dual of S_{2}(ℤ) to the module S of matrices ( \begin{matrix}a & \frac{1}{2}b \\ \frac{1}{2}b & c \end{matrix} ), a, b, c ∈ ℤ.

Let S_{2}(ℝ)^{+} be the cone of definite positive matrices in S_{2}(ℝ) and S̃_{2} the cone of semi-definite positive matrices whose kernel is ℚ-rational. A standard rational boundary component of level N is a pair (Z, ϕ : \frac{1}{N} Z/Z → (ℤ/Nℤ)^{r}) where Z is a free non zero quotient of ℤ^{2} (of rank r) and ϕ is an isomorphism. Let us view ℤ^{2} as the standard lagrangian ⟨e_{1}, e_{2}⟩ of ℤ^{4} endowed with the symplectic pairing^{t}xJy, Then, a general rational boundary component of level N is the image of a standard one by the action of Sp_{4}(ℤ) on the space of lagrangians and on the projective space of ℤ^{4}. We denote by RBC_{1}, RBC_{N}, resp. SRBC_{1}, SRBC_{N}, the set of rational boundary components, resp. the set of standard rational boundary components. We can partition S̃_{2} ∩ S_{2}(ℤ) as ⊔_{Z∈SRBC_{1}}S(Z)^{+} where S(Z)^{+} denotes the set of semidefinite symmetric matrices of S_{2}(ℤ) which induce a positive definite quadratic form on Z. Let Σ = {Σ_{Z}}_{Z∈RBC_{1}} be an Sp_{4}(ℤ)-admissible family of rational polyhedral cone decompositions Σ_{Z} of S(Z)^{+} (see [8] Chapt.I Def.5.8.2). As explained in [10] p.126, this decomposition can be used for any level N congruence subgroup Γ, since it is a fortiori Γ-admissible. To Σ, one can associate a toroidal compactification \overline{X} over ℤ[\frac{1}{N}] of X as in [10] IV.6.7; it is smooth if Σ is sufficiently fine; this is assumed in the sequel. The compactification \overline{X} carries a degenerating semi-abelian scheme G extending A (see [10] Th.IV.5.7 and IV.6.7). One still denotes by \underline{ω} the sheaf e^{∗}Ω_{G/\overline{X}} where e is the unit section of G → \overline{X}. Recall that \overline{X} is a projective smooth, geometrically connected scheme over ℤ[\frac{1}{N}]. It is endowed with a projection map b to the minimal compactification X_{ℤ[ \frac{1}{N}]}^{∗}. Let D = \overline{X}\X = b^{−1}(∂X^{∗}); it is a relative Cartier divisor with normal crossings; its irreducible components are smooth. The rank two vector bundle \underline{ω} over \overline{X} does not descend as a vector bundle on X^{∗}; however its determinant ω = det \underline{ω} descends as an ample line bundle. The GL(2)-torsor \overline{T} = Isom\overline{X}(O2 ; \overline{X}, \underline{ω}) (with structural map \overline{π} : \overline{T} → \overline{X}) allows to define “the canonical extension” of the vector bundles \underline{ω}^{κ} to \overline{X}: one can either define this extension as

\underline{ω}^{κ} = (\overline{π}_{∗}O\overline{T})^{N_{M}} [κ^{−1}]

(k ≥ ℓ). Or one can also use the ℤ-structure W_{κ} = Ind_{B_{M}}^{M} κ of the rational representation W_{κ}(ℚ) of GL(2) in order to give an equivalent definition of

\underline{ω}^{κ} as the sections of the \overline{X}-vector bundle \overline{T} ×^{GL_{2}} W_{κ}; here, as usual, the contraction product is the quotient of the product by the equivalence relation (ϕ ∘ g, w) ∼ (ϕ, g · w) for any ϕ ∈ \overline{T} , g ∈ GL_{2} and w ∈ W_{κ}. For details see [10] Chapter 4 and 6, [21] Sect.4 and [15] Sect.3.

Let \underline{ω}_{κ} = \underline{ω}^{κ}(−D) the sub-vector bundle of \underline{ω}^{κ} on \overline{X} whose sections vanish along D. Recall the Koecher principle: H^{0}(\overline{X} ⊗ ℂ, \underline{ω}^{κ}) = H^{0}(X ⊗ ℂ, \underline{ω}^{κ}). We define

Definition 3.1 For any ℤ[1/N]-algebra R one defines the R-module of arithmetic Siegel modular forms resp. cusp forms, as H^{0}(X ⊗ R, \underline{ω}^{κ}) resp. H^{0}(\overline{X} ⊗ R, \underline{ω}_{κ}) which we write also H^{0}(X ⊗ R, \underline{ω}_{κ}) by convention.

For R = ℂ, these vector spaces canonically identify to the corresponding spaces of classical Siegel modular forms of level Γ and weight κ (see [15] Th.3.1). The arithmetic q-expansion (at the ∞ cusp) is defined as follows. Let η = (Z, ϕ) ∈ SRBC_{N} with Z = ℤ^{2} and with ϕ the canonical identification \frac{1}{N} ℤ^{2}/ℤ^{2} = ℤ/Nℤ^{2} (it is called the infinity cusp). Consider the rational polyhedral cone decomposition (RPCD) Σ_{η} of S_{2}(ℝ)^{+} corresponding to η. Let D_{η} = D ∩ b^{−1}({η}). By definition, the completion of \overline{X} along D_{η} admits an open cover by affine formal schemes U_{σ} (σ ∈ Σ_{η}) with a canonical surjective finite etale cover ϕ_{σ} : S_{σ} → U_{σ} where S_{σ} = Spf ℤ[1/N][[q^{T}; T ∈ S ∩σ^{∨}]]. The morphism ϕ_{σ} is Galois; its group is the stabilizer \overline{Γ}_{σ} of σ in the image \overline{Γ} of Γ ∩ Q by the projection Q → Q/U = M. Recall that M(ℤ) = GL(2, ℤ) acts on S_{2}(ℤ)^{+} by g · S = gS^{t}g. Moreover, ϕ_{σ} is uniquely determined by the property that the pull-back by ϕ_{σ} of the restriction of G to U_{σ} is the canonical Mumford family

f_{σ} : G_{σ} → S_{σ}

deduced by Mumford’s construction (see [10] p.54) from the canonical degenerescence data in DD_{ample} on the global torus G̃_{σ} = G_{m}^{2} over S_{σ}, together with the standard level N structure μ_{N}^{2} × (ℤ/Nℤ)^{2} → G̃_{σ}[N]; Given f ∈ H^{0}(\overline{X}, ω^{κ}), for any rational polyedral cone σ, we restrict f to U_{σ} and pull it back to S_{σ} by ϕ_{σ}. The bundle \underline{ω}_{G_{σ}/S_{σ}} of the Mumford family is trivial, hence the pull-back of the torsor \overline{T} to S_{σ} is trivial too; it is isomorphic to S_{σ} × GL(2). In consequence, ϕ_{σ}^{∗}\underline{ω}^{κ} is the trivial bundle W_{κ} ⊗ S_{σ}. Hence ϕ_{σ}^{∗}f yields a series in W_{κ}[[q^{T}; T ∈ S ∩ σ^{∨}]] which is invariant by \overline{Γ} where the action of γ ∈ \overline{Γ} is given by γ · (∑_{T} a_{T} q^{T}) = ∑_{T} ρ_{κ}(γ)(a_{T})q^{γT^{t}γ}. These series are compatible when one varies the cone σ either by restricting to its faces of by letting \overline{Γ} act (this action permutes the cones in Σ_{η}); recall that ⋂_{σ∈Σ_{η}} σ^{∨} = S̃_{2}; this implies that there exists one well-defined series which

belongs to the intersection W_{κ}[[q^{T}; T ∈ S∩S̃_{2}]] of the rings W_{κ}[[q^{T}; T ∈ S∩σ^{∨}]] and which is fixed by \overline{Γ}. It is called the q-expansion or Fourier expansion (at the infinity cusp) of f:

F E(f) ∈ W_{κ}[[q^{T}; T ∈ S ∩ S̃_{2}]]\overline{Γ}

For any ℤ[1/N]-algebra R and any form f ∈ H^{0}(X × R, ω^{κ}) defined over R, one defines an analogue series F E_{R}(f) with coefficients in W_{κ}(R) = W_{κ} ⊗ R.

Proposition 3.2 1) (q-expansion principle) If f is any form defined over R, if the coefficients of its q-expansion vanish in W_{κ}(R), then f = 0.

2) The map F E sends the submodule of cusp forms over any ring R to the submodule of W_{κ}(R)[[q^{T}; T ∈ S ∩S̃_{2}]]\overline{Γ} of series whose coefficients a_{T} ∈ W_{κ}(R) vanish unless T ∈ S ∩ S_{2}(ℝ)^{+}.

The first point follows from the irreducibility of the modular scheme; the second from the examination of ϕ_{σ}^{∗}(f) along ϕ_{σ}^{∗}D.

Remark: By comparing the two definitions of ω^{κ} given above, one sees that

W_{κ}[[q^{T}; T ∈ S ∩ σ^{∨}]] = (\overline{π}_{∗}O\overline{T})^{N_{M}} [κ^{−1}] ⊗_{O\overline{X}} ℤ[[q^{T}; T ∈ S ∩ σ^{∨}]]

We shall use this when comparing q-expansion of classical forms to q-expansion of p-adic forms.

3.2 p-adic Siegel modular forms and q-expansion

Let X as in the previous subsection. We fix a fine Γ-admissible family of rational polyedral cone decompositions Σ_{ξ}; we denote by For any integer m ≥ 1, let X_{m} be the pull-back of X to ℤ/p^{m}ℤ. Let S_{m} be the ordinary locus and for each n ≥ 1, consider T_{m,n} = Isom_{S_{m}}(μ_{p^{n}}^{2}, A[p^{n}]^{0}) = Isom_{S_{m}}(A[p^{n}]^{et}, (ℤ/p^{n}ℤ)^{2}); for any n ≥ 1, T_{m,n} is a connected Galois cover of S_{m} of Galois group GL_{2}(ℤ/p^{n}ℤ) (see [10] Prop.7.2). Let V_{m,n} = H^{0}(T_{m,n}, O_{T_{m,n}}), V_{m,∞} = ⋃_{n≥1} V_{m,n}. One can define the Σ- ”toroidal compactification” \overline{S}_{m} of S_{m} as the locus of \overline{X}_{m} over which G[p]^{0} is of multiplicative type; similarly, define \overline{T}_{m,n} as

Isom_{\overline{X}⊗ℤ/p^{m}ℤ}(μ_{p^{n}}^{2}, G[p^{n}]^{0})

We still denote by D the pull-back to \overline{T}_{m,n} of the divisor at ∞. We can now define V_{!,m,n} = H^{0}(\overline{T}_{m,n}, O_{\overline{T}_{m,n}}(−D)) and V_{!,m,∞} = ⋃_{m} V_{!,m,n}. We also consider the corresponding p-adic limits: S_{∞} = \underrightarrow{lim} S_{m}, T_{∞,∞} = \underrightarrow{lim} T_{m,∞}, V = \underleftarrow{lim} V_{m,∞} and V_{!} = \underleftarrow{lim} V_{!,m,∞}. These last two spaces are respectively the space of generalized p-adic modular forms resp. cusp forms. Let M resp. N_{M} be the group of ℤ_{p}-points of M = GL_{2} resp. N_{M} the unipotent radical of the standard Borel B_{M} of M. Then, T_{∞,∞} → S_{∞} is a right M étale torsor, hence M acts on the left (by right translations) on V (and V_{!}) by m · f(ψ) = f(ψ ∘ m). Let LC(M/N_{M}, ℤ/p^{m}ℤ) resp. C(M/N_{M}, ℤ_{p}) be the ring of ℤ/p^{m}ℤ-valued locally constant, resp. ℤ_{p}-valued continous functions on M/N_{M}, viewed as a left M-module via the left translation action. In particular, these modules are \overline{Γ}-modules. Note that C(M/N_{M}, ℤ_{p}) = projlim LC(M/N_{M}, ℤ/p^{m}ℤ). Let us define now the p-adic q-expansion map. It is a ring homomorphism

FE : V^{N_{M}} → (C(M/N_{M}, ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]])\overline{Γ}

given as follows. For the infinity cusp η defined above, and for any σ ∈ Σ_{η}, we consider the base change ϕ_{σ,m} of the morphism ϕ_{σ} : S_{σ} → U_{σ} to ℤ/p^{m}ℤ. As noticed above, the canonical Mumford family f_{σ} : G_{σ} → S_{σ} admits a canonical rigidification ψ_{can} : μ_{p^{∞}}^{2} ≅ G_{σ}[p^{∞}]^{0} induced from the tautological rigidification of G̃_{σ} = G_{m}^{2}. This provides a canonical lifting Φ_{σ,m} : S_{σ} → \overline{T}_{m,∞} of ϕ_{σ,m}. These liftings are compatible when m grows, this gives rise to a lifting Φ_{σ} : S_{σ}_{(p)} → \overline{T}_{∞,∞} of ϕ_{σ} : S_{σ}_{(p)} → U_{σ}_{(p)} (the hat means p-adic completion). For f ∈ V, one can therefore take the pull-back of f mod p^{m} by Φ_{σ,m} (resp. of f by Φ_{σ}). The resulting series belongs to O_{S_{σ}} ⊗ℤ/p^{m}ℤ = ℤ/p^{m}ℤ[[q^{T}; T ∈ S∩σ^{∨}]] resp. O_{S_{σ}} ⊗ ℤ_{p} = ℤ_{p}[[q^{T}; T ∈ S ∩ σ^{∨}]]. It is however useful for further use to view it as belonging to O_{S_{σ}} ⊗ LC(M, ℤ/p^{m}ℤ) resp. to O_{S_{σ}} ˆ⊗C(M, ℤ_{p}) = C(M, ℤ_{p})[[q^{T}; T ∈ S ∩ σ^{∨}]] in the following way: the map x ∈ M ↦ Φ_{σ,m}^{∗}(x · f) is an O_{S_{σ}} ⊗ ℤ/p^{m}ℤ-valued locally constant map on M. The evaluation of this function at 1 ∈ M gives the ℤ/p^{m}ℤ[[q^{T}; T ∈ S ∩ σ^{∨}]]-valued q-expansion mentioned above. By taking the inverse limit over m, one gets the desired q-expansion with coefficients in C(M, ℤ_{p}). Both ℤ_{p}-coefficient and C(M, ℤ_{p})- coefficient q-expansions are compatible to restriction to faces; however, only the C(M, ℤ_{p})-coefficient expansion is compatible to the action of \overline{Γ}; we conclude that the functions x ∈ M ↦ Φ_{σ}^{∗}(x · f) for all σ’s give rise to an element of the submodule H^{0}(\overline{Γ}, C(M, ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]]) of \overline{Γ}- invariants of C(M, ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]]. We finally restrict our attention to f ∈ V^{N_{M}}; thus we obtain a q-expansion in

C(M/N_{M}, ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]]

We list below some well-known facts for which we refer to [15].

Proposition 3.3 1) (p-adic q-expansion principle) For any σ ∈ Σ_{η}, for any m ≥ 1, V/p^{m}V ⊂ V_{m,∞} ↪ ℤ/p^{m}ℤ[[q^{T}; T ∈ S ∩ σ^{∨}]] is injective with flat cokernel. In particular, the ring homomorphism FE is injective.

2) The restriction of F E to the ideal V_{!} of cusp forms takes values in the ideal generated by q^{T} for T ∈ S ∩ S_{2}(ℝ)^{+}. and the q-expansion principle holds for cusp forms for any cone σ and any m ≥ 1 as above.

We simply recall that the first point results from the irreducibility of \overline{T}_{m,∞} (Igusa irreducibility theorem, [10] V.7.2) and the second from direct examination of Φ_{σ}^{∗}(f).

It remains to compare the classical and p-adic modular forms resp. qexpansions. The embedding of classical forms into V comes from the canonical morphism ι : T_{∞,∞} → T |S_{∞} given by the fact that for an abelian variety A (of dimension 2) over a base S where p is nilpotent, any rigidification ψ : μ_{p^{∞}}^{2} ≅ A[p^{∞}]^{0} gives rise to an isomorphism O_{S}^{2} ≅ \underline{ω}_{A/S}. One checks easily that ι^{∗} : H^{0}(X, \underline{ω}^{κ}) → V^{N_{M}}[κ] and ι^{∗} : H^{0}(X, \underline{ω}_{κ}) → V_{!}^{N_{M}}[κ].

Thus given a classical form, we first view it as a section of (π_{∗}O_{T})^{N_{M}}, then one restricts it to the ordinary locus and one takes its pull-back by the morphism ι.

The comparison of the two definitions of \underline{ω}^{κ} provides a commutative square expressing the compatibility of classical and p-adic q-expansions:

V^{N_{M}}  → H^{0}(\overline{Γ}, C(M/N_{M}, ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]])
↑  ↑
H^{0}(X, \underline{ω}^{κ}) →  H^{0}(\overline{Γ}, W_{κ}(ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]])

In the case where κ is diagonal so that W_{κ}(ℤ_{p}) is free of rank one, one can formulate more simply the diagram by composing both horizontal maps by the evaluation of functions on M/N_{M} at 1, sending f : M/N_{M} → ℤ_{p} to f(1). We thus get a commutative square

V^{N_{M}}  →  ℤ_{p}[[q^{T}; T ∈ S ∩ S̃_{2}]])
↑  ↑
H^{0}(X, \underline{ω}^{κ}) → H^{0}(\overline{Γ}, W_{κ}(ℤ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]])

Let H ∈ H^{0}(X_{1}, det^{p−1}\underline{ω}) be the Hasse invariant on X_{1}. We fix an integer t ≥ 1 sufficiently large such that H^{t} lifts to X over ℤ_{p}. This can be achieved because det \underline{ω} is ample. We denote by E such a lifting. Recall that FE(E) ≡ 1 (mod p); this is because FE(H) = 1 in ℤ/pℤ[[q^{T}; T ∈ S ∩ S̃_{2}]].

By [15] Sect.3.6, the Hecke operators U_{p,1} = [N_{M}diag(1, 1, p, p)N_{M}] and U_{p,2} = p^{−3}[N_{M}diag(1, p, p, p^{2})N_{M}] do act on V_{!}^{N_{M}}. Let e = lim (U_{p,1}U_{p,2})^{n!} be the corresponding idempotent of End_{ℤ_{p}}V_{!}^{N_{M}}. The module eV_{!}^{N_{M}} is called the module of ordinary p-adic cusp forms (with strict Iwahori p-level). Hida’s control theorem [15] Th.1.1 says that for any weight κ (not necessarily cohomological), the cokernel of the inclusion eH^{0}(S_{∞}, \underline{ω}_{κ}) ⊂ eV_{!}^{N_{M}}[κ] is finite. Comment: Actually, Th.1.1 of [15] also contains a “classicity statement”, but only for very regular weights. Since we need in [26] an analogue of this statement including all cohomological weights (including those such that k = ℓ), we prove it there for all cohomological weights after localisation to a non-Eisenstein maximal ideal of the Hecke algebra. This theorem is crucial for us in [26] in order to produce overconvergent cusp forms g satisfying (LIM) The q-expansion of g is the p-adic limit of q-expansions of cusp eigenforms of cohomological weight. This condition provides the framework for the conjecture stated in the present paper. On the other hand, it would be very interesting to generalize Hida theorem to p-adic forms with finite slope for U_{p,1} different from 0. Such a generalization would produce new overconvergent forms satisfying (LIM).

3.3 Overconvergence

We endow ℂ_{p} with the p-adic norm such that ||_{p} = p^{−1}. For any extension L of ℚ_{p} contained in ℂ_{p} and for any real number r ∈]0, 1[, we consider the L-vector space of r-overconvergent Siegel modular forms

S_{κ}(Γ; r) = H^{0}(\overline{X}\{r\} × L, \underline{ω}_{κ})

If r is in |L^{×}|_{p}, this is a Banach space for the norm |f| = sup_{x∈\overline{X}\{r\}(L)}|f(x)|_{p} by [5] Th.4.1.6. In particular, for any r < r′ in ]p^{−a}, 1[∩|L^{×}|_{p}, the inclusions

res_{r,r′} : S_{κ}(K; r) ↪ S_{κ}(K; r′)

are completely continuous by [19] 2.4.1. It should be noted that the above fact does not require the assumption that the weight κ be cohomological (that is k_{1} ≥ k_{2} ≥ 3). In [26], we indeed apply this to κ = (2, 2).

Let a be either the Abbès-Mokrane bound (a = \frac{1}{p(p−1)}) or the Andreatta-Gasbarri’s bound (a = \frac{p−1}{2p−1}). By [1] Lemma 8.2.1 and [2], for any r ∈]p^{−a}, 1[, the canonical lifting F_{can} of the Frobenius endomorphism is defined as a rigid morphism \overline{X}\{r\} → \overline{X}^{rig}. The following two results are contained in [26] Sect.4.5

Proposition 3.4 There exists r ∈]p^{−a}, 1[∩p^{ℚ} such that F_{can} maps \overline{X}\{r\} into \overline{X}\{r^{p}\} and is finite flat of degree p^{3}. It yields a continous homomorphism of Banach spaces ϕ = F_{can}^{∗} : S_{κ}(Γ; r^{p}) → S_{κ}(K; r) and a trace homomorphism T r_{ϕ} : S_{κ}(Γ; r) → S_{κ}(Γ; r^{p}).

Corollary 3.5 There exists r ∈]p^{−a/p}, 1[∩|L^{×}|_{p}, the composition ψ = res_{r^{p},r}∘T r_{ϕ} defines a completely continuous endomorphism of the Banach space S_{κ}(Γ; r).

The evaluation on the rigid Mumford families G_{η}^{rig} → S_{σ} (for all polyedral cones σ in Σ_{η} as above) defines a L-linear homomorphism

FE : S_{κ}(K; r) ↪ L[[q^{T}; T ∈ S ∩ S_{2}(ℝ)^{+}]].

The overconvergent q-expansion principle says that FE is injective. It follows directly from the connectedness of \overline{X}^{rig}. We define U_{p,1} as p^{−3}ψ the operator corresponding to the weight κ = (2, 2). We denote by S_{2}(Γ; r) the L-Banach space of r-overconvergent forms of weight (2, 2). Then it follows immediately from Cor.3.3 that

Corollary 3.6 There exists r ∈]p^{−a/p}, 1[∩|L^{×}|_{p} such that the operator U_{p,1} is completely continuous on the Banach space S_{2}(Γ; r) of weight 2 overconvergent p-adic cusp forms.

Recall that by [24] Prop.7, one can define a Fredholm determinant P (t) = det(1 − tU_{p,1}) which is a p-adic entire function of t and such that λ ∈ \overline{ℚ}_{p} is a non-zero eigenvalue of U_{p,1} if and only if P (λ^{−1}) = 0; so that the nonzero eigenvalues of U_{p,1} form a sequence decreasing to 0. By Prop.12 and Remark 3 following this proposition in [24], each spectral subspace associated to a non-zero eigenvalue is finite dimensional (its dimension being equal to the multiplicity of the root λ^{−1} of P) and there is a direct sum decomposition of the Banach space as the sum of the (finite dimensional) spectral subspace and the largest closed subspace on which U_{p,1} − λ is invertible. In particular, for any positive number α, the set of eigenvalues λ_{i} ∈ \overline{ℚ}_{p} of U_{p,1} such that ord_{p}(λ_{i}) ≤ α is finite. Moreover one has a direct sum decomposition of the Banach space S_{2}(Γ, r) as S_{2}(Γ, r)^{≤α} ⊕ S_{2}(Γ, r)^{>α}, where the first space is finite dimensional, defined as the direct sum of the spectral subspaces for all eigenvalues λ_{i} with ord_{p}(λ_{i}) ≤ α, and the second is the (closed) largest subspace on which all the operators U_{p,1} − λ_{i} are invertible.

4 Galois representations of low weight and overconvergent modular forms

4.1 Eichler-Shimura maps

Let κ = (k, ℓ) be a cohomological weight, that is, a pair of integers such that k ≥ ℓ ≥ 3. Let k = a + 3, ℓ = b + 3. Then, (a, b), a ≥ b ≥ 0 is a dominant weight for (G, B, T); let V_{a,b} be the local system on the Siegel variety associated to the irreducible representation of G of highest weight (a, b); recall that the central character of this representation is z ↦ z^{a+b}. For any (neat) compact open subgroup L of G_{f}, for a′ ≥ b′ ≥ 0 and k′ = a′ + 3, ℓ′ = b′ + 3, there is a canonical Hecke-equivariant linear injection

H^{0}(S_{L}, \underline{ω}_{κ}) ↪ H^{3}(S_{L}, V_{a,b}(ℂ))

See Section 3.8 of [15] where it is explained how to make it canonical, and where it is called the Eichler-Shimura map. Actually the image is contained in H_{!}^{3} = Im(H_{c}^{3} → H^{3}). It follows for instance from Th.5.5, Chapter VI of [10].

4.2 Galois representation associated to a cohomological cusp eigenform

Let f be a cusp eigenform of cohomological weight κ = (k, ℓ). Let k = a + 3, ℓ = b + 3. By the EIchler-Shimura injection, the Hecke eigensystem associated to f occurs in H^{3}(X(ℂ), V_{a,b}(ℂ)). For any prime q prime to N, let P_{f,q} ∈ ℂ[X] be the degree four Hecke polynomial at q for the eigensystem of f (see [26]). Let E be the number field generated by the eigenvalues of the Hecke operators outside N. We fix a p-adic embedding ι_{p} of \overline{ℚ}; let F ⊂ \overline{ℚ}_{p} be a p-adic field containing ι_{p}(E) ( big enough but of finite degree).

The Galois representation W_{f} = H^{3}(S_{K}, V_{a,b}(F)))_{f} (largest subspace where Hecke acts as on f) is E-rational and pure of Deligne weight w = 3 + a + b. Let S be the set of prime divisors of N, and Γ be the Galois group of the maximal algebraic extension of ℚ unramified outside S and p. By a series of papers (due to R. Taylor, Laumon and Weissauer) there exists a degree four Galois representation R_{f,p} : Γ → GL_{4}(\overline{ℚ}_{p}) such that for any ℓ ∉ S ∪ \{p\}, det(X · 1_{4} − R_{f,p}(F r_{ℓ})) = P_{π,ℓ}(X). Its relation to W_{f} is: W_{f}^{4} = R_{f,p}^{m}, where m = dim W_{f}. We take F big enough for R_{f,p} to be defined over it. Remark: Let ϵ : Γ → ℤ_{p}^{×} be the p-adic cyclotomic character. With the convention above, we have ν ∘ ρ_{π,p} = ϵ^{−w} · ω_{f}, where ω_{f} is a finite order character modulo N, given as the Galois avatar of the companion character of f (this can viewed using Poincaré duality for W_{f}, see for instance [25], beginning of Sect.2). Remark: Given a classical cusp eigenform g ∈ H^{0}(X, ω_{(2,2)}), there is no geometric construction of an associated Galois representation (there is no Eichler-Shimura map to transport the eigensystem to the étale cohomology). See below for a p-adic construction, if the q-expansion of g is a p-adic limit of q-expansions of cohomological weight cusp eigenforms.

4.3 A conjecture

Let g ∈ H^{0}(\overline{X}\{r\}, ω_{(2,2)}) be an overconvergent cusp eigenform of weight (2, 2) and auxiliary level group K (unramified at p). By Prop.2.6, 2, since \overline{X}\{r\} is canonically identified to \overline{X}_{G}^{mm}\{r\} ⊂ \overline{X}_{Q}(p)^{rig}, one can view g as an element of H^{0}(\overline{X}_{G}^{mm}\{r\}, ω_{(2,2)}), where \overline{X}_{G}^{mm}\{r\} is a strict neighborhood of ]\overline{X}_{G}^{mm}[ in \overline{X}_{Q}(p)^{rig}. We shall actually need to consider the pull-back of g by π_{B,Q} ∘π_{U_{B},B} as a section of \underline{ω}_{2,2} over the quasi-compact relatively compact rigid open

(π_{B,Q} ∘ π_{U_{B},B})^{−1}(\overline{X}_{G}^{mm}\{r\})

in \overline{X}_{U_{B}}(p)^{rig}. Assume that

(LIM-EIG) there exists a sequence (g_{i}) of classical cusp eigenforms g_{i} ∈ H^{0}(\overline{X}_{U_{B}}(p), ω_{κ_{i}}) with cohomological weights κ_{i} = (k_{i}, ℓ_{i}) and level K (that is, prime to p, equal to the auxiliary level of g) such that the q-expansions of the g_{i}’s converge p-adically to that of g.

Let Π_{U_{B}} be the subgroup of matrices in G(ℤ_{p}) whose reduction modulo p belongs to U_{B}(ℤ/pℤ).

Comments: 1) Note that the key-point in this assumption is that the forms g_{i} are eigenforms. If we insist that the sequence of p-adic weights satisfies κ_{i} ≡ (2, 2) (mod p − 1)p^{i}, we cannot assume in general that the level of the g_{i}’s is prime to p; then we simply need to replace K_{p} = G(ℤ_{p}) by Π_{U_{B}} as p-component of the level group in (LIM-EIG). We can motivate the choice of the p-level group Π_{U_{B}} by recalling that both in the proof of the main theorem of [6] and in the Control Theorem for the Iwahori levels for GSp(4) of [27], it has been natural to consider the pull-back of g by π_{B,Q} ∘ π_{U_{B},B} as a section of \underline{ω}_{2,2} over the strict neighborhood

(π_{B,Q} ∘ π_{U_{B},B})^{−1}(\overline{X}_{G}^{mm}\{r\})

in \overline{X}_{U_{B}}(p)^{rig}. This is the analogue of Hida’s p-stabilization for p-adic modular forms.

2) Note also that it is a well-known theorem [15] that any p-adic cusp form is the p-adic limit (in the sense of q-expansions) of prime-to-p level classical cusp forms of weights κ_{i} satisfying κ_{i} ≡ (2, 2) (mod p − 1), where however, the forms g_{i}’s are not necessarily eigen even if g is.

Recall then that for any weight κ, there is a q-expansion map (always at the infinity cusp)

H^{0}((π_{B,Q} ∘ π_{U_{B},B})^{−1}(\overline{X}_{G}^{mm}\{r\}), \underline{ω}_{κ}) → W_{κ}(ℚ_{p})[[q^{T}; T ∈ S ∩ S̃_{2}]]\overline{Γ}

These maps are compatible with the p-adic q-expansion map via the canonical injection of H^{0}((π_{B,Q} ∘ π_{U_{B},B})^{−1}(\overline{X}_{G}^{mm}\{r\}), \underline{ω}_{κ}) into the space of p-adic cusp forms. We give below a conjectural criterion for the analytic continuation of g to \overline{X}_{U_{B}}(p)^{rig}. Let ρ_{g,p} : Gal(\overline{ℚ}/ℚ) → GL_{4}(\overline{ℚ}_{p}) be the Galois representation associated to the limit of the pseudo-representations of the g_{i}’s. We call it the Galois representation associated to g. Note that by Sen theory (Bull. Soc. Math. de France 1999), if the κ_{i} converge to (2, 2) in ℤ/(p−1)ℤ×ℤ_{p} and if ρ_{g,p} is Hodge-Tate, its p-adic Hodge-Tate weights should be 0, 0, 1, 1. Our conjecture reads as follows.

Conjecture: Let g ∈ S_{2,2}(K, r) be an overconvergent cusp eigenform satisfying (LIM-EIG); assume that there exists an abelian surface A defined over ℚ such that ρ_{g,p} is isomorphic to the contragredient ρ_{A,p}^{∨} of the representation on the p-adic Tate module of A. Then, g extends to a global section g ∈ H^{0}(\overline{X}_{U_{B}}(p)^{rig}, ω_{(2,2)}) thus defining by the rigid GAGA principle a classical cusp form of weight (2, 2) and level K^{p} × Π_{U_{B}}. If the abelian variety has good reduction at p, the cusp eigenform has level prime to p.

Remark: The minimal level group Π_{g,p} at p of the classical cusp eigenform g satisfies Π_{U_{B}} ⊂ Π_{g,p} ⊂ G(ℤ_{p}); the compatibility between gobal and local Langlands correspondences predicts that the (local) Weil-Deligne representation associated to D_{pst}(ρ_{g,p}) determines Π_{g,p}.

The main result (Theorem 4) of [26] provides under certain assumptions (primarily the assumption of near ordinarity) such pairs of an overconvergent cusp eigenform g with a converging sequence (g_{i}) of cusp eigenforms, together with an abelian surface A defined over ℚ with potential good ordinary reduction at p. Actually one starts there from an abelian surface satisfying certain condition, the most stringent being that the Galois representation ρ_{A,p}^{∨} must be congruent modulo p to the representation ρ_{f,p} associated to a cusp eigenform of level K prime to p, ordinary at p with cohomological weight. Then Hida theory ([26] Lemma 4.2) yields a sequence (g_{i}) converging to a limit g which is overconvergent of weight (2, 2) and auxiliary level K. Note that once a generalization of Coleman Families Theory to the Siegel case is available, there might be new examples of such forms g. In the situation treated in [26], the representation ρ_{g,p} = ρ_{A,p}^{∨} is potentially crystalline but not crystalline, which implies that the eigenforms g_{i} are indeed p-new of p-level Π_{U_{B}}, hence the presence of Π_{U_{B}} as conjectural p-level group of g. The conjecture above would imply that the L function of the motive h^{1}(A) is automorphic: L(h^{1}(A), s) = L_{spin}(g, s), hence, by a classical theorem of Piatetskii-Shapiro, it would have analytic continuation and functional equation.

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J. Tilouine

Département de Mathématiques UMR 7539 Institut Galilée Université de Paris 13 93430 Villetaneuse France tilouine@math.univ-paris13.fr 818

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