The moduli spaces of principally polarized abelian varieties with Γ_{0}(p)-level structure

A.J. de Jong*

Introduction

Let (A, λ) be a principally polarized abelian variety of dimension g. A Γ_{0}(p)-level structure on (A, λ) is a flag of subgroup schemes 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p] such that the order of H_{i} is p^{i} and such that H_{g} is totally isotropic for the Weil-pairing induced by λ on A[p]. We let S(g, p) be the algebraic stack over Spec(ℤ) parametrizing principally polarized abelian varieties endowed with a Γ_{0}(p)-level structure (Definition 1.1).

In this article we study the spaces S(g, p). In particular we are interested in the behavior of S(g, p) over the prime (p) in Spec(ℤ). For example: Is S(g, p) flat over Spec(ℤ_{(p)})? What are the singularities of S(g, p)?

We attack these questions by constructing a scheme X over Spec(ℤ) with the following property: for any point x ∈ S(g, p) there exists a point y ∈ X such that an étale neighbourhood of x in S(g, p) is isomorphic to an étale neighbourhood of y in X (Corollary 4.6).

The scheme X is a kind of flag variety and hence it is comparatively easy to write down equations defining open subsets of it, at least for any fixed g. In §4 we do this for the case g = 2 and we answer completely the questions above. The result is that S(g, p) → Spec(ℤ) is a locally complete intersection morphism and that the singularities are not too bad. Further we can show that the ordinary locus is dense in S(g, p) × Spec(𝔽_{p}) and that S(g, p) × Spec(\overline{𝔽}_{p}) has four components.

For any fixed g a detailed study of the scheme X should give the answer to the questions mentioned above; perhaps a systematic study of the schemes X will reveal that S(g, p) is flat over Spec(ℤ) for general g.

Starting point for the work in this article were discussions on the article [CN]. In this article C. Chai and P. Norman consider the moduli stack of triples (A, λ, H_{2} ⊂ A[p]) where A is an abelian variety of dimension two, λ is a principal polarization and H_{2} ⊂ A[p] is an isotropic subgroup scheme of order p^{2}. They compute the leading terms of the equations defining the local moduli space in characteristic p and show in this way that one gets a Cohen-Macaulay algebraic stack. Our stack S(2, p) maps to theirs simply by forgetting about the subgroup scheme H_{1}. This map is surjective and generically finite; it blows up only in the points (A, λ, H_{2}) where

* Supported by the Netherlands Organisation for Scientific Research (N.W.O.), in the project Arithmetic Algebraic Geometry H_{2} ≃ α_{p} × α_{p}. It seems that the singularities of our moduli problem are somewhat milder than theirs and that it is easier to describe them.

Indeed, the explicit description of S(2, p) allows us to say something about the action of Gal(\overline{ℚ}_{p}/ℚ_{p}) on the étale cohomology groups H^{i}(S(2, p)_{\overline{ℚ}_{p}}, ℚ_{l}). The author intends to report on this in a further article.

We would like to thank Prof. M. Rapoport for proposing to study exactly this moduli problem and Prof. T. Zink for his interest in the results.

Notation and conventions

– Our reference on algebraic stacks is [DM] §4. A stack will be a stack in groupoids over the category of schemes with the étale topology. Stacks will be denoted by calligraphic uppercase letters.

– The prime number p is fixed throughout the article.

– If A is an abelian scheme over the scheme S then A^{t} denotes its dual (it exists, see [FC] page 3).

1. The moduli problem

Fix a natural number g and a prime number p. In the sequel we will study the following type of objects.

Definition 1.1. A system of type I will be a quadruple

(S, A, λ, 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p])

where - A is an abelian scheme of relative dimension g over the scheme S.

- λ is a principal polarization on A.

- 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p] is a flag of finite flat subgroup schemes of A[p] such that H_{i} has rank p^{i} over S and H_{g} ⊂ A[p] is (totally) isotropic for the Weil pairing induced by λ on A[p].

The class of all systems of type I forms a category S(g, p). The morphisms of (S, A, λ, 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p]) into (S′, A′, λ′, 0 ⊂ H′_{1} ⊂ … ⊂ H′_{g} ⊂ A′[p]) are pairs (f, g) of morphisms of schemes which fit into a commutative diagram

A ⟶^{g} A′
↓  ↓
S ⟶^{f} S′

such that g induces an isomorphism A ≃ S ×_{S′} A′, g^{∗}(λ′) = λ and such that g(H_{i}) ⊂ H′_{i}. Consider the functor π : S(g, p) → Sch given by

π((S, A, λ, 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p])) = S.

It is clear that π makes S(g, p) into a stack in groupoids over Sch. The forgetful functor of S(g, p) into the stack of principally polarized abelian varieties A_{g} defines a 1-morphism of stacks

F : S(g, p) ⟶ A_{g},
(S, A, λ, 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p]) ↦ (S, A, λ).

Proposition 1.2. The 1-morphism F is representable. It is a proper surjective morphism. Proof. Let (S, A, λ) ∈ ObA_{g}. We will show that the fibre product S ×_{A_{g}} S(g, p) is representable by a scheme Z such that the morphism Z → S is proper.

For any scheme T over S denote by A_{T} the fibre product A×_{S} T. To give a flag of subgroup schemes

0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A_{T} [p]

is equivalent to giving the flag

O_{A_{T} [p]} ⊃ I_{1} ⊃ I_{2} ⊃ … ⊃ I_{g} ⊃ (0)

of ideals I_{j} defining H_{j} ⊂ A[p]. In this way we see that there is a 1-morphism of stacks

S ×_{A_{g}} S(g, p) ⟶ P

of S ×_{A_{g}} S(g, p) into a certain flag variety P over S. This morphism identifies S ×_{A_{g}} S(g, p) with a certain closed subscheme Z ⊂ P. It is the closed subscheme with the following universal property: any morphism of schemes T → P over S factors through Z if and only if the flag

O_{A_{T} [p]} ⊃ F_{1} ⊃ … ⊃ F_{g} ⊃ (0)

determined by the map T → P is a flag of ideals of subgroup schemes H_{i} ⊂ A[p] such that H_{g} is isotropic for the Weil pairing on A_{T} [p]. It is standard to see that Z exists.

To see that F is surjective let (A, λ) be a principally polarized abelian variety over an algebraically closed field. By definition there exists an ample line bundle L on A such that λ = ϕ_{L} (see [GIT] Definition 6.3). The Weil pairing induced by λ on A[p] is equal to the pairing e^{L^{⊗p}} on K(L^{⊗p}) = A[p] (see [Mu], p. 227). By [Mu] §23 Theorem 4 there exists an isotropic subgroup H_{g} ⊂ A[p] of order p^{g}. A filtration 0 ⊂ H_{1} ⊂ … ⊂ H_{g} of H_{g} by subgroup schemes of order(H_{i}) = p^{i} determines a lift of the point (A, λ) of A_{g} to a point of S(g, p). □

Corollary 1.3. The stack S(g, p) is a separated algebraic stack of finite type over Spec(ℤ). Proof. We know that A_{g} is a separated algebraic stack of finite type over Spec(ℤ) (see for example [FC] I 4.11). However, it is easy to see that any stack which allows a representable morphism to an algebraic stack is an algebraic stack itself. Hence S(g, p) is an algebraic stack by Proposition 1.2. In the same manner we see that S(g, p) is separated and of finite type over Spec(ℤ). □

Corollary 1.4. The stack S(g, p) is smooth over Spec(ℤ[1/p]). Proof. The morphism F : S(g, p) → A_{g} is étale over Spec(ℤ[1/p]) and A_{g} is smooth over Spec(ℤ). □

We could have defined S(g, p) also using the following type of objects.

Definition 1.5. A system of type II will be a quadruple

(S, A_{0} ⟶^{α} A_{1} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g})

where - S is a scheme.

- A_{0} →^{α} A_{1} →^{α} … →^{α} A_{g} is a sequence of morphisms of abelian schemes of dimension g over S such that Ker(A_{i} →^{α} A_{i+1}) is a (finite, flat) group scheme of rank p over S.

- λ_{0} and λ_{g} are principal polarizations on A_{0} and A_{g}. We require further that the compositions

A_{i} ⟶^{α} A_{i+1} ⟶ … ⟶A_{g} ⟶^{λ_{g}} A_{g}^{t} ⟶^{α^{t}} A_{g−1}^{t} ⟶ … ⟶ A_{0}^{t} ⟶^{λ_{0}^{−1}} A_{0} ⟶^{α} … ⟶^{α} A_{i}

are equal to p · id_{A_{i}} for all i = 0, …, g. In a formula:

p · id_{A_{i}} = α^{i} ∘ λ_{0}^{−1} ∘ (α^{t})^{g} ∘ λ_{g} ∘ α^{g−i}  (1)

for all i = 0, …, g.

Remark 1.6. We only have to require (1) for one i; the equality will then hold for all i. See the proof of the proposition below.

Exactly as before we can turn the class of systems of type II into a stack in groupoids over Sch. Let us denote this stack by S′(g, p) for the moment.

Proposition 1.7. The stacks S(g, p) and S′(g, p) are isomorphic. Proof. Let us simply indicate how to get 1-morphisms

S(g, p) ⇆ S′(g, p)

and leave it to the reader to show that their compositions are isomorphic to the identities.

For any system (S, A, λ, 0 ⊂ H_{1} ⊂ … ⊂ H_{g} ⊂ A[p]) of type I we put A_{0} = A, A_{i} = A/H_{i} for i = 1, …, g and λ_{0} = λ. The maps α are simply the canonical maps A/H_{i} → A/H_{i+1}. It follows immediately that Ker(A_{i} → A_{i+1}) has rank p over S. The fact that H_{g} ⊂ A[p] is totally isotropic for the Weil pairing induced by λ on A[p] is equivalent to the fact that there exists a principal polarization λ_{g} on A_{g} which fits into the following diagram

A_{0} ⟶^{α^{g}} A_{g}
↓^{pλ_{0}}  ↓^{λ_{g}}
A_{0}^{t} ⟵^{(α^{g})^{t}} A_{g}^{t}.

This immediately gives (1) for i = 0. To prove (1) for general i we note that (α^{i} ∘ λ_{0}^{−1} ∘ (α^{t})^{g} ∘ λ_{g} ∘ α^{g−i}) ∘ α^{i} = α^{i} ∘ (λ_{0}^{−1} ∘ (α^{t})^{g} ∘ (α^{t})^{g} ∘ λ_{g} ∘ α^{g}) = α^{i} ∘ p · id_{A_{o}} = p · id_{A_{i}} ∘ α^{i}. Since α^{i} is an isogeny we can divide it out on both sides to get α^{i} ∘ λ_{0}^{−1} ∘ (α^{t})^{g} ∘ λ_{g} ∘ α^{g−i} = p · id_{A_{i}}. Thus (S, A_{0} →^{α} … →^{α} A_{g}, λ_{0}, λ_{g}) is a system of type II.

Conversely, for any system of type II we put A = A_{0}, λ = λ_{0} and H_{i} = Ker(α^{i} : A_{0} → A_{i}). It is easy to see that this defines a system of type I. □

In the rest of this article we will identify S′(g, p) with S(g, p).

Remark 1.8. The algebraic stack S(g, p) has a natural involution defined as follows: the involution transforms the system

(S, A_{0} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g})

into the system

(S, A_{g}^{t} ⟶^{α^{t}} … ⟶^{α^{t}} A_{0}^{t}, (λ_{g}^{t})^{−1}, (λ_{0}^{t})^{−1}).

Indeed, if we apply this transformation again then we get

(S, A_{0}^{tt} ⟶^{α^{tt}} … ⟶^{α^{tt}} A_{g}^{tt}, λ_{0}^{tt}, λ_{g}^{tt})

and this is canonically isomorphic to the original system by the biduality isomorphism.

2. Crystalline cohomology and deformations of abelian schemes

In this section we recall the theorem of Grothendieck on deformations of abelian schemes. We explain how to use this theorem to deform abelian schemes with a polarization.

Let S be a scheme. We will denote by NCris(S) the nilpotent crystalline site of S relative to (Spec(ℤ), (0)). Its objects are triples (U, T, δ) where U ⊂ S is Zariski open, T is a thickening of U and δ = (δ_{n})_{n≥1} is a system of a locally nilpotent divided powers of the ideal of U in T.

Crystalline cohomology gives us a contravariant functor

𝔻 : Abelian schemes over S ⟶ locally free crystals on NCris(S).

It is defined by the formula

𝔻(A) = R^{1}f_{crys∗}(O_{A_{crys}})

where f_{crys} is the morphism on the nilpotent topoï induced by the structural morphism f : A → S. (This functor is denoted by 𝔻^{∗} in [Gr].) The canonical isomorphism

𝔻(A)_{S} = R^{1}f_{∗}(Ω_{A/S}^{·})

gives us a locally free direct summand ω_{A} ⊂ 𝔻(A)_{S}. Let S′ ↩ S be a thickening endowed with locally nilpotent divided powers on the ideal of definition of S in S′.

Theorem 2.1. (Grothendieck) The filtered Dieudonné functor gives an equivalence

The Category of abelian schemes over S′ ⟶ The category of pairs (A, F il^{1})

where A is an abelian scheme over S and F il^{1} ⊂ 𝔻(A)_{S′} is a locally direct summand which lifts ω_{A} ⊂ 𝔻(A)_{S}.

An outline of the proof is given in [Gr], pp. 116-118. The details can be found in [Me] and [Ma Me] especially II §1. □

Let us agree to denote by a ^{∨} the functor on sheaves of O_{Scrys}-modules

F ⟼ F^{∨} = Hom_{O_{Scrys}}(F, O_{Scrys}).

In [BBM] 5.1 there is constructed a functorial isomorphism ϕ_{A} : 𝔻(A)^{∨} → 𝔻(A^{t}) at least if p is locally nilpotent on S. However, it is easy to see that their definition of ϕ_{A} will work also in our situation since we are working with nilpotent divided powers. Exactly the same proofs as in [BBM] will give Theorem 5.1.8 and Proposition 5.1.9 and 5.1.10 of [BBM] 5.1 in our situation.

Let A be an abelian scheme over S with a principal polarization λ. Using ϕ_{A} we can define a nondegenerate alternating paring

\begin{aligned}e_{λ} : & 𝔻(A) × 𝔻(A) & ⟶ & O_{Scrys} \\ & (x, y) & ⟼ & (ϕ_{A}^{−1} ∘ 𝔻(λ^{−1}))(x)(y). \end{aligned}

It is nondegenerate since ϕ_{A} and 𝔻(λ) are isomorphisms. To prove it is alternating we use: \underline{1} λ = λ^{t} ∘ j \underline{2} ϕ_{A^{t}} ∘ 𝔻(λ)^{∨} = 𝔻(λ^{t}) ∘ ϕ_{A} \underline{3} −ϕ_{A}^{∨} = 𝔻(j) ∘ ϕ_{A^{t}}

The first statement is the fact that a polarization is symmetric (j is the biduality isomorphism); the second is a consequence of the functoriality of ϕ_{A} and the last statement is Proposition 5.1.9 of [BBM]. Thus

𝔻(λ) ∘ ϕ_{A} =^{3} −𝔻(λ) ∘ (ϕ_{A^{t}})^{∨} ∘ 𝔻(j)^{∨}

=^{2} −ϕ_{A}^{∨} ∘ 𝔻(λ^{t})^{∨} ∘ 𝔻(j)^{∨}

=^{1} −ϕ_{A}^{∨} ∘ 𝔻(λ)^{∨}

which is exactly what we had to prove.

Of course this proof only works on objects (U, T, δ) such that O_{T} has no 2-torsion. To prove it in general we argue as follows. Since there are no obstructions to deforming a principally polarized abelian scheme over an affine scheme we can take locally on T a lift of the pair (A_{U}, λ). Hence we are reduced to the case T = U. Locally on U in the étale topology the pair (A_{U}, λ) is the pullback of a pair (A_{Z}, λ) over a scheme Z smooth over Spec(ℤ). (This is true since A_{g} → Spec(ℤ) is smooth.) Since O_{Z} has no 2-torsion we are done.

By Proposition 5.1.10 the subsheaf ω_{A} ⊂ 𝔻(A)_{S} is totally isotropic for the pairing e_{λ,S}. This immediately gives the following criterium for the liftability of the polarization λ: let (S, S′, δ) ∈ ObNCris(S).

Corollary 2.2. The polarization λ on A lifts to the abelian scheme determined by F il^{1} ⊂ 𝔻(A)_{S′} via Theorem 2.1 if and only if

e_{λ,S′} |_{F il^{1}×F il^{1}} ≡ 0.

We finish this section with a lemma that we will need later on.

Lemma 2.3. Suppose α : A → B is an isogeny of abelian schemes over S whose kernel has rank p. Then the sheaf Cokern (𝔻(α) : 𝔻(B) → 𝔻(A)) on NCris(S) is a locally free O_{Scrys}/p · O_{Scrys}-module of rank 1. Proof. This would be a special case of [BBM] Proposition 4.3.1 if p were locally nilpotent on S. Let us simply rewrite the proof.

Let β : B → A be the isogeny such that β ∘ α = p · id_{A} and α ∘ β = p · id_{B}. We have the exact sequence

0 ⟶ 𝔻(α)(𝔻(B))/p𝔻(A) ⟶ 𝔻(A)/p𝔻(A) ⟶ 𝔻(A)/𝔻(α)(𝔻(B)) ⟶ 0

and a surjection

𝔻(B)/𝔻(β)(𝔻(A)) ↠ 𝔻(α)(𝔻(B))/p𝔻(A).

The exact sequence shows that 𝔻(A)/𝔻(α)(𝔻(B)) is annihilated by p. Suppose the dimension of A over S is g. If we can show that 𝔻(B)/𝔻(β)(𝔻(A)) and 𝔻(A)/𝔻(α)(𝔻(B)) are locally generated by at most 2g − 1 and 1 sections respectively then we are done. As in the proof of [BBM] Proposition 4.3.1 it is enough to prove this when S is the spectrum of a perfect field of characteristic p. In this case the groups H_{DR}^{1}(B)/β^{∗}H_{DR}^{1}(A) and H_{DR}^{1}(A)/α^{∗}H_{DR}^{1}(B) are isomorphic to the Dieudonné modules of Ker(β) and Ker(α). Since Ker(α) is a group scheme of order p and Ker(β) is a group scheme of order p^{2g−1} we are done. □

3. Systems of modules associated to objects of S(g, p)

Any system of type II over a scheme S is transformed into a system of crystals by the crystalline cohomology functor. In this section we will describe a normal form for such a system of modules.

Let S be a scheme and let

A. = (S, A_{0} ⟶^{α} A_{1} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g})

be a system of type II. Suppose that (U, T, δ) is an object of NCris(S) and that T = Spec(R) is affine. We get a system of R-modules and maps

M. = M(A.) = (M_{g} ⟶^{α} M_{g−1} ⟶^{α} … ⟶^{α} M_{0}, q_{0}, q_{g})

as follows

M_{i} := Γ(T, 𝔻(A_{i})_{T}),
α := the maps induced by 𝔻(α)

and for i = 0 or g

q_{i} := the pairing on M_{i} induced by e_{λ_{i}}.

We will write ^{∨} for the functor Hom_{R}(·, R) on the category of R-modules. By abuse of notation we will write q_{i} also for the map M_{i} → M_{i}^{∨} given by x ↦ q_{i}(x, ·).

Propositions 3.1. The system M(A.) has the following properties - M_{i} is a locally free R-module of rank 2g.

- The pairing q_{i} is nondegenerate and alternating.

- Coker (M_{i} ⟶^{α} M_{i−1}) is a locally free R/pR-module of rank 1.

- For each i = 0, …, g the homomorphism

α^{g−i} ∘ q_{g}^{−1} ∘ (α^{∨})^{g} ∘ q_{0} ∘ α^{i} : M_{i} ⟶ M_{i}

is equal to multiplication by p. Proof. For the first three properties see §2. The last property is a consequence of the corresponding property of systems of type II and the functoriality of ϕ_{A}. □

Definition 3.2. For any ring R a system of R-modules of type II is a triple M. = (M_{g} →^{α} M_{g−1} →^{α} … →^{α} M_{0}, q_{0}, q_{g}) such that the conclusions of Proposition 3.1 hold.

For any scheme S a system of O_{S}-modules of type II is a triple M. = (M_{g} →^{α} M_{g−1} →^{α} … →^{α} M_{0}, q_{0}, q_{g}) where M_{i} is a sheaf of O_{S}-modules and such that on affine open subsets Spec(R) of S it gives rise to a system of R-modules of type II.

Let us remark that all such systems M. (resp. M.) over a ring R (resp. a scheme S) form in a natural way a category (morphisms are isomorphisms). For any ring homomorphism R → R′ (resp. morphism of schemes f : S′ → S) we get a “pullback” functor M. ⇝ M. ⊗_{R} R′ (resp. M. ⇝ f^{∗}M.) defined in the obvious manner.

Example 3.3. Let St_{i}, i = 0, …, g be the free ℤ-module with basis e_{0}, …, e_{2g−1}. We define α : St_{i+1} → St_{i} by the formula α(e_{j}) = e_{j}, j ≠ i and α(e_{i}) = pe_{i}. We define nondegenerate alternating pairings q_{0} and q_{g} on St_{0} and St_{g} by

q_{i}(e_{k}, e_{l}) = \{ \begin{matrix}0 & if k + l ≠ 2g − 1 \\ 1 & if k + l = 2g − 1 and k < l \\ −1 & if k + l = 2g − 1 and k > l. \end{matrix}

It is left to the reader to show that these formulas define a system of ℤ-modules of type II:

St. = (St_{g} ⟶^{α} St_{g−1} ⟶^{α} … ⟶^{α} St_{0}, q_{0}, q_{g}).

For any scheme S the pullback St. ⊗ O_{S} of the system St. on Spec(ℤ) will be a system of O_{S}-modules of type II. The purpose of this section is to show that any system of O_{S}-modules of type II is isomorphic to St. ⊗ O_{S} locally on S.

Let R be a ring and suppose M. is a system of R-modules of type II. In such a situation we define M_{i} for all i ∈ ℤ by putting

M_{g+k} := (M_{g−k})^{∨} k = 1, …, g − 1

and for all i ∈ ℤ

M_{i} := M\overline{i}

where \overline{i} is the element of \{0, 1, …, 2g − 1\} such that i ≡ \overline{i} mod 2g. In fact we will think of the index set as being ℤ/2gℤ.

Furthermore, we define for all i ∈ ℤ/2gℤ a nondegenerate paring

⟨ , ⟩_{i} : M_{i} × M_{2g−i} ⟶ R

as follows: if x ∈ M_{i} and y ∈ M_{2g−i} then

⟨x, y⟩_{i} = \{ \begin{matrix}q_{0}(x, y) & if i = 0 \\ q_{g}(x, y) & if i = g \\ −y(x) & if i = 1, …, g − 1 \\ x(y) & if i = g + 1, …, 2g − 1 \end{matrix}

here y(x) in the third line indicates the evaluation of y ∈ M_{2g−i} = M_{i}^{∨} on x ∈ M_{i}. Finally, we define α : M_{i} → M_{i−1} for those i ∈ ℤ/2gℤ for which it is not defined yet by the commutativity of the following diagrams

\begin{matrix}M_{0} & ⟶^{α} & M_{2g−1} \\ ↓^{q_{0}} & & ∥ \\ M_{0}^{∨} & ⟶^{α^{∨}} & M_{1}^{∨} \end{matrix} , \begin{matrix}M_{g+1} & ⟶^{α} & M_{g} \\ ∥ & & ↓^{q_{g}} \\ M_{g−1}^{∨} & ⟶^{α^{∨}} & M_{g}^{∨} \end{matrix}

and for all k = 2, …, g − 1

M_{g+k}  ⟶^{α}  M_{g+k−1}
∥  ∥
(M_{g−k})^{∨} ⟶^{α^{∨}} (M_{g−k+1})^{∨}.

Lemma 3.4.

1) For all i ∈ ℤ/2gℤ the map α^{2g} : M_{i} → M_{i} equals multiplication by p.

2) ∀ x ∈ M_{i}, y ∈ M_{2g−i} : ⟨x, y⟩_{i} = −⟨y, x⟩_{2g−i}.

3) ∀ x ∈ M_{i+1}, y ∈ M_{2g−i} : ⟨α(x), y⟩_{i} = ⟨x, α(y)⟩_{i+1}.

Lemma 3.5. For all 1 ≤ k ≤ 2g and all i ∈ ℤ/2gℤ the module Coker (M_{i+k} →^{α^{k}} M_{i}) is a locally free R/pR-module of rank k. Proof. We may assume R is a local ring with residue characteristic p. The exact sequences

0 ⟶ α^{k−1}(M_{i+k−1})/α^{k}(M_{i+k}) ⟶ M_{i}/α^{k}(M_{i+k}) ⟶ M_{i}/α^{k−1}(M_{i+k−1}) ⟶ 0

and an induction argument give that the modules M_{i}/α^{k}(M_{i+k}) are generated by k elements. Hence the modules α^{k}(M_{i+k})/α^{2g}(M_{i}) are generated by 2g − k elements. Since by assumption M_{i}/α^{2g}(M_{i}) = M_{i}/pM_{i} ≃ (R/pR)^{2g} the result follows from the exact sequence

0 ⟶ α^{k}(M_{i+k})/α^{2g}(M_{i}) ⟶ M_{i}/α^{2g}(M_{i}) ⟶ M_{i}/α^{k}(M_{i+k}) ⟶ 0.  □

Proposition 3.6. In the above situation:

1) If R is local then there exists an isomorphism

St. ⊗ R ⟶^{∼} M.

2) If I ⊂ R is a nilpotent ideal then any isomorphism

\overline{ϕ} : St. ⊗ R/I ⟶ M. ⊗_{R} R/I

can be lifted to an isomorphism

ϕ : St. ⊗ R ⟶ M.

Proof. If R is local and has a residue characteristic other than p then the first statement is easy and is left to the reader. Let us assume that the residue characteristic is p. We construct an isomorphism in four steps. Step 1. For all i ∈ ℤ/2gℤ we take an element η_{i} ∈ M_{i} which projects onto a (free) generator of the R/pR-module M_{i}/α(M_{i+1}). It follows from Lemma 3.5 that {η_{i}, α(η_{i+1}),…, α^{2g−1}(η_{i+2g−1})} forms an R-basis of M_{i}. Step 2. As q_{0}(α^{k}(η_{k}), α^{l}(η_{l})) ∈ pR if k + l ≥ 2g (see Lemma 3.4) and since q_{0} is nondegenerate we must have that the matrix (A_{ij}){i=0,...,g−1 \atop j=0,...,g−1} with

A_{ij} = q_{0}(α^{i}(η_{i}), α^{g+j}(η_{g+j}))

is invertible. Using the inverse (B_{ij}) we can succesively change our choice of η_{i}, i = 0, …, g − 1 such that

(1)  q_{0}(α^{i}(η_{i}), α^{j}(η_{j})) = 0 ∀ 0 ≤ i, j ≤ g − 1.

Indeed, suppose we already have that

q_{0}(α^{i}(η_{i}), α^{j}(η_{j})) = 0 ∀ 0 ≤ i < i_{0}, 0 ≤ j ≤ g − 1

for some 0 ≤ i_{0} ≤ g − 1. Our new choice of η_{i_{0}} will be

η_{i_{0}}^{new} = η_{i_{0}} + Σ_{k,l}λ_{k}B_{kl}α^{g−i_{0}+l}(η_{g+l})

where λ_{k} = q_{0}(α^{i_{0}}(η_{i_{0}}), α^{k}(η_{k})) for k = 0, …, g − 1 (of course λ_{k} = 0 for k ≤ i_{0}). Obviously η_{i_{0}}^{new} still gives a generator for M_{i_{0}}/α(M_{i_{0}+1}) and ∀ j ∈ {0, …, q − 1}, j ≠ i_{0} we have:

\begin{aligned}q_{0}(α^{i_{0}}(η_{i_{0}}^{new}), α^{j}(η_{j})) & = q_{0}(α^{i_{0}}(η_{i_{0}}) + Σ_{k,l}λ_{k}B_{kl}α^{g+l}(η_{g+l}), α^{j}(η_{j})) \\ & = λ_{j} − Σλ_{k}B_{kl}A_{lj} = λ_{j} − Σλ_{k}δ_{kj} = λ_{j} − λ_{j} = 0. \end{aligned}

Step 3. Exactly the same method as in step 2 can be used to alter our choice of η_{g}, …, η_{2g−1} such that

(2)  q_{g}(α^{j}(η_{g+i}), α^{j}(η_{g+j})) = 0  0 ≤ i, j ≤ g − 1.

Needless to say that (1) is fulfilled also. Step 4. For all i, j ∈ {0, …, g − 1} we let (∗)_{ij} be the condition:

(∗)_{ij} \{ \begin{matrix}If & i + j ≤ g − 1 then we demand that \\ & q_{0}(α^{i}(η_{i}), α^{g+j}(η_{g+j})) = \{ \begin{matrix}0 & if i + j < g − 1 \\ 1 & if i + j = g − 1 \end{matrix} \\ If & i + j > g − 1 then we demand that \\ & ⟨η_{i}, α^{i+j−g}(η_{g+j})⟩_{i} = 0. \end{matrix}

In this step we will choose new elements η_{0}, …, η_{g−1} such that (1) and (2) remain valid and such that

(3)  ∀i, j ∈ {0, …, g − 1} (∗)_{ij} holds.

We proceed by induction on j. Suppose we already have (∗)_{ij} for all i ∈ {0, …, g − 1} and all j, 0 ≤ j < j_{0}. (This is true for j_{0} = 0.) Put i_{0} = g − 1 − j_{0}. All pairings q_{0}(x, y) of elements x, y of the set

{α^{i_{0}}(η_{i_{0}}), α^{g}(η_{g}), α^{g+1}(η_{g+1}), …, α^{2g−1}(η_{2g−1})}

are zero or in pR except for possibly

u = q_{0}(α^{i_{0}}(η_{i_{0}}), α^{g+j_{0}}(η_{g+j_{0}}))

and

−u = q_{0}(α^{g+j_{0}}(η_{g+j_{0}}), α^{i_{0}}(η_{i_{0}})).

As q_{0} is nondegenerate u must be an invertible element of R. Our new choice of the elements η_{0}, …, η_{g−1} will be:

- for 0 ≤ i < i_{0} η_{i}^{new} = η_{i} + u^{−1} · q_{0}(α^{g+j_{0}}(η_{g+j_{0}}), α^{i}(η_{i})) · α^{i_{0}−i}(η_{i_{0}})

- for i = i_{0} η_{i_{0}}^{new} = u^{−1} · η_{i_{0}}

- for i_{0} < i ≤ g − 1 η_{i}^{new} = η_{i} − u^{−1} · ⟨η_{i}, α^{i+j_{0}−g}(η_{g+j_{0}})⟩_{i} · α^{2g+i_{0}−i}(η_{i_{0}}).

It is a trivial, though tedious, calculation to see that with these choices of η_{0}, …, η_{g−1} the formulas (1) and (2) are still valid and that (∗)_{ij} will hold for all i ∈ {0, …, g − 1} and all j, 0 ≤ j ≤ j_{0}.

Having found η_{i} as in step 1 satisfying (1), (2) and (3), we can define an isomorphism

ϕ : St. ⊗ R ⟶ M.

as follows. We define ϕ_{i} : St_{i} ⊗_{ℤ} R → M_{i} for i = 0, …, g by putting

ϕ_{i}(e_{j}) = \{ \begin{matrix}α^{2g−i+j}(η_{j}) & if j < i \\ α^{j−i}(η_{j}) & if j ≥ i. \end{matrix}

Properties (1), (2) and (3) imply that this defines an isomorphism of systems of R-modules of type II.

In order to prove the second statement of the proposition we note that the isomorphism \overline{ϕ} : St.⊗R/I → M.⊗_{R} R/I already gives us elements \overline{η}_{i} ∈ M_{i} ⊗_{R} R/I satisfying (1), (2) and (3). Let us take arbitrary lifts η_{i} ∈ M_{i} of the elements \overline{η}_{i} ∈ M_{i} ⊗_{R} R/I. The procedure described in steps 2, 3 and 4 will work to give lifts η_{i} ∈ M_{i} satisfying (1), (2) and (3). For example the matrix (A_{ij}) in step 2 will be congruent to the identity matrix mod I and hence invertible, the elements η_{i}^{new} in step 2 will be lifts of \overline{η}_{i} also since λ_{k} ∈ I, etc. Obviously the formula giving ϕ will then produce a lift of \overline{ϕ}. □

Remarks 3.7. a) In part 2) of Proposition 3.6 the ideal I may be any ideal contained in the Jacobson radical of R.

b) Let G over Spec(ℤ) be the group scheme representing the functor T ⇝ Aut(St. ⊗ O_{T}). It follows from Proposition 3.6 that G is a smooth group scheme over Spec(ℤ).

4. The singularities of S(g, p)

In this section we will compare the singularities of S(g, p) with the singularities of a scheme X which can be described as a certain flag variety. We will show that any complete local ring of S(g, p) at a geometric point is isomorphic to a complete local ring of X at some geometric point of X.

Definition 4.1. Let S be an algebraic stack and x : Spec(k) → S a geometric point of S (i.e. x is a 1-morphism and k is an algebraically closed field). The complete local ring Ô_{S,x} of S at x is the completion of the local ring at x of any étale neighbourhood of x on S.

Definition 4.2. We denote by W the stack in groupoids over Sch whose category of sections over a scheme S is the category of quintuples

(S, A_{0} ⟶^{α} … ⟶ A_{g}, λ_{0}, λ_{g}, ϕ)

where A. = (S, A_{0} ⟶^{α} … ⟶ A_{g}, λ_{0}, λ_{g}) is a system of type II and

ϕ : St. ⊗ O_{S} ⟶ M(A.)

is an isomorphism of systems of O_{S}-modules of type II. Here M(A.) denotes the system induced by A. on S.

Proposition 4.3. The morphism of stacks

π : W ⟶ S(g, p)

induced by the forgetful functor is a representable, separated, smooth, surjective morphism of finite type. In particular W is a separated algebraic stack of finite type over ℤ. Proof. Clearly W is representable as a closed subscheme of a product of frame bundles over S. Hence it is separated and of finite type over S. Proposition 3.6 shows that π is surjective and formally smooth. Since S(g, p) is of finite type and separated over Spec(ℤ) the assertions follow. □

Let X over Spec(ℤ) be the scheme representing the functor

Sch ⟶ Sets
S ⟼ The set of all (g + 1)-tuples of subsheaves

ω_{i} ⊂ St_{i} ⊗ O_{S}, i = 0, …, g such that:

- ω_{i} is a locally free direct factor of rank g.

- α(ω_{i+1}) ⊂ ω_{i}

- ω_{0} (resp. ω_{g}) is totally isotropic for q_{0} (resp. ω_{g})

The scheme X is represented by a closed subscheme of

G(g, St_{0}) × G(g, St_{1}) × … × G(g, St_{g})

where G(g, St_{i}) denotes the Grassmanian of all ℙ^{g−1} in Proj(ℤ[St_{i}]). In particular X is a projective scheme over Spec(ℤ).

Let us denote by (W, A_{0} → … → A_{g}, λ_{0}, λ_{g}, ϕ) the universal object over W. The sheaves 𝔻(A_{i})_{W} have the natural locally free direct summands

ω_{A_{i}} ⊂ 𝔻(A_{i})_{W}.

We have 𝔻(α)(ω_{A_{i+1}}) ⊂ ω_{A_{i}} and we have that ω_{A_{0}} and ω_{A_{g}} are isotropic for e_{λ_{0}} and e_{λ_{g}} (see §2). Therefore, if we put ω_{i} := ϕ_{i}^{−1}(ω_{A_{i}}) ⊂ St_{i} ⊗ O_{W} then we see that we get a W-valued point of X. More precisely: a morphism of stacks

f : W ⟶ X.

Remark 4.4. In this case f is not representable. Indeed, f is representable if and only if W is a scheme. However, if x : Spec(k) → W is a geometric point in characteristic 2 then there are nontrivial 2-morphisms x ⟶^{ψ} x : if X is given by the object (Spec(k), A_{0} → … → A_{g}, λ_{0}, λ_{g}, ϕ) then take ψ_{i} : A_{i} → A_{i} to be −id_{A_{i}}.

Since W is an algebraic stack we know what it means to say that f is smooth.

Proposition 4.5. The morphism f is smooth. Proof. Again it is enough to prove that f is formally smooth.

Let R be a ring and I ⊂ R an ideal with I^{2} = (0). Let S = Spec(R/I) and S′ = Spec(R). Suppose we are given a commutative diagram

x : S ⟶ W
↓ ↓^{f}
y : S′ ⟶ X

and x is given by (A., ϕ) = (S, A_{0} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g}, ϕ).

We can find nilpotent powers γ on I simply by taking γ_{n} = 0 ∀ n ≥ 2. This gives a system of R-modules M(A.) of type II (see Proposition 3.1). By Proposition 3.6 we can take an isomorphism

ϕ′ : St. ⊗ R ⟶ M(A.)

which lifts the isomorphism

St. ⊗ R/I ⟶ M(A.) ⊗_{R} R/I

defined by ϕ.

Let us consider the submodules ϕ′_{i}(ω_{i}) ⊂ M(A.)_{i} where ω_{i} ⊂ St_{i} ⊗ R are the submodules given by our point y : S′ → X. As ϕ′ lifts ϕ and y lifts f ∘x, the submodules ϕ′_{i}(ω_{i}) ⊂ M(A.)_{i} = Γ(S′, 𝔻(A_{i})_{S′}) will lift ω_{A_{i}} ⊂ 𝔻(A_{i})_{S}. Hence by Theorem 2.1 we get abelian schemes A′_{i} → S′ lifting the abelian schemes A_{i} → S. By Theorem 2.1 and Corollary 2.2 the fact that y is a point of X is exactly equivalent to the fact that the maps α and the polarizations λ_{o} and λ_{g} lift to give maps A′_{i} → A′_{i+1} and polarizations on A′_{0} and A′_{g}. By the rigidity lemma ([GIT] Proposition 6.1) any relations between the morphisms of the system (S, A_{0} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g}) will also hold between the corresponding morphisms of the system (S′, A′_{0} → … → A′_{g}, λ_{0}, λ_{g}). Hence (S′, A′_{0} → … → A′_{g}, λ′_{0}, λ′_{g}) will be a system of type II and (S′, A′_{0} → … → A′_{g}, λ′_{0}, λ′_{g}, ϕ′) will define the morphism S′ → W we were looking for. □

Corollary 4.6. For any geometric point x : Spec(k) → S(g, p) there exists a geometric point y : Spec(k) → X such that

Ô_{S(g,p),x} ≃ Ô_{X,y}.

In other words: x and y have isomorphic étale neighbourhoods in S(g, p) and X. (See [Ar] Corollary 2.6.) We can take y = f(z) where z : Spec(k) → W is any geometric point such that π(z) = x.

Proof. Take a geometric point z : Spec(k) → W with π(z) = x. Put y = f(z). By Proposition 4.3 we know that

Ô_{W,z} ≃ Ô_{S(g,p),x}[[t_{1}, …, t_{m}]]

for some m and by Proposition 4.5 we know that

Ô_{W,z} ≃ Ô_{X,y}[[s_{1}, …, s_{l}]]

for some l. If we show that

dim_{k} T_{x}S(g, p) = dim_{k} T_{y}X  (1)

then it follows that m = l. Repeated application of the lemma below will then give the result.

For any k-vectorspace V, k ⊕ V will be the ring in which V is a square zero ideal. On the ideal V we take the nilpotent divided power structure γ_{V} which has γ_{V,n} = 0 ∀ n ≥ 2. For any abelian variety A over Spec(k) we have the isomorphism

𝔻(A)_{Spec(k⊕V)} ≃ 𝔻(A)_{Spec(k)} ⊗_{k} (k ⊕ V)

since we can take the trivial lift of A to Spec(k⊕V). This isomorphism is functorial with respect to both A and V.

Suppose the point z is given by (Spec(k), A_{0} ⟶^{α} … ⟶^{α} A_{g}, λ_{0}, λ_{g}, ϕ) = (A., ϕ). By the remarks above, the isomorphism ϕ will induce for any k-vector space V an isomorphism M_{V} (A.) ≃ St. ⊗ (k ⊕ V). (Here M_{V} (A.) denotes the system of k ⊕ V -modules of type II induced by A. on Spec(k ⊕ V).) Hence the same reasoning as in the proof of Proposition 4.5 gives us 1-1 correspondences:

morphisms  morphisms
Spec(k ⊕ V) ⟶ S(g, p) ⟷ Spec(k ⊕ V) ⟶ X
reducing to x mod V  reducing to y mod V

functorial in the k-vectorspace V. This is equivalent to giving an isomorphism of tangent spaces T_{x}S(g, p) ≃ T_{y}X (see [Sch] Lemma 2.10). □

Lemma 4.7. Suppose (R_{1}, m_{1}, k_{1}) and (R_{2}, m_{2}, k_{2}) are Noetherian complete local rings. If R_{1}[[t]] ≃ R_{2}[[s]] then R_{1} ≃ R_{2}. Proof. Remark that R_{1}[[t]] ≃ R_{2}[[s]] implies k_{1} ≃ k_{2} = k. If we denote by m̃_{1} (resp. m̃_{2}) the maximal ideal of R_{1}[[t]] (resp. R_{2}[[s]]) then we see that ∀n ∈ ℕ

m̃_{1}^{n}/m̃_{1}^{n+1} ≃ m_{1}^{n}/m_{1}^{n+1} ⊕ t · m_{1}^{n−1}/m_{1}^{n} ⊕ … ⊕ t^{n} · k

and similar for m̃_{2}/m̃_{2}^{n+1}. Hence we get for all n ∈ ℕ

dim_{k} m_{1}^{n}/m_{2}^{n+1} = dim_{k} m̃_{1}^{n}/m̃_{1}^{n+1} − dim_{k} m̃_{1}^{n−1}/m̃_{1}^{n}
= dim_{k} m̃_{2}^{n}/m̃_{2}^{n+1} − dim_{k} m̃_{2}^{n−1}/m̃_{2}^{n} = dim_{k} m_{2}^{n}/m_{2}^{n+1}.  (2)

Let us study the composition

π : R_{1} ⟶ R_{1}[[t]] ⟶^{∼} R_{2}[[s]] ⟶ R_{2}[[s]]/(s − x) ⟶^{∼} R_{2}

for some x ∈ m_{2}. On cotangent spaces we get

\begin{matrix}\overline{π} : & m_{1}/m_{1}^{2} ↪ m_{1}/m_{1}^{2} ⊕ k · t ⟶^{∼} m_{2}/m_{2}^{2} ⊕ k · s ⟶ \\ ⟶ (m_{2}/m_{2}^{2} ⊕ k · s)/k · (s − \overline{x}) ⟶^{∼} m_{2}/m_{2}^{2}. \end{matrix}

where \overline{x} denotes the class of x in m_{2}/m_{2}^{2}. From this we see that we can choose x ∈ m_{2} such that \overline{π} is an isomorphism of m_{1}/m_{1}^{2} onto m_{2}/m_{2}^{2}. Hence π will be surjective. It follows from (2) that π is an isomorphism. □

5. The structure of S(g, p) for g = 2

In this section we study the scheme X for g = 2. The results of Section 4 will allow us to determine the structure of S(2, p) fairly completely.

Let us consider the point x_{0} ∈ X(𝔽_{p}) corresponding to the flags

\overline{ω}_{0} = ⟨e_{2}, e_{3}⟩ ⊂ St_{0} ⊗ 𝔽_{p}
\overline{ω}_{1} = ⟨e_{0}, e_{3}⟩ ⊂ St_{1} ⊗ 𝔽_{p}
\overline{ω}_{2} = ⟨e_{0}, e_{1}⟩ ⊂ St_{2} ⊗ 𝔽_{p}.

In a suitable Zariski open neighbourhood U of x_{0} in X the universal flags ω_{i} over X are given by

(1) \{ \begin{matrix}ω_{0} = ⟨e_{2} + a_{11}e_{0} + a_{12}e_{1}, e_{3} + a_{21}e_{0} + a_{22}e_{1}⟩ ⊂ St_{0} ⊗ O_{U} \\ ω_{1} = ⟨e_{0} + b_{11}e_{1} + b_{12}e_{2}, e_{3} + b_{21}e_{1} + b_{22}e_{2}⟩ ⊂ St_{1} ⊗ O_{U} \\ ω_{2} = ⟨e_{0} + c_{11}e_{2} + c_{12}e_{3}, e_{1} + c_{21}e_{2} + c_{22}e_{3}⟩ ⊂ St_{2} ⊗ O_{U} \end{matrix}

for certain a_{ij}, b_{ij}, c_{ij} ∈ Γ(U, O_{X}).

Since ω_{0} is isotropic for q_{0} we must have that

q_{0}(e_{2} + a_{11}e_{0} + a_{12}e_{1}, e_{3} + a_{21}e_{0} + a_{22}e_{1}) = 0

that is a_{11} = a_{22}. Since α(ω_{1}) ⊂ ω_{0} we must have that

pe_{0} + b_{11}e_{1} + b_{12}e_{2} ∈ ⟨e_{2} + a_{11}e_{0} + a_{12}e_{1}, e_{3} + a_{21}e_{0} + a_{22}e_{1}⟩

hence

pe_{0} + b_{11}e_{1} + b_{12}e_{2} = b_{12}(e_{2} + a_{11}e_{0} + a_{12}e_{1})

and we get the equations p = b_{12}a_{11} and b_{11} = b_{12}a_{12}. Continuing in this manner (using that α(ω_{2}) ⊂ ω_{1} and that ω_{2} is isotropic for q_{2}) we derive the system of equations:

(2) \{ \begin{matrix}a_{11} & = a_{22} \\ p & = b_{12}a_{11} \\ b_{11} & = b_{12}a_{12} \\ 0 & = a_{21} + b_{22}a_{11} \\ b_{21} & = a_{22} + b_{22}a_{12} \end{matrix} \begin{aligned}0 & = b_{11} + c_{12}b_{21} \\ c_{11} & = b_{12} + c_{12}b_{22} \\ p & = c_{22}b_{21} \\ c_{21} & = c_{22}b_{22} \\ c_{11} & = c_{22} \end{aligned}

Conversely, over the ring

R = ℤ[a_{ij}, b_{ij}, c_{ij}]/ the ideal generated by the system of equations (2)

we get a universal system of flags which can be written in the form (1). These arguments give that U = Spec(R) is an open subscheme of X.

If we write x = a_{11}, y = b_{12}, a = c_{12}, b = a_{12} and c = b_{22} then it is easy to see that R is isomorphic to

ℤ[x, y, a, b, c]/(xy − p, ax + by + abc).

We see that U is smooth over Spec(ℤ[1/p]) and that the fibre U_{0} of U → Spec(ℤ) over Spec(𝔽_{p}) consists of four components:

U_{0} = Z_{00} ∪ Z_{01} ∪ Z_{10} ∪ Z_{11}

defined by

ideal of Z_{00} is (y, a)
ideal of Z_{01} is (y, x + bc)
ideal of Z_{10} is (x, y + ac)
ideal of Z_{11} is (x, b).

In particular the dimension of U_{0} is 3 and hence the morphism U → Spec(ℤ) is a locally complete intersection morphism.

We remark that U ⊂ X is dense in every fibre of X → Spec(ℤ). The group scheme G (see Remark 3.7) acts on X and the morphism G × U → X is surjective. Thus any point of X has an open neighbourhood isomorphic to an open subset of U.

Finally let us give a list of the isomorphism types of the complete local rings at geometric points x : Spec(k) → X in characteristic p:

- smooth: W (k)[[x_{1}, x_{2}, x_{3}]]

- normal crossings:

\{ \begin{matrix}W (k)[[x_{1}, x_{2}, x_{3}, x_{4}]]/(x_{1}x_{2} − p) \\ W (k)[[x_{1}, x_{2}, x_{3}, x_{4}]]/(x_{1}x_{2}x_{3} − p) \end{matrix}

- fibre products of normal crossings:

W (k)[[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}]]/(x_{1}x_{2} − p, x_{3}x_{4} − p)

- If x = x_{0} then W (k)[[x, y, a, b, c]]/(xy − p, ax + by + abc).

Using the above what can we say about S(2, p)? Of course by Corollary 4.6 we see that the complete local rings of S(2, p) at geometric points are of the simple types listed above. In particular S(2, p) → Spec(ℤ) is a locally complete intersection morphism of relative dimension three.

Let k be a perfect field of characteristic p. Suppose ω_{i} ⊂ St_{i} ⊗ k, i = 0, 1, 2 define a geometric point x : Spec(k) → X. Let us define the following numbers:

\begin{aligned}σ_{i}(x) & = dim_{k}(ω_{i}/α(ω_{i+1})) \\ τ_{i}(x) & = dim_{k}(M_{i}/α(M_{i+1}) + ω_{i}) \end{aligned}

for i = 0, 1. The possibilities for (σ_{i}(x), τ_{i}(x)) are (1,0), (0,1) and (1,1). Assume x is the image under f of a point z : Spec(k) → W given by (Spec(k), A_{0} → A_{1} → A_{2}, λ_{0}, λ_{2}, ϕ). The group scheme G_{i} = Ker(A_{i} → A_{i+1}), i = 0, 1 is isomorphic to μ_{p}, ℤ/pℤ or α_{p}. The relation with the numbers σ_{i}(x) and τ_{i}(x) is the following:

isomorphism type of G_{i}  μ_{p}  ℤ/pℤ  α_{p}
(σ_{i}(x), τ_{i}(x))  (1,0)  (0,1)  (1,1)

This is true since ω_{G_{i}} ≃ ω_{A_{i}}/ω_{A_{i+1}} and similarly τ_{i}(x) is equal to the dimension of the space of invariant differentials on the dual of G_{i}.

In this way we see that we can define the ordinary locus X^{ord} ⊂ X_{0} = fibre of X over Spec(𝔽_{p}) as follows: X^{ord}(k) is the set of points x of X_{0} such that (σ_{i}(x), τ_{i}(x)) is either (1, 0) or (0, 1) for i = 0, 1. The preceding discussion shows that

π^{−1} (ordinary locus of S(2, p) × Spec(𝔽_{p})) = f^{−1}(X^{ord}) ⊂ W × Spec(𝔽_{p})

A glance at (1) shows that:

generic value of  generic value of
Component of X_{0}  (σ_{0}(x), τ_{0}(x))  (σ_{1}(x), τ_{1}(x))
Z_{00}  (1,0)  (1,0)
Z_{01}  (1,0)  (0,1)
Z_{10}  (0,1)  (1,0)
Z_{11}  (0,1)  (0,1)

In particular we see that the ordinary locus X^{ord} is dense in X_{0} and hence the same is true for S(2, p).

The ordinary locus of S(2, p) × Spec(𝔽_{p}) is the disjoint union of 4 parts:

ordinary locus of S(2, p) × Spec(𝔽_{p}) = S_{00} ⨿ S_{01} ⨿ S_{10} ⨿ S_{11}.

The part S_{00} can be defined as the image of f^{−1}(Z_{00} ∩X^{ord}) under π or as the locus where G_{0} ≃ μ_{p} and G_{1} ≃ μ_{p}. Similar for S_{01}, S_{10} and S_{11}. The above arguments show that S(2, p)×Spec(𝔽_{p}) is the union of the closures of the S_{ij}. Hence if we show that S_{ij} is irreducible then these closures are the components of S(2, p) × Spec(𝔽_{p}). A standard argument using the theorem on the monodromy of the p-torsion points of the universal abelian scheme over A_{2} × Spec(𝔽_{p}) (see [Ek] or [FC]) shows that S_{ij} is indeed irreducible.

Let us conclude the discussion of S(2, p) by exhibiting a geometric point s_{0} : Spec(\overline{𝔽}_{p}) → S(2, p) which corresponds to the point x_{0} ∈ X(\overline{𝔽}_{p}). To do this we remark that the point x_{0} ∈ X(\overline{𝔽}_{p}) is characterized by the following property: The map α^{2} | \overline{ω}_{i} : \overline{ω}_{i} → St_{i−2} ⊗ \overline{𝔽}_{p} is zero for i = 0, 1, 2. Here we used the constructions of §3 to get modules and maps St_{2} ⊗ \overline{𝔽}_{p} → St_{1} ⊗ \overline{𝔽}_{p} → St_{0} ⊗ \overline{𝔽}_{p} → St_{−1} ⊗ \overline{𝔽}_{p} → St_{−2} ⊗ \overline{𝔽}_{p} = St_{2} ⊗ \overline{𝔽}_{p}. This means that a geometric point z : Spec(\overline{𝔽}_{p}) → W given by (Spec(\overline{𝔽}_{p}), A_{0} →^{α} A_{1} →^{α} A_{2}, λ_{0}, λ_{2}, ϕ) projects to x_{0} if and only if

\begin{matrix}Ker(α^{2} : A_{0} ⟶ A_{2}) ≃ α_{p} × α_{p} \\ Ker(A_{1} ⟶^{α} A_{2} ⟶^{λ_{2}} A_{2}^{t} ⟶^{α^{t}} A_{1}^{t}) ≃ α_{p} × α_{p}. \end{matrix}

We can construct such a system (A_{0} → A_{1} → A_{2}, λ_{0}, λ_{1}) as follows. Let E be a supersingular elliptic curve over \overline{𝔽}_{p}. Put A_{0} = E × E and λ_{0} equal to the product polarization. We put A_{2} = E × E/α_{p} × α_{p} and A_{1} = A_{0}/α_{p} where we embed α_{p} ↪ E × E = A_{0} by the map (1, γ) : α_{p} ↪ α_{p} × α_{p} ⊂ E × E for some γ ∈ \overline{𝔽}_{p}. As in the proof of Proposition 1.7 we get λ_{2} on A_{2} for free. If γ^{p+1} = −1 then Ker(A_{1} → A_{2} →^{∼} A_{2}^{t} → A_{1}^{t}) will be isomorphic to α_{p} × α_{p} (see [MB] or [KO]).

For general g we have the following questions:

– If g ≥ 3 is it still true that S(g, p) is flat over Spec(ℤ)?

– If g ≥ 3 is it still true that the ordinary locus is dense in the fibre of S(g, p) over Spec(𝔽_{p})? We expect that the answer to both questions is yes. If this is so then the same reasoning as above will show that S(g, p) × Spec(𝔽_{p}) has 2^{g} components.

For general g also, it should be possible to construct a compactification of S(g, p) using the methods of [FC].

Finally, let us report on some of the calculations we did for the case g = 3. The point x_{0} ∈ X(𝔽_{p}) is also characterized by the fact that it is stabilized by the whole group G×Spec(𝔽_{p}). We computed the equations defining a neighbourhood of a point x ∈ X(𝔽_{p}) stabilized by the group G × Spec(𝔽_{p}) for the case g = 3. These computations suggest that X is flat over Spec(ℤ) and that the ordinary locus is dense in X × Spec(𝔽_{p}) (but we did not give a complete proof). However, it is definitely not true that X is a locally complete intersection over Spec(ℤ) or that X × Spec(𝔽_{p}) is the union of smooth components as is the case for g = 2.

References

[Ar] M. Artin, Algebraic approximation of structures over complete local rings. Publ. Math.

I.H.E.S. 36 (1969), 23-34.

[BBM] P. Berthelot, L. Breen, W. Messing, Théorie de Dieudonné cristalline II. Lecture Notes in

Math. 930, Springer-Verlag (1982).

[CN] C.-L. Chai, P. Norman, Bad reduction of the Siegel moduli scheme of genus two with Γ_{0}(p)-level structure. Amer. Journal of Math. 112 (1990), 1003-1071.

[DM] P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S. 36 (1969), 75-109.

[Ek] T. Ekedahl, The action of monodromy on torsion points of Jacobians. In: Arithmetic Algebraic Geometry, ed.: G. van der Geer, F. Oort, J.H.M. Steenbrink. Progress in Mathematics Vol. 89, Birkhäuser (1991), 41-49.

[FC] G. Faltings, C.-L. Chai, Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer grenzgebiete, 3. Folge, Band 22, Springer-Verlag (1990).

[GIT] D. Mumford, Geometric invariant theory, second enlarged edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag (1982).

[Gr] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné. Séminaire de Mathématiques Supérieur 45, Presses de l’Université de Montréal (1974).

[KO] T. Katsura, F. Oort, Families of supersingular abelian surfaces. Compos. Math. 62 (1987),

107-167.

[Ma Me] B. Mazur, W. Messing, Universal extensions and one-dimensional crystalline cohomology.

Lecture Notes in Math. 370, Springer-Verlag (1974).

[MB] L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur ℙ^{1} II. In: Séminaire sur les pinceaux de courbes de genre au moins deux. Editor: L. Szpiro. Astérisque 86, société mathématique de France, Paris (1981).

[Me] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, (1972).

[Mu] D. Mumford, Abelian Varieties. Tata Inst. Fund. Research Studies in Math. Vol. 5, Oxford Univ. Press, Oxford, 1970.

[Sch] M. Schlessinger, Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208-222.