The moduli spaces of polarized abelian varieties

A.J. de Jong*

Introduction

In this article we study the moduli spaces A_{g,d} over Spec(ℤ) parametrizing abelian varieties of dimension g with a polarization of degree d^{2}. We use the language of algebraic stacks since it seems the most natural language to describe moduli spaces with.

Although these moduli spaces are in a sense the most basic moduli spaces of abelian varieties not much seems to be known about their local and global structure. In [Mu] the components of A_{g,d} are determined using the spaces A_{g,δ} parametrizing the polarized abelian varieties with a fixed sequence of ”elementary divisors” δ = (δ_{1}, …, δ_{g}) of positive integers δ_{1} | δ_{2} … | δ_{g} such that ∏_{i=1}^{g} δ_{i} = d associated to the polarization. In [Cr] the local structure of A_{g,d} at a point of A_{g,δ} is determined. Finally, in [NO] the stratification by p-rank V_{0} ⊂ V_{1} ⊂ … ⊂ V_{g−1} ⊂ V_{g} = = A_{g,d} ⊗ 𝔽_{p} is studied; it is shown there that the dimension of V_{g−i} is equal to \frac{1}{2}g(g + 1) − i ([NO] Theorem 4.1).

In §1 we (re)define the locally closed substacks A_{g,δ} ⊂ A_{g,d} whose closures are the components of A_{g,d}. We prove that all geometric fibres of the morphisms A_{g,δ} → Spec(ℤ) are irreducible; this answers a question of Mumford ([Mu], top of page 458; [GIT], page 189). It follows that the closures of A_{g,δ} ⊗ \overline{𝔽}_{p} are the irreducible components of A_{g,d} ⊗ \overline{𝔽}_{p}.

In §2 we determine how the fibres of the components of A_{g,d} over the finite primes split up into irreducible components. It turns out that this question is equivalent to a question in the theory of finite abelian groups (Proposition 2.3).

In §3 we continue the study of the stratification by p-rank of A_{g,d} ⊗ \overline{𝔽}_{p} which was started in [NO]. The locus Z := V_{g−1}\V_{g−2} ⊂ A_{g,d}⊗\overline{𝔽}_{p} corresponding to abelian varieties of p-rank exactly g − 1 is decomposed into irreducible components. Finally we determine which components of Z are contained in a fixed component of A_{g,d} ⊗ \overline{𝔽}_{p}; again this turns out to be equivalent to a question in the theory of finite abelian groups (Proposition 3.4).

To go further it seems necessary to determine whether or not the locus V_{g−2} ⊂ A_{g,1} ⊗ \overline{𝔽}_{p} is irreducible for g large. For g = 2 this is not the case ([KO]), while for g = 3 it is (Oort, private communication). More generally, we would like to know the answer to the same question for the substacks V_{f} (f arbitrary).

The author would like to thank B. Edixhoven for his help with the proof of Proposition 1.5 and Prof. F. Oort for many stimulating conversations.

* Supported by the Netherlands Organisation for Scientific Research (N.W.O.), in the project Arithmetic Algebraic Geometry

Notation and conventions

– The letter p will always denote a prime number.

– 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.

– A (locally) closed substack of an algebraic stack S is a full subcategory T ⊂ S which is an algebraic stack, which contains together with any t ∈ Ob(S) all isomorphic t′ and such that the inclusion T → S is representable by (locally) closed immersions. In this case (T )^{c} will denote the closure of T , a closed substack of S.

– If T is an algebraic stack and k is a field then T ⊗ k denotes the 2-fibre product T ×_{Spec(ℤ)} Spec(k).

– 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 stacks of polarized abelian varieties

In this section we follow the exposition of Mumford in [Mu] §1 and [GIT] Appendix 7A. Further we prove that A_{g,δ} ⊗ k is irreducible for any field k.

Definition 1.1. For any positive integers g, d let A_{g,d} be the category of triples (S, X, λ) where S is a scheme, X is an abelian scheme over S of relative dimension g and λ : X → X^{t} is a polarization of degree d^{2}. The natural functor

\begin{matrix}A_{g,d} & ⟶ & Sch \\ (S, X, λ) & ⟼ & S \end{matrix}

makes A_{g,d} into a stack.

We collect a number of results on A_{g,d} in a series of remarks.

Remarks 1.2.

1) Our notations will be consistent with those of [GIT] Appendix 7A with the difference that our symbols A_{g,d}, A_{g,δ}, etc. will always denote algebraic stacks (cf. Remark 1.10 and Remark 1.13.1).

2) The stack A_{g,d} is a separated algebraic stack of finite type over Spec(ℤ). This can be proved using the theory of Hilbert schemes as is done in [DM] §5 in the case of the stack classifying stable curves of genus g.

3) There exists a coarse moduli scheme A_{g,d,1} for the algebraic stack A_{g,d} (see [GIT] Definition 7.4 and Theorem 7.10). The scheme A_{g,d,1} is quasi-projective over Spec(ℤ) (see [MB] Theorem 4.2).

4) The morphism A_{g,d} → Spec(ℤ) is a locally complete intersection morphism of relative dimension \frac{1}{2}g(g + 1). This follows from a theorem of Mumford ([Oo] Theorem 2.3.3) since we know that every geometric fibre of A_{g,d} → Spec(ℤ) has pure dimension \frac{1}{2}g(g + 1) (see [NO] Theorem 3.1).

5) The morphism A_{g,d} → Spec(ℤ) is smooth over the open subscheme Spec(ℤ[1/d]). This follows from the theorem of Grothendieck that there are no obstructions to deforming an abelian variety with a separable polarization (see [Oo] Theorem 2.4.1 and Errata; see also [Cr] Theorem 12).

For any object (S, X, λ) of A_{g,d} the kernel of λ is a finite flat group scheme of rank d^{2} over S. We will construct a nondegenerate alternating pairing

e^{λ} : Ker(λ) ×_{S} Ker(λ) ⟶ 𝔾_{m,S}.

Locally in the étale topology on S we can choose a rigidified line bundle (i.e. a line bundle trivialized along the origin) L on X such that Λ(L) = λ (see [GIT] Definition 6.2). This follows from [Oo] Lemma 2.3.2 and the representability of P ic_{X/S}. The commutator on the theta-group G(L) induces a nondegenerate alternating pairing e^{L} on K(L) = Ker(λ) (see [AV] §23 or [MB2] §3). Any other such line bundle will be of the form L ⊗ M for some M ∈ P ic_{X/S}^{0}. But e^{L⊗M} = e^{L} · e^{M} and e^{M} ≡ 1 (since K(M) = X; see [MB2] Appendice). Hence e^{L} does not depend on the choice of L and it will descend to give the paring e^{λ} as desired.

This gives certain restrictions on the group scheme Ker(λ). Let us construct some candidates for it. For every sequence δ = (δ_{1}, …, δ_{g}) of positive integers δ_{1} | δ_{2} … | δ_{g} such that ∏_{i=1}^{g} δ_{i} = d we put

K(δ) = ∏_{i=1}^{g} μ_{δ_{i},Spec(ℤ)} × ℤ/δ_{i}ℤ_{Spec(ℤ)}.

On K(δ) → Spec(ℤ) we have the nondegenerate alternating pairing

\begin{matrix}e_{δ} : K(δ) × K(δ) & ⟶ & 𝔾_{m,Spec(ℤ)} \\ ((ζ_{i}, n_{i})_{i=1...g}, (η_{i}, m_{i})_{i=1...g}) & ⟼ & ∏_{i=1}^{g} ζ_{i}^{m_{i}} · η_{i}^{−n_{i}}. \end{matrix}

Definition 1.3. Let A_{g,δ} be the full subcategory of A_{g,d} consisting of those objects (S, X, λ) of A_{g,d} such that there exists a faithfully flat T → S such that Ker(λ)_{T} ≃ K(δ)_{T}.

Remarks 1.4.

1) The category A_{g,δ} endowed with the natural functor A_{g,δ} → Sch is a stack.

2) We would have gotten the same subcategory A_{g,δ} ⊂ A_{g,d} if, in Definition 1.3, we had asked for the existence of an isomorphism (Ker(λ)_{T} , e_{T}^{λ}) ≃ (K(δ)_{T} , e_{δ,T}) after a faithfully flat base change T → S.

Proposition 1.5. The stack A_{g,δ} is a separated algebraic stack of finite type over Spec(ℤ). The morphism A_{g,δ} → A_{g,d} is representable by locally closed immersions; i.e. A_{g,δ} is a locally closed substack of A_{g,d}.

We will show that the morphism of stacks A_{g,δ} → A_{g,d} is representable by locally closed immersions. The proposition will then follow from the following lemma.

Lemma 1.6. If F : S_{1} → S_{2} is a representable morphism of stacks and S_{2} is an algebraic stack then so is S_{1}. If, in addition, F is separated (resp. of finite type) and S_{2} is separated (resp. of finite type over Spec(ℤ)) then S_{1} is separated (resp. of finite type over Spec(ℤ)). Proof. The proof is formal and is left to the reader. □

To prove that A_{g,δ} → A_{g,d} is representable by locally closed immersions we have to prove the following general fact: Suppose G is a group scheme finite locally free of rank d^{2} over a scheme S, then there exists a locally closed subscheme Z ⊂ S such that a morphism S′ → S factors through Z if and only if G_{S′} becomes isomorphic to K(δ)_{S′} after a faithfully flat base extension T′ → S′.

By taking primary parts of G and K(δ) we may assume that d is a prime power, say d = p^{n}. As a first approximation let Z_{1} be the locally closed subscheme of S where the rank of G[p^{l}] is constant and equal to the rank of K(δ)[p^{l}] for all l. If p is invertible on S then Z_{1} already works. In Z_{1} ⊗ 𝔽_{p} we have the closed subset Z_{2} defined by

(G_{1}[F])^{(p)} ∩ G_{1}^{(p)}[V] ≠ 0.

Here F : G_{1} → G_{1}^{(p)} (resp. V : G_{1}^{(p)} → G_{1}) denotes the Frobenius morphism (resp. Verschiebung) of G_{1} := G_{|Z_{1}⊗𝔽_{p}} over Z_{1} ⊗ 𝔽_{p}. Thus Z_{2} is the locus of Z_{1} ⊗ 𝔽_{p} where the local-local part of G is nonzero.

Put Z_{3} = Z_{1} \ Z_{2}. Take an algebraically closed field k of characteristic p and a point z ∈ Z_{3}(k). By the above we have

G_{z} ≃ ∏ μ_{p^{n_{i}}} × ∏ ℤ/p^{m_{j}}ℤ

for some n_{i}, m_{j} ∈ ℕ. In a connected étale neighbourhood U(z) of z in Z_{3} the surjection G_{z} → ∏ ℤ/p^{m_{j}}ℤ extends:

G_{U(z)} ⟶ ∏ ℤ/p^{m_{j}}ℤ_{U(z)} ⟶ 0.

Similarly for the injection ∏ μ_{p^{n_{i}}} → G_{z} and we see that G_{U(z)} becomes an extension

0 ⟶ ∏ μ_{p^{n_{i}} ,U(z)} ⟶ G_{U(z)} → ∏ ℤ/p^{n_{i}}ℤ_{U(z)} ⟶ 0.  (∗)

Since the ranks of G[p^{l}] are constant on Z_{3} we see that

0 ⟶ (∏ μ_{p^{n_{i}} ,U(z)}) [p^{l}] ⟶ G_{U(z)}[p^{l}] ⟶ (∏ ℤ/p^{n_{i}}ℤ_{U(z)}) [p^{l}] ⟶ 0

is exact also. Hence after a faithfully flat base extension T (z) → U(z) the extension (∗) becomes split:

G_{T (z)} ≃ ∏ μ_{p^{n_{i}} ,T (z)} × ∏ ℤ/p^{m_{j}}ℤ_{T (z)} .

These arguments show that Z_{3} ⊗ 𝔽_{p} is the disjoint union of subschemes where the numbers n_{i} and m_{j} are constant. Finally to get Z we throw out of Z_{3} the components of Z_{3} ⊗ 𝔽_{p} where {p^{n_{i}}} ≠ {δ_{1}, …, δ_{g}} or {p^{m_{j}}} ≠ {δ_{1}, …, δ_{g}} . It follows from the considerations above that Z works: Any such morphism S′ → S clearly factors through Z and the morphisms T (z) → Z (for z ∈ Z(k)) are flat and cover a neighbourhood of Z ⊗ 𝔽_{p} in Z. □

Suppose k is an algebraically closed field of characteristic p > 0 and suppose x : Spec(k) → A_{g,δ} is a geometric point given by the polarized abelian variety (X_{0}, λ_{0}) over Spec(k). In [Cr] the complete local ring of A_{g,d} at x is computed. We will describe the result.

Let us consider the moduli functor P[X_{0}, λ_{0}] of the pair (X_{0}, λ_{0}). See [Cr] Definition 8. We have

Ker(λ_{0}) = M_{0} ⊕ M_{0}^{D} ⊕ Ker(λ_{0})_{ét,ét}

where

M_{0} = ∏_{i=1}^{g} ℤ/p^{a_{i}}ℤ_{Spec(k)}

and a_{i} = ord_{p}(δ_{i}). Let R be an Artinian local algebra with residue field k and let (X, λ) be a lift over Spec(R) of the pair (X_{0}, λ_{0}) over Spec(k). We have a decomposition

Ker(λ) = A ×_{Spec(R)} B

where A is the p-part of Ker(λ) and B is a group scheme of exponent prime to p. We have a natural surjection

π : A ⟶ ∏_{i=1}^{g} ℤ/p^{a_{i}}ℤ_{Spec(R)}.

Indeed, this is simply the map to the group of components of A. Dually, we get an injection

i : ∏_{i=1}^{g} μ_{p^{a_{i}} ,Spec(R)} ⟶ A

such that

e^{λ}(i(ζ_{1}, …, ζ_{g}), x) = ∏_{i=1}^{g} ζ_{i}^{n_{i}}

if π(x) = (\overline{n}_{1}, …, \overline{n}_{g}) ∈ ∏_{i=1}^{g} ℤ/p^{a_{i}}ℤ. Further Im(i) = Ker(π) and Ker(π) is isotropic for e^{λ}.

Let e_{1}, …, e_{g} denote the natural generators of ∏_{i=1}^{g} ℤ/p^{a_{i}}ℤ. After a faithfully flat base extension T → Spec(R) we can find ẽ_{1}, …, ẽ_{g} ∈ A(T) such that π(ẽ_{i}) = e_{i}. We define elements η_{ij} ∈ 𝔾_{m}(T) by the formula

η_{ij} = e^{λ}(p^{a_{j}} · ẽ_{j}, ẽ_{i})

for all 1 ≤ i < j ≤ g.

Proposition 1.7.

1) The element η_{ij} lies in μ_{p^{a_{i}}} (R).

2) (Spec(R), X, λ) ∈ ObA_{g,δ} ⟺ η_{ij} = 1 ∀ i, j, 1 ≤ i < j ≤ g.

3) η_{ij} = ε_{g−j,g−i} if e_{i} maps to h_{g+1−i} in the notation of [Cr], page 859. Proof. Since Ker(π) is isotropic for e^{λ} we have

η_{ij}^{p^{a_{i}}} = e^{λ}(p^{a_{j}} ẽ_{j}, p^{a_{i}} ẽ_{i}) = 1.

Suppose we change ẽ_{i_{0}} into ẽ_{i_{0}} + α for some α ∈ Ker(π)(T). If i < i_{0} then

e^{λ}(p^{a_{i_{0}}} (ẽ_{i_{0}} + α), ẽ_{i}) = e^{λ}(p^{a_{i_{0}}} ẽ_{i_{0}}, ẽ_{i}) · e^{λ}(α, p^{a_{i_{0}}} ẽ_{i}) = η_{ii_{0}}

since both p^{a_{i_{0}}} ẽ_{i} and α are in Ker(π)(T). If i_{0} < j then

e^{λ}(p^{a_{j}} ẽ_{j}, ẽ_{i_{0}} + α) = e^{λ}(p^{a_{j}} ẽ_{j}, ẽ_{i_{0}}) · e^{λ}(p^{a_{j}} ẽ_{j}, α) = η_{i_{0}j}

since both p^{a_{j}} ẽ_{j} and α are in Ker(π)(T). We have shown that η_{ij} does not depend on the choice of either ẽ_{i} or ẽ_{j} and hence η_{ij} will be in μ_{p^{a_{i}}} (R).

Suppose (Spec(R), X, λ) ∈ ObA_{g,δ}. This means that after a faithfully flat base change T → Spec(R) we get an isomorphism Ker(λ)_{T} ≃ K(δ)_{T}. Hence the map π_{T} has a section since the corresponding map for K(δ)_{T} has a section. Thus we can take ẽ_{i} such that p^{a_{i}} ẽ_{i} = 0 and in particular η_{ij} = 1 ∀ i, j, 1 ≤ i < j ≤ g.

Conversely, suppose η_{ij} = 1 ∀ i, j, 1 ≤ i < j ≤ g. Let us take a faithfully flat base change T → Spec(R) and ẽ_{i} ∈ A(T) such that π(ẽ_{i}) = e_{i}. The element p^{a_{g}} ẽ_{g} lies in Ker(π)(T) and

e^{λ}(p^{a_{g}} ẽ_{g}, ẽ_{i}) = 1

for i = 1, …, g − 1 by assumption and for i = g since e^{λ} is alternating. Thus we see that p^{a_{g}} ẽ_{g} is zero. Let H_{g} ⊂ A_{T} be the subgroup scheme given by

H_{g} = ⟨i({1}^{g−1} × μ_{p^{ag} ,T}), ẽ_{g}⟩ ≃ μ_{p^{ag} ,T} ×_{T} ℤ/p^{a_{g}}ℤ_{T} .

By the description of e^{λ} above we get

A_{T} = H_{g} × H_{g}^{⊥}.

We may assume that the elements ẽ_{g−1}, …, ẽ_{1} are in H_{g}^{⊥}(T). Thus we can give the same argument to prove that H_{g}^{⊥} = H_{g−1} × H_{g−1}^{⊥} with H_{g−1} ≃ μ_{p^{a_{g−1}} ,T} ×_{T} ℤ/p^{a_{g−1}}ℤ_{T} . Continuing like this we see that

A_{T} ≃ H_{g} × … × H_{1} ≃ ∏_{i=1}^{g} μ_{p^{a_{i}} ,T} × ℤ/p^{a_{i}}ℤ_{T} .

Therefore, we may conclude that Ker(λ)_{T} ≃ K(δ)_{T} as there is no difficulty in extending the isomorphism of Ker(λ_{0})_{ét,ét} with (K(δ)_{Spec(k)})_{ét,ét} to an isomorphism of B with the corresponding part of K(δ)_{Spec(R)}.

The elements ε_{ij} of [Cr], page 859 are defined as follows. Let Y be the abelian variety X/Im(i). By [Cr] Proposition 16 we get a polarization μ : Y → Y^{t} such that the following diagram commutes

X ⟶^{λ} X^{t}
↓ ↑
Y ⟶^{μ} Y^{t}.

As Ker(π) = Im(i) we also get the following diagram

∏_{i=1}^{g} μ_{p^{a_{i}} ,Spec(R)} ≃ Im(i)
↓^{i} ↓
A  ↪ X
↓^{π} ↓
∏_{i=1}^{g} ℤ/p^{a_{i}}ℤ_{Spec(R)} ↪ Y.

If we take h_{i} to be e_{g+1−i} then ε_{ij} for i < j is defined by the formula

ε_{ij} = e_{p^{ν_{i}}} (h_{i}, μ(h_{j}))

where e_{p^{ν_{i}}} denotes the Weil pairing Y [p^{ν_{i}}] × Y^{t}[p^{ν_{i}}] → μ_{p^{ν_{i}} ,Spec(R)} and ν_{i} = a_{g+1−i}.

Let us take a line bundle L on Y such that Λ(L) = μ (use [Oo] Lemma 2.3.2). Let M be the pullback of L to X (and thus Λ(M) = λ). Finally, let us take a faithfully flat T → Spec(R) and ẽ_{i} ∈ A(T) such that π(ẽ_{i}) = e_{i}. We have for i < j:

ε_{ij} = e_{p^{ν_{i}}} (h_{i}, μ(h_{j})) = e^{L^{⊗p^{ν_{i}}}} (h_{i}, h_{j}) =
= e^{L^{⊗p^{a_{g+1−i}}}} (e_{g+1−i}, e_{g+1−j}) = e^{M^{⊗p^{a_{g+1−i}}}} (ẽ_{g+1−i}, ẽ_{g+1−j})
= e^{M}(p^{a_{g+1−i}} ẽ_{g+1−i}, ẽ_{g+1−j})
= e^{λ}(p^{a_{g+1−i}} ẽ_{g+1−i}, ẽ_{g+1−j}) = η_{g+1−j,g+1−i}.

Here we have used various formulas of [Od] Section 1. (See also [AV], page 228.) □

Let R be the complete local ring of A_{g,d} at the point x defined by our pair (X_{0}, λ_{0}). In R we have the elements η_{ij} constructed using the universal deformation of (X_{0}, λ_{0}) over Spec(R). By [Cr] Corollary 23 we conclude that

R ≃ W (k)[[t_{1}, …, t_{\frac{1}{2} g(g+1)}]][η_{ij}, 1 ≤ i < j ≤ g]/(η_{ij}^{p^{a_{i}}} − 1).

Proposition 1.8. The morphism A_{g,δ} → Spec(ℤ) is surjective and smooth of relative dimension \frac{1}{2}g(g + 1). Proof. By Proposition 1.5 the complete local ring of A_{g,δ} at the point x above is given by R/I for some ideal I ⊂ R. By Proposition 1.7.2 this ideal is generated by the elements η_{ij} − 1. Hence R/I ≃ W (k)[[t_{1}, …, t_{\frac{1}{2} g(g+1)}]] and the morphism A_{g,δ} → Spec(ℤ) is smooth of relative dimension \frac{1}{2}g(g + 1). To prove that it is surjective we construct for every prime p an object (Spec(\overline{𝔽}_{p}), X, λ) of A_{g,δ}. We can take X = E^{g} where E is an ordinary elliptic curve over Spec(\overline{𝔽}_{p}) and λ = δ_{1} · μ × … × δ_{g} · μ where μ is the canonical principal polarization on E. □

Construction 1.9. We need the algebraic stacks A_{g,1,n}^{∗} parametrizing abelian schemes with a principal polarization and a “sympletic level-n structure”. The objects of the underlying category of A_{g,1,n}^{∗} are quadruples (S, X, λ, α) where

- S is a scheme,

- X is an abelian scheme over S of relative dimension g,

- λ : X → X^{t} is a principal polarization,

- α is an isomorphism of group schemes

α : X[n] ⟶ (μ_{n,S})^{g} ×_{S} (ℤ/nℤ_{S})^{g}

which transforms the Weil paring induced by λ on X[n] into the following paring

\begin{matrix}(μ_{n,S})^{g} ×_{S} (ℤ/nℤ_{S})^{g} ×_{S} (μ_{n,S})^{g} ×_{S} (ℤ/nℤ_{S})^{g} & ⟶ & μ_{n,S} \\ ((ζ_{i})_{i=1...g}, (n_{i})_{i=1...g}, (η_{i})_{i=1...g}, (m_{i})_{i=1...g}) & ⟼ & ∏_{i=1}^{g} ζ_{i}^{m_{i}} · η_{i}^{−n_{i}}. \end{matrix}

It is standard to check that A_{g,1,n}^{∗} is a stack. The forgetful morphism A_{g,1,n}^{∗} → A_{g,1} is representable, affine and of finite type. Hence A_{g,1,n}^{∗} is a separated algebraic stack of finite type over Spec(ℤ) (see Lemma 1.6).

Remarks 1.10.

1) In [GIT], page 189 and [Mu], page 458 the algebraic stack A_{g,1,n}^{∗} is defined as a stack (or moduli space) over Spec(ℤ[ζ_{n}]). Our construction gives A_{g,1,n}^{∗} over Spec(ℤ) and if we make the base change Spec(ℤ[ζ_{n}]) → Spec(ℤ) then we get the stack of Mumford. Usually the moduli space of principally polarized abelian varieties with level-n structure is defined as the moduli space (algebraic stack or coarse moduli space) A over Spec(ℤ[ζ_{n}, 1/n]) classifying principally polarized abelian schemes (X, λ) endowed with a sympletic isomorphism X[n] ≃ (ℤ/nℤ)^{2g}. Here the Weil pairing is thought of as a pairing X[n] × X[n] → ℤ/nℤ where we have used the identification μ_{n} = ℤ/nℤ · ζ_{n} over Spec(ℤ[ζ_{n}, 1/n]). The relation with A_{g,1,n}^{∗} is that there is a canonical isomorphism

A_{g,1,n}^{∗} × Spec(ℤ[ζ_{n}, 1/n]) ≅ A

also given by the identification μ_{n} = ℤ/nℤ · ζ_{n}.

2) The morphism A_{g,1,n}^{∗} → Spec(ℤ) is surjective and smooth of relative dimension \frac{1}{2}g(g + 1). Over Spec(ℤ[1/n]) this is easy. To prove that A_{g,1,n}^{∗} is smooth over Spec(ℤ_{(p)}) we can use the deformation theory of ordinary abelian varieties (see [Ka]): If (Spec(\overline{𝔽}_{p}), X, λ, α) ∈ ObA_{g,1,n}^{∗} and p | n then X must be an ordinary abelian variety. Details are left to the reader.

The geometric fibres of A_{g,1,n}^{∗} → Spec(ℤ) are all irreducible. If the characteristic of the base field does not divide n then this is [FC] IV 6.8. If the characteristic does divide n then this follows from [FC] V 7.1. We can also use Theorems 2.1 and 3.1 of [Ek]. Indeed, Theorem 3.1 of [Ek] asserts that a similar moduli stack M_{g,n}^{∗} of genus g curves has the property that all geometric fibres of M_{g,n}^{∗} → Spec(ℤ) are irreducible. But, since this irreducibility is a question of monodromy of torsion points over M_{g} (resp. A_{g,1}) (see introduction to [Ek]), the result for M_{g,n}^{∗} implies the result for A_{g,1,n}^{∗}.

Consider the morphism (see [GIT], p. 190 and [Mu], p. 458)

\begin{matrix}f : A_{g,1,δ_{g}}^{∗} & ⟶ & A_{g,δ} \\ (S, X, λ, α) & ⟼ & (S, Y, μ) \end{matrix}

where Y is the étale cover of X defined by the diagram

Y ⟶^{μ} Y^{t}
↓^{π}  _{π^{t}}↑
X ⟶^{λ} X^{t}

such that

Ker(π^{t}) = λ (α^{−1} (∏_{i=1}^{g} μ_{δ_{i},S} × (0))) .

For any field k we denote by f ⊗ k the morphism obtained by base extension

f ⊗ k : A_{g,1,δ_{g}}^{∗} ⊗ k ⟶ A_{g,δ} ⊗ k.

If the characteristic of k does not divide δ_{g} then f ⊗k is surjective. If the characteristic of k does divide δ_{g} then the image of f ⊗ k is exactly the ordinary locus of A_{g,δ} ⊗ k. As the ordinary locus is dense in A_{g,d} ⊗ k (see [NO] Theorem 3.1) and hence is dense in A_{g,δ} ⊗ k, we can conclude the following.

Proposition 1.11. All geometric fibres of A_{g,δ} → Spec(ℤ) are irreducible.

If we use once more the fact that the ordinary locus is dense in A_{g,d} ⊗ \overline{𝔽}_{p} for every prime p and the fact that any ordinary point of A_{g,d} lies in one of the substacks A_{g,δ} then we get the following theorem.

Theorem 1.12. (Decomposition into irreducible components.)

1) The irreducible components of the algebraic stack A_{g,d} are the closures (A_{g,δ})^{c} of the locally closed substacks A_{g,δ}.

2) If p does not divide d then A_{g,d} ⊗ \overline{𝔽}_{p} is the disjoint union of its irreducible components A_{g,δ} ⊗ \overline{𝔽}_{p}.

3) The irreducible components of A_{g,δ} ⊗ \overline{𝔽}_{p} (if p | d) are the closures (A_{g,δ} ⊗ \overline{𝔽}_{p})^{c} of the locally closed substacks A_{g,δ} ⊗ \overline{𝔽}_{p}.

(A_{g,d} ⊗ \overline{𝔽}_{p})_{red} = ⋃_{δ}(A_{g,δ} ⊗ \overline{𝔽}_{p})^{c}

Remarks 1.13.

1) Let us denote by A_{g,d} the coarse moduli scheme associated to A_{g,d} (see Remark 1.2.3). In [GIT] Appendix 7A and [Mu] the definition of A_{g,δ} contains the statement that the set of points x ∈ A_{g,d} corresponding to abelian varieties (X, λ) such that Ker(λ) ≃ K(δ) is open in A_{g,d}. We see from the equations η_{ij} = 1 defining A_{g,δ} ⊂ A_{g,d} that this is false as soon as δ_{g−1} ≠ 1. To be explicit let us take g = 2 and δ = (p, p). The complete local ring R of A_{g,d} at a geometric point of A_{g,δ} in characteristic p is in this case equal to

R = W (k)[[t_{1}, t_{2}, t_{3}]][η_{12}]/(η_{12}^{p} − 1).

Over the subscheme

Spec(W (k)[η_{12}]/(η_{12}^{p−1} + η_{12}^{p−2} + … + 1)) ↪ Spec(R)

we have a polarized abelian scheme (X, λ) whose special fibre gives a point of A_{g,d} lying in the set defined above, but whose generic fibre gives a point of A_{g,d} which does not lie in the set defined above (by Proposition 1.7.2). Hence this set is not open. It is true, however, that the set of points described above is open in each fibre of the map A_{g,d} → Spec(ℤ).

2) If p | d but p^{2}∖| d then the union in part 3 of the theorem is still disjoint. Indeed, if (A_{g,δ}⊗\overline{𝔽}_{p})^{c} intersects (A_{g,δ′} ⊗ \overline{𝔽}_{p})^{c} then δ_{i} is equal to δ′_{i} up to powers of p for all i = 1, …, g. But p can only divide δ_{g} (resp. δ′_{g}) and hence δ = δ′. If p^{2} | d and g > 1 then the union will not be disjoint any longer (see §3).

3) If g > 1 then we see from the equations preceding Proposition 1.8 that A_{g,d} ⊗ 𝔽_{p} is reduced if and only if p^{2}∖| d.

2. The fibres of the components of A_{g,d}

From Theorem 1.12 it follows that we have

((A_{g,δ})^{c} ⊗ 𝔽_{p})_{red} = ⋃_{δ′∈S(p,δ)}(A_{g,δ′} ⊗ 𝔽_{p})^{c}

where the union is over a certain set S(p, δ). In this section we will give another description of the sets S(p, δ).

Suppose d = p^{n} · d′, p∖| d′ and δ = (δ_{1}, …, δ_{g}) = (p^{a_{1}} · ν_{1}, …, p^{a_{g}} · ν_{g}) with Π_{i=1}^{g}ν_{i} = d′. Put δ_{0} = (p^{a_{1}}, …, p^{a_{g}}). In this case we have

S(p, δ) = {(ν_{1} · δ′_{1}, …, ν_{g} · δ′_{g}) | δ′ = (δ′_{1}, …, δ′_{g}) ∈ S(p, δ_{0})} .

This allows us to restrict ourselves to the case that d = p^{n} for some n ∈ ℕ. The statement above can be proved either by using an appropriate correspondence linking A_{g,d} and A_{g,d′} or by modifying the arguments of this section to get S(p, δ) for d ≠ p^{n}.

As of now d = p^{n}. Let us introduce some notations. Put

A(δ) = ℤ/δ_{1}ℤ × … × ℤ/δ_{g}ℤ,
A(δ)^{∨} = Hom_{ℤ_{p}}(A(δ), ℚ_{p}/ℤ_{p})

and

B(δ) = A(δ) × A(δ)^{∨}.

On B(δ) we define the following nondegenerate alternating pairing

\begin{matrix}q : B(δ) × B(δ) & ⟶ & ℚ_{p}/ℤ_{p} \\ ((x, x̂), (y, ŷ)) & ⟼ & ŷ(x) − x̂(y). \end{matrix}

Finally, we define

T (δ) = The set of δ′ such that there exists a totally

isotropic subgroup of B(δ) isomorphic to A(δ′).

Remark 2.1. Equivalently, δ′ ∈ T (δ) if and only if there exists a surjection B(δ) → A(δ′) with totally isotropic kernel.

Examples 2.2.

1) Suppose g = 2, d = p^{n} and δ = (p^{l}, p^{n−l}) with l ≤ \frac{1}{2}n. In this case we have T (δ) = \{(p^{m}, p^{n−m}) | \frac{1}{2}n ≥ m ≥ l\}.

2) Suppose g = 3, d = p^{6} and δ = (p, p^{2}, p^{3}). In this case we have T (δ) = {(p, p^{2}, p^{3}), (p^{2}, p^{2}, p^{2})}.

Proposition 2.3. T (δ) = S(p, δ). Proof. If δ′ ∈ S(p, δ) then there exist:

- A discrete valuation ring R of mixed characteristic with algebraically closed residue field of characteristic p. Put S = Spec(R) and let η (resp. s) be the generic (resp. special) point of S.

- An object (S, X, λ) of A_{g,d} such that (η, X_{η}, λ_{η}) is an object of A_{g,δ} and (s, X_{s}, λ_{s}) is an

object of A_{g,δ′}. We may assume that Ker(λ)(S) ≃ B(δ) after replacing R by a sufficiently large, finite extension. We can choose this isomorphism in such a way that the pairing e^{λ} on Ker(λ)(S) corresponds to the pairing q on B(δ). The reduction map

Ker(λ)(S) ⟶ Ker(λ)(s)

is surjective and its kernel is isotropic for the pairing e^{λ}. By assumption Ker(λ)(s) ≃ A(δ′). Hence we see that δ′ ∈ T (δ) (use Remark 2.1).

Conversely, suppose δ′ ∈ T (δ). Let us choose a surjection π : B(δ) → A(δ′) with totally isotropic kernel. Choose lifts ẽ_{i} ∈ B(δ) of the standard generators e_{i} ∈ A(δ). Put a_{ij} = q(δ_{j}ẽ_{j}, ẽ_{i}) for i < j. We remark that the elements a_{ij} ∈ δ_{i}^{−1} · ℤ_{p}/ℤ_{p} ⊂ ℚ_{p}/ℤ_{p} are independent of the choice of the elements ẽ_{i} and ẽ_{j}. We leave it to the reader to show that they determine the isomorphism type of the abelian group B(δ).

Let R be a finite extension of W (\overline{𝔽}_{p}) which contains a primitive δ′_{g}-th root of unity ζ. Choose an object (Spec(\overline{𝔽}_{p}), X_{0}, λ_{0}) of A_{g,δ′} and an isomorphism Ker(λ) ≃ K(δ′)_{Spec(\overline{𝔽}_{p})}. By [Cr] Corollary 24 we can find a lift (X, λ) of (X_{0}, λ_{0}) over Spec(R) with η_{ij} = ζ^{a_{ij}} for all i, j, 1 ≤ i < j ≤ g. After replacing R by a finite extension, we have a surjection

Ker(λ)(Spec(R)) ⟶ Ker(λ_{0})(\overline{𝔽}_{p}) ≃ A(δ′)

with isotropic kernel. Further, we get structural constants a′_{ij} ∈ ℚ_{p}/ℤ_{p} computed from this using the pairing e^{λ} and the identification μ_{δ′_{g}}(Spec(R)) ≃ (δ′_{g})^{−1}·ℤ_{p}/ℤ_{p}·ζ. However, by the definition of η_{ij}, we see that a_{ij} = a′_{ij} and thus, by the remark above, that Ker(λ)(Spec(R)) ≃ B(δ). Therefore, we conclude that (η, X_{η}, λ_{η}) is an object of A_{g,δ} (η is the generic point of Spec(R)) and that δ′ ∈ S(p, δ). □

3. The fibres of A_{g,d} ⟶ Spec(ℤ) over Spec(𝔽_{p})

We continue to assume that d = p^{n}. As in Section 2 analogous results will hold for general d. To any allowed sequence δ there is associated the function

f_{δ} : ℕ ⟶ ℕ,
f_{δ}(m) = log_{p} rank (K(δ)[p^{m}]) = log_{p} order (B(δ)[p^{m}]) .

If δ ≠ δ′ then f_{δ} ≠ f_{δ′}. If k is a field and (Spec(k), X, λ) is an object of A_{g,d} then we have the function

f(X, λ) : ℕ ⟶ ℕ
f(X, λ)(m) = log_{p} rank(Ker(λ)[p^{m}]).

Definition and proposition 3.1. For any (Spec(k), X, λ) ∈ ObA_{g,d}, where k is a field, there exists a sequence δ = δ(X, λ) such that f(X, λ) = f_{δ}. Proof. The assertion is not trivial only if the characteristic of k is p. Further, it is enough to prove it after a field extension. Thus we assume that k is a perfect field of characteristic p. The polarization λ will induce an alternating pairing

a : M ⊗_{W (k)} M ⟶ W (k)

on the Dieudonné module M of the p-divisible group X[p^{∞}] of X. This pairing is nondegenerate in the sense that the cokernel of the map M → Hom_{W (k)}(M, W (k)) derived from it is a finite length W (k)-module. (This module is in fact isomorphic to the Dieudonné module of Ker(λ).) Hence we can choose a basis for M such that a is given by the matrix

( \begin{matrix}& & & p^{a_{1}} \\ & 0 & & & ... \\ & & & & & p^{a_{g}} \\ −p^{a_{1}} \\ & ... & & & 0 \\ & & −p^{a_{g}} \end{matrix} )

for some a_{i} ∈ ℤ_{≥0} with a_{1} ≤ … ≤ a_{g}. Using the general properties of Dieudonné modules we get that f(X, λ) = f_{δ} with δ = (p^{a_{1}}, …, p^{a_{g}}). □

Let x : Spec(\overline{𝔽}_{p}) → A_{g,d} be a geometric point defined by the polarized abelian variety (X_{0}, λ_{0}) over \overline{𝔽}_{p}. We would like to answer the following question: If δ = δ(X_{0}, λ_{0}), does it follow that x lies in the closure of A_{g,δ} ⊗ 𝔽_{p}? Unfortunately, we do not know the answer for p ≤ 3.

Proposition 3.2. If p > 3 then x lies in (A_{g,δ(X_{0},λ_{0})} ⊗ 𝔽_{p})^{c}. Proof. We use that (X_{0}, λ_{0}) can be lifted to (X, λ) over the ring R = W (k)[\sqrt{p}] (see [No] Main Theorem). Since p − 1 > 2 the group schemes Ker(λ)[p^{m}] are flat over Spec(R) (see [Ra] Corollaire 3.3.6). Hence f(X_{0}, λ_{0}) = f(X_{η}, λ_{η}) and (η, X_{η}, λ_{η}) ∈ ObA_{g,δ} (here η denotes the generic point of Spec(R)). This proves that x lies in (A_{g,δ})^{c}. Let us take any point y in the ordinary locus of (A_{g,δ})^{c} ⊗𝔽_{p} which specializes to x and a point z in A_{g,δ} which specializes to y. Let us denote by f_{x}, f_{y} and f_{z} the functions associated to the corresponding polarized abelian varieties. Since ranks of finite group schemes can only jump up under specialization, we have f_{δ} = f_{x} ≥ f_{y} ≥ f_{z} = f_{δ}. Hence y is a point of A_{g,δ} ⊗ 𝔽_{p}. □

Let us prove that for every f ∈ {0, 1, …, g} there exists a reduced closed substack

V_{f} ⊂ A_{g,d} ⊗ 𝔽_{p}

whose geometric points correspond to abelian varieties with p-rank less than or equal to f.

To this end consider the fine moduli scheme X = A_{g,d,l}⊗𝔽_{p} where l is a large prime different from p (see [GIT] Theorem 7.9). The 1-morphism X → A_{g,d} ⊗ 𝔽_{p} is étale. In fact, A_{g,d} ⊗ 𝔽_{p} can be identified with [X/G] (see [DM] Example 4.8) where G = GL (2g, ℤ/lℤ). The set of points of X which correspond to abelian varieties with p-rank less or equal to f is closed since the p-rank can only decrease under specialization. Hence there exists a reduced closed subscheme V_{f} ⊂ X whose set of points is exactly this set. It is easy to see that V_{f} ⊂ X is invariant under the action of G on X. Therefore, we can define V_{f} as the reduced closed substack [V_{f}/G] of [X/G] = A_{g,d} ⊗ 𝔽_{p}.

Of course V_{0} ⊂ V_{1} ⊂ … ⊂ V_{g−1} ⊂ V_{g} = A_{g,d} ⊗ 𝔽_{p} and the dimension of V_{g−i} is equal to \frac{1}{2}g(g + 1) − i ([NO] Theorem 4.1). Let us study the locally closed substack Z := V_{g−1}\V_{g−2} of A_{g,d} ⊗ 𝔽_{p} which is the locus parametrizing abelian varieties of p-rank exactly g − 1.

Suppose x : Spec(k) → Z is a geometric point defined by (X, λ) over Spec(k). The p-divisible group X[p^{∞}] of X will be isomorphic to

E[p^{∞}] × (𝔾_{m}[p^{∞}])^{g−1} × (ℚ_{p}/ℤ_{p})^{g−1}

for some supersingular elliptic curve E over Spec(k). Hence we see that

Ker(λ) ≃ E[p^{m}] ×_{Spec(k)} K(\overline{δ})_{Spec(k)}

for some m and \overline{δ} = (\overline{δ}_{1}, …, \overline{δ}_{g−1}) such that \overline{δ}_{i} | \overline{δ}_{i+1} and p^{m} · Π_{i=1}^{g−1}\overline{δ}_{i} = d. More generally, let (S, X, λ) be an object of Z and consider the toric part X[p^{∞}]_{tor} of the p-divisible group of X. Locally in the fpqc topology on S it is isomorphic to (𝔾_{m}[p^{∞}])^{g−1}. The same holds true for X^{t}[p^{∞}]_{tor} and locally in the fpqc topology on S the map X[p^{∞}]_{tor} → X^{t}[p^{∞}]_{tor} will look like

\overline{δ}_{1} · id × … × \overline{δ}_{g−1} · id : (𝔾_{m}[p^{∞}])^{g−1} ⟶ (𝔾_{m}[p^{∞}])^{g−1}.

Therefore, we conclude that the sequences \overline{δ} = (\overline{δ}_{1}, …, \overline{δ}_{g−1}) associated to the geometric points of Z are locally constant on Z. In other words: Z is a disjoint union

Z = ∐_{(p^{m},\overline{δ})} Z(p^{m}, \overline{δ}) ⊂ A_{g,d} ⊗ 𝔽_{p}

of substacks Z(p^{m}, \overline{δ}) where these sequences are constant and equal to \overline{δ}.

Proposition 3.3. The algebraic stacks Z(p^{m}, \overline{δ}) are geometrically irreducible. Proof. Let Z′ ⊂ A_{g,1} ⊗ 𝔽_{p} be the locally closed substack which is the analog of Z in the case of principally polarized abelian schemes. We use the fact that the algebraic stack Z′ is geometrically irreducible (see [EO]).

Write \overline{δ} = (p^{a_{1}}, …, p^{a_{g−1}}) and let t = max(m, a_{g−1}). Let Z′_{t} be the stack with objects (S, X, λ, α) where (S, X, λ) ∈ ObZ′ and α is an injection of group schemes

α : (μ_{p^{t},S})^{g−1} ×_{S} (ℤ/p^{t}ℤ_{S})^{g−1} ↪ X[p^{t}]

such that α is compatible with the Weil pairing induced by λ on X[p^{t}] and the standard pairing on the group scheme on the left (see 1.9). The forgetful morphism Z′_{t} → Z′ is representable, affine and of finite type and hence Z′_{t} is a separated algebraic stack of finite type over Spec(ℤ) (see Lemma 1.6).

If (S, X, λ, α) ∈ ObZ′_{t} then we put G = Im(α)^{⊥}; i.e. G is the orthogonal complement of Im(α) for the Weil pairing on X[p^{t}]. Thus X[p^{t}] = G×_{S} Im(α) and if S = Spec(k) where k is an algebraically closed field then G ≃ E[p^{t}] for some supersingular elliptic curve E over Spec(k). The group scheme G[F^{m}] is finite and flat over S.

Consider the morphism

\begin{matrix}f : Z′_{t} & ⟶ & Z(p^{m}, \overline{δ}) \\ (S, X, λ, α) & ⟼ & (S, Y, μ) \end{matrix}

where Y is the finite flat cover of X defined by the diagram

Y ⟶^{μ} Y^{t}
↓^{π}  _{π^{t}}↑
X ⟶^{λ} X^{t}

such that

Ker(π^{t}) = λ (G[F^{m}] ×_{S} α (∏_{i=1}^{g−1} μ_{δ_{i},S} × (0))) .

It is straightforward to prove that (S, Y, μ) is an object of Z(p^{m}, \overline{δ}) and that f is surjective. Therefore, it suffices to prove that Z′_{t} ⊗ \overline{𝔽}_{p} is irreducible. To do this we only need to show that the p-adic monodromy

π_{1}(Z′ ⊗ \overline{𝔽}_{p}) ⟶ GL(g − 1, ℤ_{p})

is surjective. Let us denote by A^{o} the ordinary locus of A_{g−1,1} ⊗ \overline{𝔽}_{p}. We already know that the p-adic monodromy π_{1}(A^{o}) → GL(g − 1, ℤ_{p}) is surjective (see [FC] V 7.1 or [Ek] Theorem 3.1 & our remarks following 1.10). We can see A^{o} as a closed substack of Z′ ⊗ \overline{𝔽}_{p} via the immersion

\begin{matrix}A^{o} & ↪ & Z′ ⊗ \overline{𝔽}_{p} \\ (S, X, λ) & ↦ & (S, X ×_{S} E_{S}, λ × λ_{E}) \end{matrix}

where E is a supersingular elliptic curve over Spec(\overline{𝔽}_{p}). Hence the p-adic monodromy over Z′ ⊗ \overline{𝔽}_{p} is surjective too. □

Using the stacks Z(p^{m}, \overline{δ}) we can say something about the intersections of the components of A_{g,d} ⊗ 𝔽_{p}, at least in codimension 1. Indeed, suppose we have two components (A_{g,δ} ⊗ 𝔽_{p})^{c} and (A_{g,δ′} ⊗ 𝔽_{p})^{c} and assume that some component W of the intersection

(A_{g,δ} ⊗ 𝔽_{p})^{c} ∩ (A_{g,δ′} ⊗ 𝔽_{p})^{c}

has codimension 1 in A_{g,d} ⊗ 𝔽_{p}. The abelian variety corresponding to the generic point of W is not ordinary. Hence W ⊂ V_{g−1} = Z^{c}. Since the dimension of W is equal to the dimension of Z, we get that W is equal to one of the components of Z^{c}, i.e.

W = Z(p^{m}, \overline{δ})^{c}

for some m and \overline{δ} = (\overline{δ}_{1}, …, \overline{δ}_{g−1}) such that \overline{δ}_{i} | \overline{δ}_{i+1} and p^{m} · Π_{i=1}^{g−1}\overline{δ} = d. Thus to see which components of A_{g,d} ⊗ 𝔽_{p} intersect in codimension 1 we have to determine for which pairs (δ, \overline{δ}) we have that

Z(p^{m}, \overline{δ}) ⊂ (A_{g,d} ⊗ 𝔽_{p})^{c}.

The following proposition gives an answer to this question.

Proposition 3.4. The following are equivalent - Z(p^{m}, \overline{δ}) ⊂ (A_{g,δ} ⊗ 𝔽_{p})^{c},

- there exists an exact sequence of abelian groups

0 ⟶ ℤ/p^{m}ℤ ⟶ A(δ) ⟶ A(\overline{δ}) ⟶ 0.

Proof. If Z(p^{m}, \overline{δ}) ⊂ (A_{g,δ} ⊗ 𝔽_{p})^{c} then there exist:

- A complete discrete valuation ring R of characteristic p. Put S = Spec(R), let η be the generic point of S, \overline{η} a geometric point lying over η and let s be the special point of S.

- An object (S, X, λ) of A_{g,d} such that (η, X_{η}, λ_{η}) ∈ Ob(A_{g,δ}) and (s, X_{s}, λ_{s}) ∈ Ob(Z(p^{m}, \overline{δ})). We assume further that the residue field of R is algebraically closed and hence

Ker(λ_{s}) ≃ E[p^{m}] × ∏_{i=1}^{g−1} μ_{\overline{δ}_{i},s} × ∏_{i=1}^{g−1} ℤ/\overline{δ}_{i}ℤ_{s}

for some supersingular elliptic curve E over s.

The rigidity of multiplicative group schemes gives us a unique injection

α : ∏_{i=1}^{g−1} μ_{\overline{δ}_{i},S} ⟶ Ker(λ)

lifting the injection over s. Dually, using our pairing e^{λ}, we get a surjection

β : Ker(λ) ⟶ ∏_{i=1}^{g−1} ℤ/\overline{δ}_{i}ℤ_{S}
↓_{e^{λ}}  ∥
Ker(λ)^{D}  ⟶^{α^{D}} (∏_{=1}^{g−1} μ_{\overline{δ}_{i},S})^{D} .

Let us look at the cohomology H of the complex (β ∘ α = 0 since e^{λ} is alternating)

0 ⟶ ∏_{i=1}^{g−1} μ_{\overline{δ}_{i},S} ⟶^{α} Ker(λ) ⟶^{β} ∏_{i=1}^{g−1} ℤ/\overline{δ}_{i}ℤ_{S} ⟶ 0.

It is a finite, flat group scheme H over S which lifts the group scheme E[p^{m}] over s. Thus H has order p^{2m} and H is not killed by p^{m−1}. By construction the pairing e^{λ} induces a nondegenerate alternating pairing on H. Finally, H_{η} has local-local part equal to zero since this is true for Ker(λ)_{η}. Therefore, we conclude that H\overline{η} ≃ μ_{p^{m},\overline{η}} × ℤ/p^{m}ℤ\overline{η} as this is the only isomorphism type of a finite group scheme over \overline{η} which is selfdual, has local-local part equal to zero, has order p^{2m} and is not killed by p^{m−1}. Remark that G ⇝ G(\overline{η}) is an exact functor on the category of finite group schemes over \overline{η}; hence we get the exact sequence

0 ⟶ Ker(β)(\overline{η}) ⟶ Ker(λ)(\overline{η}) ⟶^{β} ∏_{i=1}^{g−1} ℤ/\overline{δ}_{i}ℤ ⟶ 0
↓^{≃}
H(\overline{η})  ↓^{≃}  ∥
↓^{≃}
0 ⟶ ℤ/p^{m}ℤ ⟶  A(δ)  ⟶  A(\overline{δ})  ⟶ 0.

Conversely, suppose there exists an exact sequence

0 ⟶ ℤ/p^{m}ℤ ⟶ A(δ) ⟶ A(\overline{δ}) ⟶ 0.

We write δ_{i} = p^{a_{i}}, i = 1, …, g and \overline{δ}_{i} = p^{b_{i}}, i = 1, …, g − 1. Let \overline{e}_{1}, …, \overline{e}_{g−1} denote the standard generators of A(\overline{δ}). Choose lifts e_{1}, …, e_{g−1} ∈ A(δ) of the elements \overline{e}_{i} ∈ A(\overline{δ}) and write

p^{b_{i}} · e_{i} = \overline{c}_{i} ∈ ℤ/p^{m}ℤ ⊂ A(δ).

It is a simple exercise to show that the isomorphism type of A(δ) is determined by the sequence \overline{c}_{1}, …, \overline{c}_{g−1}. We will use later the fact that p^{a_{g}−m−b_{i}}c_{i} ∈ ℤ, this is true since p^{a_{g}} · e_{i} = 0 and hence p^{a_{g}−b_{i}} · c_{i} ∈ p^{m}ℤ.

Let x : Spec(k) → Z(p^{m}, \overline{δ}) be a geometric point given by (X_{0}, λ_{0}) over Spec(k). Let (Y_{0}, μ_{0}) be a principally polarized abelian variety over Spec(k) such that there exists a diagram

X_{0} ⟶^{λ_{0}} X_{0}^{t}
↓^{π}  _{π^{t}}↑
Y_{0}  ⟶^{μ_{0}} Y_{0}^{t}

with

Ker(π^{t}) ≃ E_{0}[F^{m}] × ∏_{i=1}^{g−1} ℤ/\overline{δ}_{i}ℤ_{Spec(k)}

for some supersingular elliptic curve E_{0} over Spec(k) (cf. proof of Proposition 3.3). We know that

Y_{0}[p^{∞}] ≃ E_{0}[p^{∞}] × (𝔾_{m}[p^{∞}])^{g−1} × (ℚ_{p}/ℤ_{p})^{g−1}

and we can choose this isomorphism in such a way that the polarization μ_{0} induces the same quasi-polarization on E_{0}[p^{∞}] as the canonical principal polarization on E_{0} induces on E_{0}[p^{∞}].

Let us take a deformation E of E_{0} over S = Spec(k[[T]]) such that E is ordinary over the generic point η of S. In this way the polarized p-divisible group Y_{0}[p^{∞}] is deformed into

E[p^{∞}] × (𝔾_{m,S}[p^{∞}])^{g−1} × (ℚ_{p}/ℤ_{p})_{S}^{g−1}

with quasi-polarization

polarization of E induced on E[p^{∞}] × (μ_{0}|_{(𝔾_{m}[p^{∞}])^{g−1}×(ℚ/ℤ_{p})^{g−1}})_{S} .

Now we apply the Serre-Tate Theorem ([Me] V 2.3) to get a deformation (Y, μ) over S of the pair (Y_{0}, μ_{0}) such that the p-divisible group Y [p^{∞}] is isomorphic to the one described above. Finally, after a finite flat base extension such that S is still the spectrum of a complete local ring with residue field k, we may assume that

E[p^{a_{g}}](S) ≃ ℤ/p^{a_{g}}ℤ

and hence

Y [p^{a_{g}}](S) ≃ ℤ/p^{a_{g}}ℤ × (ℤ/p^{a_{g}}ℤ)^{g−1}.

Let G ⊂ Y [p^{a_{g}}] be the flat closure of the subgroup of Y [p^{a_{g}}](S) generated by

(p^{a_{g}−m}, 0, …, 0)

and

(p^{a_{g}−m−b_{1}} · c_{1}, p^{a_{g}−b_{1}}, 0, …, 0)
⋮
(p^{a_{g}−m−b_{g−1}} · c_{g−1}, 0, …, 0, p^{a_{g}−b_{g−1}}) .

Clearly, the group scheme G_{η} is isomorphic to A(δ). The group scheme G_{0} (the fibre of G over the special point of S) certainly contains

∏_{i=1}^{g−1} ℤ/p^{b_{i}}ℤ ⊂ E_{0}[p^{a_{g}}] × (μ_{p^{ag}})^{g−1} × (ℤ/p^{a_{g}}ℤ)^{g−1}

and E_{0}[F^{m}]. Hence μ_{0}(G_{0}) = Ker(π^{t}) ⊂ Y_{0}. Thus the pair (X, λ), where X^{t} = Y^{t}/μ(G) and λ is the polarization fitting into the diagram

X ⟶^{λ} X^{t}
↓ ↑
Y ⟶^{μ} Y^{t},

is a lift over S of the pair (X_{0}, λ_{0}). The triple (η, X_{η}, λ_{η}) is an object of A_{g,δ} ⊗ 𝔽_{p}. We have shown that any geometric point of Z(p^{m}, \overline{δ}) lies in the closure of A_{g,δ} ⊗ 𝔽_{p} and so we are done. □

Examples 3.5.

1) For g = 2 the first example of an intersection of (A_{g,δ} ⊗ 𝔽_{p})^{c} with (A_{g,δ′} ⊗ 𝔽_{p})^{c} was given by P. Norman in [No2]. He showed that (A_{2,(p,p)} ⊗ 𝔽_{p})^{c} intersects (A_{2,(1,p^{2})} ⊗ 𝔽_{p})^{c} by looking at the universal deformation space of (E × E, p · (λ × λ)) where (E, λ) is a principally polarized supersingular elliptic curve over an algebraically closed field of characteristic p.

More generally, suppose g = 2 and d = p^{n}. We remark that we have an exact sequence of abelian groups

0 ⟶ ℤ/p^{m}ℤ ⟶ A ⟶ ℤ/p^{n−m}ℤ ⟶ 0

if and only if A ≃ ℤ/p^{l}ℤ ⊕ ℤ/p^{n−l}ℤ for some l (l ≤ ⌊\frac{n}{2}⌋) with l ≤ m ≤ n − l. Hence we see that

(A_{2,(p^{l},p^{n−l})} ⊗ 𝔽_{p})^{c} ⊃ Z (p^{m}, (p^{n−m}))

if and only if l ≤ m ≤ n − l and we see that

(A_{2,(p^{l_{1}} ,p^{n−l_{1}} )} ⊗ 𝔽_{p})^{c} ∩ (A_{2,(p^{l_{2}} ,p^{n−l_{2}} )} ⊗ 𝔽_{p})^{c}

contains Z(p^{m}, (p^{n−m})) if and only if l_{1} ≤ m ≤ n − l_{1} and l_{2} ≤ m ≤ n − l_{2}. Finally, we note that each component (A_{2,δ} ⊗ 𝔽_{p})^{c} contains Z(p^{⌊ \frac{n}{2} ⌋}, (p^{n−⌊ \frac{n}{2} ⌋})).

2) Suppose that g = 3 and d = p^{3}. In this case the possibilities for δ are: δ = (1, 1, p^{3}), δ = (1, p, p^{2}) and δ = (p, p, p). The substack Z(p^{m}, \overline{δ}) can only be contained in two or more components (A_{3,δ} ⊗ 𝔽_{p})^{c} if m is either 1 or 2. We have the following list of possibilities for exact sequences

0 ⟶ ℤ/p^{m}ℤ ⟶ A(δ) ⟶ A(\overline{δ}) ⟶ 0.

- If m = 1, \overline{δ} = (1, p^{2}) then δ = (1, 1, p^{3}) or δ = (1, p, p^{2}).

- If m = 1, \overline{δ} = (p, p) then δ = (1, p, p^{2}) or δ = (p, p, p).

- If m = 2, \overline{δ} = (1, p) then δ = (1, 1, p^{3}) or δ = (1, p, p^{2}). We conclude that the components corresponding to δ = (1, 1, p^{3}) and δ = (p, p, p) do not intersect in codimension 1 whilst the other two pairs of components do meet in codimension 1. It seems probable, though we dit not prove it, that the two components mentioned above will meet in a codimension 2 locus of A_{3,p^{3}} ⊗ 𝔽_{p}.

References

EGA A. Grothendieck, J. Dieudonné, Eléments de géométrie algébriques. Springer-Verlag, Berlin

(1971) and Publ. Math. I.H.E.S. 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967). SGA A. Grothendieck et al., Séminaire de géométrie algébrique. Springer Lecture Notes and

Masson-North Holland. Especially: SGA 3: M. Demazure, A. Grothendieck, Schémas en groupes I,II,III. Lecture Notes in Math. 151,152,153, Springer-Verlag (1971).

[An] S. Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques. Bull. de la Société Mathématique de France, Mém. 33 (1973), 5-79.

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

[Cr] S.E. Crick, Local moduli of abelian varieties. Amer. J. of Math. 97 (1975), 851-861.

[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.

[EO] T. Ekedahl, F. Oort, Connected subsets of a moduli space of abelian varieties. To appear.

[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).

[Ka] N. Katz, Serre-Tate local moduli. Surfaces algébriques (édité par J. Giraud, L. Illusie et M. Raynaud), Exposé 5bis, pp. 138- 202. Lecture notes in Math. 868, Springer-Verlag, 1981.

[KO] T. Katsura, F. Oort, Families of supersingular abelian surfaces. Compos. Math. 62 (1987), 107-167.

[MB] L. Moret-Bailly, Pinceaux de variétés abéliennes. Astérisque 129, Soc. Math. de France (1985).

[MB2] L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur ℙ^{1} I, II. In: Séminaire sur les pinceaux de courbes de genre au moins deux. Editor: L. Szpiro. Astérisque 86, Soc. Math. de France (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, The structure of the moduli spaces of curves and abelian varieties. Actes, Congrès international math. 1970 Tome 1, Gauthier Villars, Paris (1971), 457-465.

[No1] P. Norman, Lifting abelian varieties. Inven. Math. 64 (1981), 431-443.

[No2] P. Norman, Intersections of the components of the moduli space of abelian varieties. Journal of Pure and Applied Algebra 13 (1978), 105-107.

[No3] P. Norman, An algorithm for computing local moduli of abelian varieties. Annals of Math. 101 (1975), 499-509.

[NO] P. Norman, F. Oort, Moduli of abelian varieties. Ann. of Math. 112 (1980), 413-439.

[Od] T. Oda, The first de Rham cohomology groups and Dieudonné modules. Ann. Sci. Ec. Norm. Superieur 2 (1959), 63-135.

[Oo] F. Oort, Finite group schemes, local moduli for abelian varieties and lifting problems. In: Algebraic Geometry, Oslo, 1970, 223-254. Also in: Compositio Math. 23 (1972), 265-296.

[Ra] M. Raynaud, Schemas en groupes de type (p, …, p). Bull. Soc. Math. de France, t. 102 (1974), 241-280.