LIFTING ABELIAN VARIETIES WITH ADDITIONAL STRUCTURE

CHIA-FU YU

Introduction

This note is to investigate the moduli spaces of polarized O_{B}-abelian varieties with a determinant condition (see Section 1 for the definition). The main result is that when p is unramified in B and greater than 2, the moduli space is a complete intersection relative to the mixed characteristic base. See Theorem 4.5 and Section 4 for details. The main tools are Dieudonné modules and Grothendieck-Messing’s deformation theory [G][Me].

The main point of this work, same as that in [N], is to deal with inseparable polarizations. The result of this work concludes: the naive moduli problem provides a integral model of a Shimura varieties Sh_{K}(G, X) of PEL type, which is flat and the reduction is a locally complete intersection at the places where G is unramified and the level subgroup K is maximal. In the case that p is ramified in B, G. Pappas [P] showed that there are some examples (unitary Shimura varieties) in which the integral models constructed by the naive modular interpretation are not flat over the base.

Section 1 contains the definition of the moduli spaces we considered. Section 2 contains some well known classical results. Section 3 provides a certain degree classification of the Dieudonné modules in question. Section 4 uses Grothendieck-Messing and Fontaine’s theory to lift and deform abelian varieties with additional structure.

Acknowledgments. This work began with a discussion with J. Achter, who drew my attention to this possible generalization. C.-L. Chai clarified a few inaccuracies of the first draft and gave a valuable suggestion which improves the previous results. The author would like to thank them both heartily. The final revision of this paper was done during my staying at the Max-Planck-Institut für Mathematik in Bonn in the summer of 2000. I thank the Institute for the kind hospitality and the good working environment.

1. Moduli spaces

Let B be a semi-simple ℚ-algebra with a positive involution ∗, O_{B} a maximal order stable under the involution ∗. Fix a ℚ-valued skew-Hermitian B-module (V, ψ) with ℚ-dimension 2g and an ℝ-linear representation h of ℂ on V_{ℝ} such that the symmetric bilinear form (x, y) → ψ(x, h(i)y) is positive definite. Let E be the field of the definition of the isomorphic class of the complex representation of B on V_{1} := V_{ℂ}^{−1,0}, on which h(z) acts as z. In fact, E = ℚ(Tr(b; V_{1}))_{b∈B}. We denote the representation V_{1} by Φ(h). We will define a condition on the Lie algebra Lie(A)_{S} for an O_{B}-abelian scheme A over a base O_{E}-scheme S. This condition will be called the Kottwitz determinant condition defined by Φ(h), [K].

Let {e_{i}} be a ℤ-basis of O_{B} and X_{i} be some indeterminants. Define f_{h}(X) := det(∑ e_{i}X_{i}; V_{1}). The polynomial f_{h} is independent of the choice of basis and can be shown to have coefficients in O_{E}. For an O_{B}-abelian scheme A over an O_{E}-scheme S, we say that the Lie algebra Lie(A)_{S} satisfies the Kottwitz determinant condition if det(∑ e_{i}X_{i}; Lie(A)) = f_{h}(X) in O_{S}[X]. Now we can formulate the moduli problems:

Let A_{g,d,Φ(h)}^{O_{B}} be the moduli stack over Spec O_{E} whose objects are (S, A, λ, ι), where:

• S is a locally noetherian O_{E}-scheme.

• A is an abelian scheme of dimension g over S.

• λ : A → A^{t} is a polarization with deg λ = d^{2}.

• ι : O_{B} → End_{S}(A) is a ring mono-morphism such that λ ι(b^{∗}) = ι(b)^{t}λ.

• Lie(A)_{S} satisfies the Kottwitz determinant condition defined by Φ(h). The morphisms between two objects are isomorphisms as polarized O_{B}-abelian schemes over S. Let A_{g,d}^{O_{B}} be the moduli stack over Spec ℤ parameterizing polarized O_{B}-abelian schemes (A, λ, ι)_{S} of relative dimension g with deg λ = d^{2}.

Proposition 1.1. (1) The forgetful morphism A_{g,d}^{O_{B}} → A_{g,d} over Spec ℤ is representable, finite and unramified.

(2)The forgetful morphism A_{g,d,Φ(h)}^{O_{B}} → A_{g,d}^{O_{B}} over Spec O_{E} is representable and closed immersion.

Therefore A_{g,d,Φ(h)}^{O_{B}} and A_{g,d}^{O_{B}} are separated (Deligne-Mumford) algebraic stacks over Spec O_{E} and Spec ℤ, respectively.

Proof. (1) The relative representability follows from a consequence of Hilbert schemes: the functor T → End_{T} (A_{T}) is representable by a disjoint union of projective schemes over S. Quasi-finiteness is due to the following lemma. Properness comes from the Néron mapping properties. Unramfiedness is due to the rigidity of the endomorphisms of abelian schemes.

(2) It is clear that to have the Kottwitz condition is a closed condition and the representability is standard.

Lemma 1.2. Let (B, ∗), (E, ′) be semi-simple ℚ-algebras with positive involutions, and O_{B}, O_{E} orders which are stable under the involutions ∗ and′, respectively. Then the set Hom_{∗−alg}(O_{B}, O_{E}) of the ring homomorphisms compatible with the involutions is finite.

Proof. Let n be the number such that n Tr_{B/ℚ}(1) = Tr_{E/ℚ}(1). For any ℚ-algebra homomorphism φ : B → E, we have n Tr_{B/ℚ}(b) = Tr_{E/ℚ}(φ(b)) for all b ∈ B. To see this, by tensoring with ℂ, we get a product of complex matrix rings mapping into a product of complex matrix rings. Since there is a unique irreducible representation of M_{n}(ℂ), up to conjugation there is only one algebra homomorphism from M_{n}(ℂ) to M_{mn}(ℂ), the diagonal embedding. Therefore, Tr_{E/ℚ}(φ(b)) is a multiple of Tr_{B/ℚ}(b). Define a metric on B ⊗ ℝ, E ⊗ ℝ by (x, y) → n Tr_{B/ℚ}(xy^{∗}), Tr_{E/ℚ}(xy′), respectively. Then any ∗-homomorphism preserves the metrics.

Since any algebra homomorphism φ is determined by the image of a finite set of generators {ξ_{i}}, it suffices to show that the choice of each φ(ξ_{i}) is finite. If φ preserves involutions, the image φ(ξ_{i}) must lie in the set O_{E} ∩ {x ∈ E ⊗ ℝ | |x| = |ξ_{i}|} which is discrete and compact, hence finite.

We can choose an isomorphism B ≃ ∏_{i} M_{n_{i}}(D_{i}) where D_{i} are division algebras with positive involutions ∗_{i} such that 1) the involution ∗ induces the conjugate transpose: (d_{kl})_{i} → (d_{lk}^{∗_{i}})_{i} and 2) O_{B} is sent to ∏_{i} M_{n_{i}}(O_{D_{i}}) with O_{D_{i}} maximal and stable under involutions. Let (A, λ, ι)_{S} be an object of A_{g,d,Φ}^{O_{B}}. Since all idempotents of O_{B} are fixed by the involution ∗, we can write (A, λ, ι)_{S} = ∏_{i}(A_{i}, λ_{i}, ι_{i})_{S}^{n_{i}}, where (A_{i}, λ_{i}, ι_{i}) is a polarized O_{D_{i}}-abelian scheme over S with a certain condition Φ_{i} on the Lie algebras. So we have A_{g,d,Φ}^{O_{B}} = ∏_{i} A_{g_{i},d_{i},Φ_{i}}^{O_{D_{i}}}. Therefore, in practice it is enough to consider that case that B is a division algebra.

Remark. 1) The assumption that the order O_{B} is maximal and stable under the involution ∗ is unnecessary for A_{g,d,Φ}^{O_{B}} to be a separated algebraic stack. However, we are mainly interested in such cases. While studying the reduction of A_{g,d,Φ}^{O_{B}} at one prime over p, we may choose a weaker assumption: O_{B} ⊗ ℤ_{p} is maximal and stable under the involution ∗.

2) The generic fiber of A_{g,d,Φ}^{O_{B}} in general is not the canonical model of a single Shimura variety of PEL type. Rather the generic fiber of A_{g,d,Φ}^{O_{B}} is the disjoint union of a finite number of those due to elementary divisors and failure of Hasse principle. We refer the reader to the works of de Jong [dJ], Kottwitz [K], and Rapoport-Zink [RZ] for definitions and detailed discussions.

3) The generic fiber A_{g,d}^{O_{B}} ⊗ ℚ = ⨆_{Φ} A_{g,d,Φ}^{O_{B}} ⊗ E_{Φ}, where Φ runs through all possible representation type Lie(A) of polarized O_{B}-abelian varieties of dimension g over ℂ. However, it is not true that A_{g,d}^{O_{B}} = ⨆_{Φ} A_{g,d,Φ}^{O_{B}}. There are examples in characteristic p in which the Lie algebra does not satisfy any determinant condition. Moreover, such an example can form a continuous family. Therefore it is certainly not true that A_{g,d}^{O_{B}} is flat over Spec ℤ, because X_{R} is flat if and only if the generic fiber X_{K} is scheme-theoretically dense in X_{R}, where R is a DVR.

2. The classical results

Let B be a division algebra with a positive involution ∗. Denote by F the center Z(B) of B if the involution ∗ is of the first kind. If the involution ∗ is of second kind, denote by K the center Z(B) of B and let F be the subfield of K fixed by σ, where σ ∈ Gal(K/F) is the element induced by the involution ∗. It is well known that F is a totally real field and K/F is a CM extension.

According to Albert’s classification [Mu], they are divided into 4 cases:

(C) Type I. B = F.

(C) Type II. B is a totally indefinite quaternion algebra over F. Let x̄ = tr x − x be the canonical involution of B. Then any positive involution has the form a x̄ a^{−1} for some a ∈ B, a^{2} ∈ F and totally negative. We can also choose an isomorphism

ℝ ⊗_{ℚ} B ≃ M_{2}(ℝ) × ⋯ × M_{2}(ℝ)

such that the involution extends to the product of transpositions on the right side.

(D) Type III. B is a totally definite quaternion algebra over F. The canonical involution x → tr x − x is positive and the unique positive one. There is an isomorphism

ℝ ⊗_{ℚ} B ≃ ℍ × ⋯ × ℍ

carrying the involution to the product of the standard involutions of ℍ.

(A) Type IV. B is a central simple algebra over K and it will satisfy the following properties (i) if v is a finite place fixed by σ, Inv_{v}(B) = 0, and (ii) for any finite place v of K, Inv_{v}(B) + Inv_{σv}(B) = 0.

For any positive involution ′, there is an isomorphism

ℝ ⊗_{ℚ} B ≃ M_{q}(ℂ) × ⋯ × M_{q}(ℂ)

which carries ′ to the product of the complex conjugate transpositions. If ∗ is another positive involution, then x^{∗} = a x′ a^{−1} for some a in B whose image in the right side is (A_{1}, …, A_{f}) with A_{i} Hermitian positive definite.

Let V be as in Section 1. Let 2g := dim_{ℚ} V, f := [F : ℚ], q^{2} := [B : K]. Let r be the number in the following cases:

r := \begin{cases}\frac{g}{f} & (Type I) \\ \frac{g}{2f} & (Type II or III) \\ \frac{g}{q^{2}f} & (Type IV) \end{cases} .

From [S], we know that for Type C or D, V_{1} is a multiple of the reduced representation of B over ℚ. For Type A, there are a few possibilities for V_{1}: suppose that {τ_{1}, …, τ_{f}} is a CM type of K and let n_{i}, n̄_{i} be the multiplicity of the standard representation of B ⊗_{K,τ_{i}} ℂ and B ⊗_{K,στ_{i}} ℂ in V_{1}, respectively. Then n_{i} + n̄_{i} = qr for all i = 1, …, f.

We have the corresponding Hermitian symmetric domains:

Type C 𝔖 = ℌ_{r} × ⋯ × ℌ_{r}. (f copies)

Type D 𝔖 = ℌ′_{r} × ⋯ × ℌ′_{r}. (f copies)

Type A 𝔖 = ℌ_{n_{1},n̄_{1}} × ⋯ × ℌ_{n_{f} ,n̄_{f}}.

where ℌ_{r} is the Siegel space of degree r; ℌ′_{r} is the space of all complex alternating matrices z of size r such that 1 − z̄^{t}z is positive hermitian; and ℌ_{r,s} is the space of complex matrices z with r rows and s columns such that 1 − z̄^{t}z is positive hermitian.

We conclude that

1. For Type C or D, there is only one Kottwitz determinant condition on Lie algebras, so E = ℚ. For Type A, all the possible Kottwitz determinant conditions on Lie algebras correspond to the set { (n_{i}, n̄_{i}) | n_{i} + n̄_{i} = rq, i = 1 … f } and E is the reflex field of these pairs.

2. The dimension of A_{g,d,Φ}^{O_{B}} ⊗ ℂ is (C) \frac{1}{2}r(r + 1)f, (D) \frac{1}{2}r(r − 1)f, and (A) ∑_{i=1}^{f} n_{i}n̄_{i}.

Classically, the moduli space attached to a division algebra of Type I is called a Hilbert-Siegel modular variety. Quaternionic Shimura varieties of PEL type are corresponding to the division algebras of Type II or Type III. Picard modular varieties are of Type IV.

3. Preparation on Dieudonné modules

In this section, k will be a perfect field of characteristic p > 0 containing the residue fields of O_{K} (over p). Denote by W := W (k) the Witt vectors over k and B(k) its fraction field. Let W (1) := D^{∗}(Ĝ_{m}), where D^{∗} is the classical contravariant Dieudonné functor. For any abelian variety A over k, define D_{∗}(A) := Hom_{W} (D^{∗}(A[p^{∞}]), W (1)). This is not usual covariant Dieudonné module defined by p-typical curves in Cartier theory, e.g. in [Z]. The reason we do this linear dual simply is that it is easier to handle when more and more structure comes in. For example, the classical contravariant Dieudonné functor will give us a Dieudonné module with (left) O_{B^{opp}} action instead of O_{B} action, also the polarization will induces an pairing on M^{t} (weight -1) instead of on M. Keep in mind that we still have the crystalline theory, it just behaves in the covariant way instead. For example, we have D_{∗}(A) ⊗ R = H_{1}^{cris}(A/R) for any nilpotent PD-extension R of k.

We shall classify the Dieudonné module attached to a polarized O_{B}-abelian variety by putting it in a normal form. Assume that B is a division algebra, p is unramified in B and p ≠ 2 if it is of type D. To each polarized O_{B}-abelian variety (A, λ, ι) over k, we associate a quasi-polarized O_{B} ⊗ℤ_{p}-Dieudonné module M. Let v_{1}, v_{2}, …, v_{s} be the primes of F lying over p. Then M = ⊕M_{v}, where the M_{v}’s are quasi-polarized O_{B,v}-Dieudonné modules. Thus we can focus on each piece, drop the subscript v, and replace O_{B}, O_{K}, O_{F} , … by their completions at v. Redefine g, f, q, r as in Section 2.

Fix an embedding ν : ℚ̄ → ℚ̄_{p}. We index the set of embeddings {σ_{i}} = Hom(O_{F} , W) in a way that σ_{i+1} = σσ_{i}, for i ∈ ℤ/fℤ, where σ is the Frobenius map on W. The chosen embeddings {τ_{i}} of the global CM field, denoted by K as in Section 2, into ℚ̄ associates a set of embeddings (again denoted by) {τ_{i}} ⊂ Hom(O_{K}, W) with respect to ν, if we are in the case of type A.

Let {τ_{i}} ⊂ Hom(O_{K}, W) be embeddings associated to the chosen CM type {τ_{i}} (in Section 2), by abuse of language, with respect to ν. Set M^{i} := {x ∈ M |ax = σ_{i}(a)x, a ∈ O_{F} } and N^{i}, N̄^{i} the eigenspaces on which O_{K} acts via τ_{i}, τ̄_{i} = σ^{f}τ_{i}, resp. Corresponding to the classical classification of the division algebra B, we have the following cases: (C) (I) O_{B} = O_{F}, M = ⊕M^{i} is a quasi-polarized O_{F} -Dieudonné module such that

– F : M^{i} → M^{i+1}, V : M^{i} → M^{i−1}, (M^{f} = M^{0}, M^{−1} = M^{f−1})

– ⟨M^{i}, M^{j}⟩ = 0 if i ≠ j, and

– ⟨ax, y⟩ = ⟨x, ay⟩ ∀x, y ∈ M, a ∈ O_{F} .

(II) O_{B} ≃ M_{2}(O_{F}), the involution ∗ : (a_{ij}) → (a_{ji}), M = ⊕M^{i}. Everything is the same as (I) except

⟨ax, y⟩ = ⟨x, a^{∗}y⟩ ∀ x, y ∈ M, a ∈ O_{B}.

(D) O_{B} ≃ M_{2}(O_{F}), ∗ is the standard adjoint involution and everything is the same as (II). (A) O_{B} ≃ M_{q}(O_{K}), the involution ∗ : (a_{ij}) → (ā_{ji}), M = ⊕(N^{i} ⊕ N̄^{i}). The nonzero pairing

only occurs in ⟨N^{i}, N̄^{i}⟩ for all i. The operators F and V act on M of degree 1 and -1, respectively. We have 2 cases: (split) O_{K} ≃ W (𝔽_{p^{f}})^{2}; N^{f} = N^{0}, N̄^{f} = N̄^{0} N^{−1} = N^{f−1}, N̄^{−1} = N̄^{f−1}. (inert) O_{K} ≃ W (𝔽_{p^{2f}}); N^{f} = N̄^{0}, N̄^{f} = N^{0}, N^{−1} = N̄^{f−1}, N̄^{−1} = N^{f−1}.

Observe that the idempotents of O_{B} in type A and C are fixed by the involution ∗. Using Morita’s equivalence, it reduces to considering O_{F} or O_{K}-Dieudonné modules. For type D, consider the symmetric form ⟨x, y⟩′ := ⟨x, Cy⟩, where C is the Weyl element. Now Morita’s equivalence works well for this symmetric pairing and it reduces to consider anti-polarized O_{F} -Dieudonné modules. Combining with the determinant condition on the Lie algebra, we conclude that (reduced version): (C) M = ⊕M^{i} is a ℤ/fℤ-graded quasi-polarized O_{F} -Dieudonné module with rank_{W} M^{i} = 2r

and dim_{k}(M/V M)^{i} = r. (D) M = ⊕M^{i} is a ℤ/fℤ-graded quasi-anti-polarized O_{F} -Dieudonné module with rank_{W} M^{i} =

2r and dim_{k}(M/V M)^{i} = r. (A) (split) M = ⊕(N^{i} ⊕ N̄^{i}) is a ℤ/fℤ-graded quasi-polarized O_{K}-Dieudonné module with

rank_{W} N^{i} = rank_{W} N̄^{i} = qr and dim_{k}(N/V N)^{i} = n_{i} (and with bar). The nonzero pairing only occurs in ⟨N^{i}, N̄^{i}⟩.

(inert) M = ⊕(N^{i} ⊕ N̄^{i}) is a ℤ/2fℤ-graded quasi-polarized O_{K}-Dieudonné module with rank_{W} N^{i} = rank_{W} N̄^{i} = qr and dim_{k}(N/V N)^{i} = n_{i}. The nonzero pairing only occurs in ⟨N^{i}, N̄^{i}⟩.

We will do normal forms for type C and A together, and then deal with the case type D separately.

Lemma 3.1. Let M be W -modules of rank 2g with a non-degenerate alternating form ⟨·, ·⟩, and N a W -submodule of M with pM ⊂ N. Then there exist W -bases {e_{i}}, {f_{i}} for N, M respectively such that

• e_{i} ∈ {f_{i}, p f_{i}}.

• ⟨e_{i}, e_{j}⟩ ≠ 0 ⟺ |i − j| = g.

Proof. Set ord⟨x, y⟩ := ord_{p}⟨x, y⟩ and ord⟨S_{1}, S_{2}⟩ := min_{x∈S_{1},y∈S_{2}} ord⟨x, y⟩, and denote by [x, y] the W -submodule generated by x, y. Note that

M = [x, y] ⊕ [x, y]^{⊥} ⟺ ord⟨x, y⟩ = ord⟨x, M⟩ = ord⟨M, y⟩.

Choose f_{1}, f_{g+1} ∈ M s.t. ord⟨f_{1}, f_{g+1}⟩ = ord⟨M, M⟩. We have three cases:

(i) If f_{1} ∈ N, f_{g+1} ∈ N, then set e_{1} = f_{1}, e_{g+1} = f_{g+1}. We have

ord⟨e_{1}, e_{g+1}⟩ = ord⟨N, N⟩

this implies

N = [e_{1}, e_{g+1}] ⊕ [e_{1}, e_{g+1}]^{⊥}, M = [f_{1}, f_{g+1}] ⊕ [f_{1}, f_{g+1}]^{⊥}

and

p [f_{1}, f_{g+1}]^{⊥} ⊂ [e_{1}, e_{g+1}]^{⊥}.

This reduces to the lower rank case and by induction one can find symplectic bases.

(ii) If f_{1} ∈ N, f_{g+1} ∉ N, then set e_{1} := f_{1}, e_{g+1} := p f_{g+1}. If ord⟨e_{1}, e_{g+1}⟩ = ord⟨e_{1}, N⟩ = ord⟨e_{g+1}, N⟩, then one can decompose N, M simultaneously as in case (i). Suppose not.

(a) If ∃ e′_{1} ∈ N s.t. ord⟨e′_{1}, e_{g+1}⟩ < ord⟨e_{1}, e_{g+1}⟩, then

ord⟨e′_{1}, f_{g+1}⟩ < ord⟨f_{1}, f_{g+1}⟩, contradiction.

(b) If ∃ e′_{g+1} ∈ N s.t. ord⟨e_{1}, e′_{g+1}⟩ < ord⟨e_{1}, e_{g+1}⟩, then

ord⟨e_{1}, e′_{g+1}⟩ = ord⟨f_{1}, f_{g+1}⟩. ∵ ord⟨f_{1}, f_{g+1}⟩ ≤ ord⟨e_{1}, e′_{g+1}⟩

Replacing f_{g+1} by e′_{g+1}, we are back to case (i).

(iii) If f_{1} ∉ N, f_{g+1} ∉ N, then set e_{1} := p f_{1}, e_{g+1} := p f_{g+1}. If ord⟨e_{1}, e_{g+1}⟩ = ord⟨e_{1}, N⟩ = ord⟨e_{g+1}, N⟩, then one can decompose N, M simultaneously as in case (i). Suppose not. Then ∃ e′_{1} ∈ N s.t.

ord⟨e′_{1}, e_{g+1}⟩ < ord⟨e_{1}, e_{g+1}⟩
⇒ ord⟨e′_{1}, f_{g+1}⟩ + 1 < ord⟨f_{1}, f_{g+1}⟩ + 2
⇒ ord⟨e′_{1}, f_{g+1}⟩ ≤ ord⟨f_{1}, f_{g+1}⟩
⇒ ord⟨e′_{1}, f_{g+1}⟩ = ord⟨f_{1}, f_{g+1}⟩.

Replacing f_{1} by e′_{1}, we are back to case (ii).

Apply this lemma to N = F M^{i−1}, M = M^{i} in the case type C and A (in the reduced version). We get normal forms:

Corollary 3.2. Type C. There exist 2 W -bases {e_{k}^{i}}, {f_{k}^{i}} of M^{i} for each i = 1, …, f such that

• F e_{k}^{i} ∈ {f_{k}^{i+1}, pf_{k}^{i+1}}.

• {e_{k}^{i}, e_{r+k}^{i}}_{k=1...r} is a symplectic basis for M^{i}. Type A. There exist 2 W -bases {e_{k}^{i}}, {f_{k}^{i}} of N^{i} and {ē_{k}^{i}}, {f̄_{k}^{i}} of N̄^{i} for each i = 1, …, f such that

• F e_{k}^{i} ∈ {f_{k}^{i+1}, pf_{k}^{i+1}}, F ē_{k}^{i} ∈ {f̄_{k}^{i+1}, pf̄_{k}^{i+1}}.

• {e_{k}^{i}, ē_{k}^{i}}_{k=1...qr} is a symplectic basis for N^{i} ⊕ N̄^{i}.

Now we are in type D. For a W -module M, denote by [e_{1}, e_{2}, …, e_{n}] the W -submodule generated by the elements {e_{i}}.

Lemma 3.3. Let M be a W -module with a non-degenerate symmetric form ⟨·, ·⟩, N a W - submodule of M such that pM ⊂ N. Then we can decompose M and N w.r.t. ⟨·, ·⟩:

M = [f_{1}, f_{s+1}] ⊕ [f_{2}, f_{s+2}] ⊕ ⋯ ⊕ [f_{2s+1}] ⊕ ⋯ ⊕ [f_{n}]
N = [e_{1}, e_{s+1}] ⊕ [e_{2}, e_{s+2}] ⊕ ⋯ ⊕ [e_{2s+1}] ⊕ ⋯ ⊕ [e_{n}]

such that • e_{k} ∈ {f_{k}, pf_{k}} for k ≥ 2s + 1; (e_{k}, e_{s+k}) = (pf_{k}, f_{s+k}) for k ≤ s;

• ⟨e_{k}, e_{k}⟩ = ⟨e_{s+k}, e_{s+k}⟩ = 0 for k ≤ s.

Proof. We may assume that ord⟨N, N⟩ > ord⟨M, M⟩ =: n, otherwise there exists e_{1} ∈ N s.t. ord⟨e_{1}, e_{1}⟩ = n, then M = [e_{1}] ⊕ [e_{1}]^{⊥}, N = [e_{1}] ⊕ [e_{1}]^{⊥} and we are done.

Let f_{1} ∈ M s.t. ord⟨f_{1}, f_{1}⟩ = n, and set e_{1} := pf_{1}. If ord⟨e_{1}, e_{1}⟩ = ord⟨e_{1}, N⟩, then M = [f_{1}] ⊕ [f_{1}]^{⊥}, N = [e_{1}] ⊕ [e_{1}]^{⊥} and we are done.

Suppose not, i.e. ∃ e_{2} ∈ N, s.t. ord⟨e_{1}, e_{2}⟩ < ord⟨e_{1}, e_{1}⟩, then ord⟨e_{1}, e_{2}⟩=1+n. Set f_{2} := e_{2}, then we have ord⟨f_{1}, f_{2}⟩ = n, which implies that ord⟨e_{1}, e_{2}⟩ = 1 + n = ord⟨N, N⟩. Therefore, M = [f_{1}, f_{2}] ⊕ [f_{1}, f_{2}]^{⊥}, N = [e_{1}, e_{2}] ⊕ [e_{1}, e_{2}]^{⊥}. Moreover, the matrix (⟨f_{i}, f_{j}⟩) has the form

(\begin{matrix}p^{n}u_{1} & p^{n} \\ p^{n} & p^{n+b}u_{2} \end{matrix}) ,

but we may assume that it is (\begin{matrix}0 & p^{n} \\ p^{n} & 0 \end{matrix}) by the next lemma. This completes the proof.

Lemma 3.4. Suppose {x_{1}, x_{2}} is a basis of M such that the matrix (⟨x_{i}, x_{j}⟩) is

(\begin{matrix}p^{n}u_{1} & p^{n} \\ p^{n} & p^{n+b}u_{2} \end{matrix}) ,

where the u_{i}’s are units in W and b is a positive integer. Then we can choose a basis {y_{1}, y_{2}} of M such that

• the subspace spanned by ȳ_{2} and that spanned by x̄_{2} in M/pM are the same, and

• the matrix (⟨y_{i}, y_{j}⟩) is (\begin{matrix}0 & p^{n} \\ p^{n} & 0 \end{matrix}).

Proof. Set y_{2} := x_{2} + tx_{1}, then the condition for ⟨y_{2}, y_{2}⟩ = 0 is

t^{2}u_{1} + 2t + p^{b}u_{2} = 0. (1)

Let s := tu_{1} and u := u_{1}u_{2}. One yields

s^{2} + 2s + p^{b}u = 0 (2)

By Hensel’s Lemma, s has a solution in pW to (2), and so as t to (1).

Set y_{1} := x_{1} + ay_{2}, then we can choose a in W such that ⟨y_{1}, y_{1}⟩ = 0. Modifying y_{1} by a multiple of y_{1}, one may assume ⟨y_{1}, y_{2}⟩ = p^{n}. We are done.

Apply Lemma 3.3 to N = F M^{i−1}, M = M^{i}. We have a normal form for type D (in the reduced version).

Corollary 3.5. Type D. There are an integer s_{i} and 2 W -bases {e_{k}^{i}}, {f_{k}^{i}} of M^{i} for each i = 1 … f such that

• F e_{k}^{i} ∈ {f_{k}^{i+1}, pf_{k}^{i+1}} for k > 2s_{i};

• (F e_{k}^{i}, F e_{s_{i}+k}^{i}) = (f_{k}^{i+1}, pf_{s_{i}+k}^{i+1}) for k ≤ s_{i};

• ⟨e_{j}^{i}, e_{k}^{i}⟩ ≠ 0 ⟺ (j = k > 2s_{i}) or (j, k < 2s_{i} and |j − k| = s_{i}).

4. On Liftings and Deformations

In this section, we use Grothendieck-Messing’s deformation theory to lift polarized O_{B}- abelian varieties and to investigate their local moduli spaces. Let (B, O_{B}, E, O_{E}, Φ) be the setting in Section 1, Δ be the product of rational primes at which B is ramified. The ground field k is perfect in characteristic p > 0. Assume that p does not divide Δ and p ≠ 2 if B contains a factor of type D. See Theorem 4.1 and 4.5 for the main results.

Theorem 4.1. Let k be an algebraically closed field of characteristic p > 2. The reduction map A_{g,d,Φ}^{O_{B}}(W [\sqrt{p}]) → A_{g,d,Φ}^{O_{B}}(k) is surjective.

Proof. Let (A, λ, ι) be a point of A_{g,d,Φ}^{O_{B}}(k) and M its covariant Fontaine’s Dieudonné module, the linear dual of the Dieudonné module in the sense of Fontaine (see [F], p.218; also [N], sec. 1). In order to lift (A, λ, ι) to R := W [\sqrt{p}], we need to construct a free direct summand F il of M ⊗_{W} R, as an R-module, such that

• F il ⊗ R/𝔪_{R} = V M/pM;

• F il is stable under the action of O_{B};

• ⟨F il, F il⟩ = 0.

Since p is unramified in B, we only need to consider the reduced version of types A,C and D. Note that H_{1}(A/R) and any such an ‘admissible‘ lifting F il have natural decompositions H^{i} and F il^{i} into eigenspaces with respect to the embeddings of O_{F} into W. Any lifting of (V M/pM)^{i} in H^{i} will be stable under the action of O_{F} or O_{K}, and ⟨H^{i}, H^{j}⟩ = 0 if i ≠ j.With these, it is enough to construct F il^{i} in H^{i} for each i such that

• F il^{i} ⊗ R/𝔪_{R} = (V M/pM)^{i};

• ⟨F il^{i}, F il^{i}⟩ = 0. Therefore we may drop the index i.

By the corollaries in Section 3, we find a solution for F il by cases:

Type A and C. M is generated, as a W -module, by {X_{i}, Y_{i}}_{i=1..s}, {X_{i}, X′_{i}, Y_{i}, Y′_{i}}_{i=s+1..s+t} (W -basis) s.t.

• Y_{i}, Y′_{i} ∈ V M for all i;

• \begin{cases}⟨X_{i}, Y_{i}⟩ = p^{a_{i}} & for 1 ≤ i ≤ s. \\ ⟨Y_{i}, Y′_{i}⟩ = p⟨X_{i}, X′_{i}⟩ = p^{a_{i}+1} & for i > s. \\ all other pairings are zero. \end{cases}

Take F il to be the R-submodule of M⊗R generated by {Y_{i}}_{i=1..s}, {Y_{i}+πX_{i}, Y′_{i}−πX′_{i}}_{i=s+1..s+t}, where π := \sqrt{p}.

Type D. M is generated, as a W -module, by {X_{i}, Y_{i}}_{i=1..s}, {X_{i}, Y_{i}}_{i=s+1..s+t} s.t.

• Y_{i} ∈ V M for all i;

• \begin{cases}⟨X_{i}, Y_{i}⟩ = p^{a_{i}} & for 1 ≤ i ≤ s. \\ ⟨Y_{i}, Y_{i}⟩ = −p⟨X_{i}, X_{i}⟩ = p^{a_{i}+1} & for i > s. \\ all other pairings are zero. \end{cases}

Take F il to be the R-submodule of M ⊗ R generated by {Y_{i}}_{i=1..s}, {Y_{i} + πX_{i}}_{i=s+1..s+t}. This completes the proof.

Note that we only need k to be perfect in type A and C. In type D one needs some finite extension of k to have such a normal form. We make this stronger assumption of k to make the result simpler. We also need to use Fontaine’s theory in the proof since Grothendieck-Messing’s theory requires topologically nilpotent PD-structure on the base ring R, which limits p > e+1=3.

Lemma 4.2. Let (A_{0}, λ_{0}, ι_{0}) be a polarized O_{B}-abelian variety over k (not necessary admitting a determinant condition defined by some Φ). The deformation functor D := Def[A_{0}, ι_{0}] (without the polarization) is pro-presentable by W [[x_{1}, …, x_{t}]],where t is the dimension of the k-vector space Hom_{O_{B}}(H^{0}(A_{0}^{t}, Ω_{A_{0}^{t}}^{1}), H^{1}(A_{0}^{t}, O_{A_{0}^{t}})).

Proof. D is pro-representable because Def[A_{0}] is pro-representable. We identify M/pM := D_{∗}(A_{0}) ⊗ k with H_{DR}^{1}(A_{o}^{t}/k), then V M/pM = H^{0}(A_{0}^{t}, Ω_{A_{0}^{t}}^{1}) and M/V M = H^{1}(A_{o}^{t}, O_{A_{o}^{t}}). Then one can see that D(k[ϵ]) = Hom_{O_{B}}(V M/pM, M/V M) = Hom_{O_{B}}(H^{0}(Ω_{A_{0}^{t}}^{1}), H^{1}(O_{A_{0}^{t}})).

The smoothness of D is clear and left to the reader.

We need some notation to state the following proposition. Let V be a k-vector space with a k-linear O_{B} action. Denote by ⋀_{O_{B}}^{2} V the k-vector space of ⋀_{k}^{2} V modulo the subspace spanned by elements (b x ∧ y − x ∧ b^{∗}y) for all b ∈ O_{B}, x, y ∈ V.

Proposition 4.3. Let (A_{0}, λ_{0}, ι_{0}) be a polarized O_{B}-abelian variety over k. Let Obs be the space ⋀_{O_{B}}^{2}(V M/pM), where M is the covariant Dieudonné module of A_{0}. Let O := W [[x_{1}, …, x_{t}]] and o := dim_{k} Obs. Then there exist elements a_{1}, …, a_{o} of O such that Def[A_{0}, λ_{0}, ι_{0}] is prorepresentable by Spf O/(a_{1}, …, a_{o}).

Proof. First of all, we may assume that B is a division algebra by the same idempotent reduction in Section 1.

By the rigidity on the endomorphisms of abelian varieties, hence on the polarizations, Def[A_{0}, λ_{0}, ι_{0}] is pro-representable by a closed formal subscheme of Spf O, say Spf O/𝔞.

By Artin-Rees lemma, there exists an n > 0 such that

𝔪^{n} ∩ 𝔞 = 𝔪(𝔪^{n−1} ∩ 𝔞) = 𝔪^{n} ∩ 𝔪𝔞

then we have a short exact sequence

0 → I → R → R′ → 0

where R := O/𝔪𝔞 + 𝔪^{n}, R′ := O/𝔞 + 𝔪^{n} and I := 𝔞/𝔪𝔞 is an ideal of R with I^{2} = 0, 𝔪I = 0. The canonical homomorphisms

O ⟶ O/𝔞
↓  ↓
R ⟶ R′

give a O_{B}-abelian scheme (A′, ι′) over R′ and a polarized O_{B}-abelian scheme (A, λ, ι) over R along with (A, ι)⊗R′ ≃ (A′, ι′). We apply Morita’s equivalence on H_{1}^{dR}(A/R) and H_{1}^{dR}(A′/R′) and reduce to modules with O_{K} or O_{F} action. Denote by F il and H_{1}(R) (resp. F il′ and H_{1}(R′)) be the free R (resp. R′)-modules corresponding to Lie(A)_{R} and H_{1}^{dR}(A/R) (resp. Lie(A)′_{R} and H_{1}^{DR}(A′/R′)). We have natural isomorphism H_{1}(R) ⊗ R′ ≃ H_{1}(R′), and the polarization λ′ induces a symplectic (resp. symmetric) form on H_{1}(R) in types A and C (resp. in type D).

Morita’s equivalence also gives

dim_{k} ∧_{O_{B}}^{2}(V M/pM) = rank_{R} ∧_{O_{K}}^{2} (F il) (A)
dim_{k} ∧_{O_{B}}^{2}(V M/pM) = rank_{R} ∧_{O_{F}}^{2} (F il) (C)
dim_{k} Sym_{O_{B}}^{2}(V M/pM) = rank_{R}Sym_{O_{F}}^{2} (F il) (D)

and the right hand sides are

rank_{R} ⊕_{i=1}^{f} F il^{i} ⊗_{R} \overline{F il}^{i} (A)
rank_{R} ⊕_{i=1}^{f} ∧_{R}^{2}F il^{i} (C)
rank_{R} ⊕_{i=1}^{f} Sym_{R}^{2}F il^{i} (D)

(A) Let \{x_{1}^{i}, x_{2}^{i}, …, x_{n_{i}}^{i}\} and \{x̄_{1}^{i}, x̄_{2}^{i}, …, x̄i ; n̄_{i}\} be R-bases of F il^{i} and \overline{F il}^{i} resp.

(CD) Let {x_{1}, x_{2}, …, x_{r_{i}}} be a R-basis of F il^{i}.

Set

b̄_{jk}^{i} := \begin{cases}⟨x_{j}^{i}, x̄_{k}^{i}⟩ & (A) \\ ⟨x_{j}^{i}, x_{k}^{i}⟩ & (CD) \end{cases}

Since ⟨F il′, F il′⟩ = 0 in R′, we have b̄_{jk}^{i} ∈ I. Let b_{jk}^{i} be an element of 𝔞 such that b_{jk}^{i} ≡ b̄_{jk}^{i} mod 𝔪𝔞 for all i, j, k, and 𝔟 be the ideal of O generated by all b_{jk}^{i}.

Set R′′ := O/𝔟 + 𝔪𝔞 + 𝔪^{n}. By the construction of R′′, ⟨F il, F il⟩ = 0 in R′′. Hence the polarization λ′ lifts to R′′ and there is a map O/𝔞 → R′′. This yields 𝔞 ⊂ 𝔟 + 𝔪𝔞 + 𝔪^{n}, therefore 𝔟 = 𝔞. This completes the proof.

Lemma 4.4. Let (R, K, k = R/(π)) be a complete discrete valuation ring of mixed characteristic (0, p). Let A be a finitely generated commutative R-algebra.

(1) If for any homomorphism f : A → L with L algebraically closed of characteristic p, there exists a lifting f̃: A → W (L)[\sqrt{p}] such that the following diagram commutes

\xymatrix{ & W (L)[\sqrt{p}] \ar[d] \\ A \ar[ur]^{f̃} \ar[r]^{f} & L}

Then dim A ⊗ κ = dim A ⊗ K.

(2) If B = R[[X_{1}, …, X_{t}]]/(a_{1}, …, a_{o}) and dim B ⊗ k = t − o, then B is flat over R and is a complete intersection relative to R ([EGA],19.3.6).

Proof. (1) Let C be the image of the map A → A ⊗ K. The closed subscheme Spec C is the scheme-theoretic image of the morphism j : Spec A ⊗ K → Spec A, hence C is flat over R and the underlying topological space of Spec C is the Zariski closure of the image of j. We will show that the image of j is a Zariski dense open subset of Spec A. Then Spec C → Spec A is a homeomorphism, hence dim A ⊗ k = dim C ⊗ k = dim C ⊗ K = dim A ⊗ K.

For every 𝔭 ∈ Spec A ⊗ k, we have the following diagram

\xymatrix{ & R′ \ar[r] \ar[d] & W (\overline{k(𝔭)})[\sqrt{p}] \ar[d] \\ A \ar[urr]^{π̃} \ar[ur]^{π̃} \ar[r]_{π} & A/𝔭 \ar[r] & \overline{k(𝔭)}}

where R′ is the image of the lifting π̃ : A → W (\overline{k(𝔭)})[\sqrt{p}]. Let 𝔓_{K} be the kernel of the morphism π̃_{K} : A ⊗ K → R′ ⊗ K. We have that 𝔓 := j(𝔓_{K}) is the kernel of π̃, whose Zariski closure contains 𝔭, as 𝔓 ⊂ 𝔭. We are done.

(2) The dimension of B is t − o + 1 since dim B ≥ t − o + 1 and dim B/(π) = t − o. Since R[[X_{1}, …, X_{t}]] is Cohen-Macaulay, {a_{1}, …, a_{o}, π} is a regular sequence in R[[X_{1}, …, X_{t}]] and π is not a zero divisor of B. Therefore B is flat over R and is a complete intersection relative to R.

Theorem 4.5. The structure morphism A_{g,d,Φ}^{O_{B}} → Spec O_{E}[1/2Δ] is flat and a complete intersection morphism ([EGA], 19.3.6).

Proof. The problem is at least local in the base, so we may replace O_{E}[1/2Δ] by one of its local rings. Call this local ring R and its fraction field K. Choosing a suitable level structure, we may assume X := A_{g,d,n,Φ}^{O_{B}} ⊗ R is a scheme and may further assume that the residue field k is an algebraic closure of a finite field and R is complete by faithfully flat base change. By Theorem 4.1 and Lemma 4.4 (1), we have dim X_{k} = dim X_{K}, i.e. every geometric fiber outside 2Δ has the same dimension.

By Proposition 4.3 and Lemma 4.4 (2), it is enough to show that dim X_{k} = t − o. We may assume that B is a division algebra. From Section 2, the dimensions of these geometric fibers are (A) ∑_{i=1}^{f} n_{i}n̄_{i}, (C) \frac{1}{2}r(r + 1)f, and (D) \frac{1}{2}r(r − 1)f. On the other hand, we compute in the proof of Proposition 4.3 that (A) t = ∑_{i=1}^{f} 2n_{i}n̄_{i}, o = ∑_{i=1}^{f} n_{i}n̄_{i}; (C) t = r^{2}f, o = \frac{1}{2}r(r − 1)f; and (D) t = r^{2}f, o = \frac{1}{2}r(r+1)f. One verifies that dim X_{k} = t−o in each case. This completes the proof.

Remark. Proposition 4.3 is inspired by a theorem of Mumford ([O], Theorem 2.3.3). We modified the proof there by the crystalline theory. The original proof could work by describing the space of the obstruction classes of the polarization.

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Department of Mathematics, Columbia University, New York, NY 10027

  E-mail address: chiafu@math.columbia.edu