On reduction of Hilbert-Blumenthal varieties

Chia-Fu Yu

§1. Introduction

Let F be a totally real number field of degree g and O_{F} be its ring of integers. A Hilbert-Blumenthal variety parameterizes the isomorphism classes of O_{F} -abelian varieties of dimension g with a certain condition (and with certain additional structure). The purpose of the condition is to exclude some bad points in characteristic p such that the integral model becomes flat. In [R], Rapoport used the condition that the Lie algebra of the O_{F} -abelian scheme over a base scheme S is a locally free O_{F} ⊗ S-module. We will call it the Rapoport condition and the Rapoport locus for the defined moduli space. The condition was modified later by Deligne and Pappas [DP] in order to confirm the properness of the compactification constructed in [R] in the case of bad reduction. We will call the modified condition the Deligne-Pappas condition, and the Deligne-Pappas space for corresponding moduli space, see (2.4) and (5.1) for the precise definitions. Over ℤ_{(p)} the Deligne-Pappas condition is equivalent to that the O_{F} -abelian scheme admits a prime-to-p polarization.

This paper is based on the important work of Rapoport [R], and that of Deligne-Pappas [DP]. It serves several purposes. All of them go to understand the geometry of reduction of Hilbert-Blumenthal varieties. In this introduction we assume that there is only one prime v of O_{F} lying over p. Let O := O_{F} ⊗ ℤ_{p}, k(v) be the residue field of O_{F} at v, and π ∈ O_{F} be a uniformizer of O. Let k be an algebraically closed field of characteristic p.

One purpose of this paper is to study various conditions and invariants of O_{F} -abelian varieties using Dieudonné modules. Three invariants–the Lie types (see (2.3)), the slopes (Newton polygons), and a-numbers (see (2.10))–are discussed and three conditions–the Rapoport condition, the Deligne-Pappas condition, and the Kottwitz determinant condition–are also discussed. In [DP], Deligne and Pappas have defined a stratification on the reduction ℳ_{v}^{DP} of the Deligne-Pappas space by the Lie types. In this paper we define a stratification on the reduction ℳ of the Rapoport locus by the a-numbers. We did not discuss the Ekedahl-Oort stratification as it was the content of the work by F. Andreatta and E. Goren. The readers interested in the Ekedahl-Oort stratification are referred to their work.

Along this direction we obtain the following results

1. The Deligne-Pappas condition is equivalent to the Kottwitz determinant condition for O_{F} -abelian varieties of characteristic p.

2. The Kottwitz determinant condition is an open and close condition in the moduli space of polarized O_{F} -abelian varieties of characteristic p.

3. The Rapoport locus is the smooth locus of the Deligne-Pappas space.

4. The set slope(A) of slopes of an O_{F} -abelian variety A is either \{\frac{i}{g}, …, \frac{i}{g}, \frac{g−i}{g}, …, \frac{g−i}{g}\} for some 0 ≤ i ≤ g/2, or \{\frac{1}{2}, …, \frac{1}{2}\}.

5. Every isogeny class of polarized O_{F} -abelian varieties over k contains one which satisfies the Rapoport condition.

The next problem is whether these invariants arising from the Dieudonné modules in question can be realized by abelian varieties. We solve this problem by establishing the following result (Theorem 7.5)

Theorem Any quasi-polarized O-p-divisible group (H, λ, ι) over k is isomorphic to the pdivisible attached to a polarized O_{F} -abelian variety.

This is the integral analogue of the Manin’s problem. The result implies that the strata defined by these invariants are all not empty. As an application we construct a polarized O_{F} -abelian surface of characteristic p in the complement of the Rapoport locus which is specialized by a point of characteristic 0. To the author’s knowledge, this is the first example in the literature that shows invalidity of the valuation criterion for the Rapoport locus at good reduction, even though the statement has been known, due to Deligne and Pappas [DP, Introduction], for almost ten years.

The proof of the algebraization theorem goes as follows. Using the weak Grothendieck conjecture, we algebrise the p-divisible groups in the Rapoport locus. Then we algebrise the rest of p-divisible groups by classifying the formal isogeny classes explicitly. The classification uses a result of Rapoport-Zink [RZ] and of Rapoport-Richartz [RR].

Another purpose of this paper is to generalize the main results of E. Goren and F. Oort [GO] on the a-number stratification, the slope stratification and their relation, removing the assumption of p. To describe our results, we introduce some notations. Let S(g) denote the set \{i ∈ ℤ; 0 ≤ i ≤ \frac{g}{2}\} ∪ \{\frac{g}{2}\} with the order induced from ℚ. For each i ∈ S(g), we denote by s(i) the set of slopes \{\frac{i}{g}, …, \frac{i}{g}, \frac{g−i}{g}, …, \frac{g−i}{g}\}. Let ⌈m⌉ denote the smallest integer not less than the real number m. For each O_{F} -abelian variety A over k, the reduced (usual) a-number t(A) of A is defined to be the dimension of the k-vector space Hom(α_{p}, A[π]). In this direction we establish the following results

1. For each a-number \underline{a}, we define a closed subscheme ℳ_{≥\underline{a}} of ℳ. It has the property that x ∈ ℳ_{≥\underline{a}}(k) if and only if the a-number of x ≥ \underline{a}. We show (Theorem 5.4) that the closed subscheme ℳ_{≥\underline{a}} is smooth over Speck(v) of pure dimension g − | \underline{a} |.

2. An a-number \underline{a} = (a^{i}) is called spaced if a^{i}a^{i+1} = 0, ∀ i ∈ ℤ/fℤ (cf. [GO, p.112]). If \underline{a} is spaced, then the points in ℳ_{≥\underline{a}} of slope s(|\underline{a}|) are dense in ℳ_{≥\underline{a}} (Theorem 6.8).

3. The generic point of each irreducible component of ℳ_{≥\underline{a}} has slope ≥ s(λ(\underline{a})), where λ(\underline{a}) := max\{|\underline{b}|; \underline{b} ⊂ \underline{a}, \underline{b} is spaced\} (cf. [GO, p.112]) (Corollary 6.9).

4. Let x : Speck → ℳ be a geometric point of reduced a-number one. Then for each m ∈ S(g), s(m) ≤ slope(x), the reduced closed subscheme ℳ_{x}^{≥s(m)} of slope ≥ s(m) in ℳ_{x} is formally smooth of codimension ⌈m⌉ and its generic point has slope s(m) (Theorem 6.12).

5. The weak Grothendieck conjecture (see [GO. Sec. 5.1]) for ℳ holds (Corollary 6.15).

6. The strong Grothendieck conjecture (see [GO. Sec. 5.1]) for ℳ_{v}^{DP} holds when p is totally ramified in F (Theorem 6.19).

The methods are based on the previous works of F. Oort [O2], E. Goren and F. Oort [GO], and the author [Y]. Using the explicit deformation method developed by Norman [N], Norman-Oort [NO], T. Zink [Z2], we construct the universal deformation of the Dieudonné module in the Rapoport locus. Then we study systematically the a-number stratification and the slope stratification on the formal neighborhood around the point. The results above are extracted from the systematic iteration of Frobenius map. It is possible to extract finer information beyond the reduced a-number one from the result of our computation. We leave this possible generalization in the future when the finer information becomes useful. We follow the approach of [O2] and [Y], which is different from that of Goren and Oort [GO]. Therefore, we give a new proof of several results of theirs: Thm. 2.3.4, Cor. 2.3.11–13, Thm. 5.2.1, Thm. 5.4.4, and the key cases of Thm. 5.4.11 of [GO]. Andreatta and Goren earlier obtained similar results in the case when p is totally ramified. Our work is independent from their results.

In [C], C.-L. Chai studied the combinatoric properties of Newton polygons for connected reductive quasi-split algebraic groups, inspired by earlier works of R. Kottwitz [K1], K.-Z. Li and F. Oort [LO], M. Rapoport and M. Richartz [RR]. He gave a group-theoretic conjectural description of the dimensions of Newton strata of good reduction of Shimura varieties. The main motivation of this work is to examine whether his group-theoretic description of Newton strata for quasi-split groups applies for the simplest case of bad reduction. The results of this work support his description even when the reductive group G in question is no longer unramified. One may expect that Chai’s description applies for the bad reduction case as well, under the assumption of the existence of good integral models of Shimura varieties for certain special subgroups K_{p} ⊂ G(ℚ_{p}). We confirm (Corollary 6.20) a question of Chai [C, Question 7.6, p.984] on the dimensions of Newton strata in the Hilbert-Blumenthal cases when p is totally ramified.

The results of this work reveal an important feature: the stratifications on the smooth locus of bad reduction of Hilbert-Blumenthal varieties behave very similarly as those on the good reduction. The reader can compare the results 1–6 above and those of [GO] for the good reduction case. A fundamental question is whether this feature holds for more general PEL-type Shimura varieties. We have no idea but expect that the cases of type C still hold.

The following is the structure of this paper. In Section 2 we describe the structure of Dieudonné module of general polarized O_{F} -abelian varieties of dimension g. In Section 3 we classify the formal isogeny classes explicitly. In Section 4 We provide the normal forms of the Dieudonné modules in the Rapoport locus. In Section 5 we give the natural generalization the a-number stratification on the Rapoport locus and study its properties. In Section 6 we study systematically the a-number stratification and the slope stratification on formal neighborhoods in the Rapoport locus, by the methods of [N], [NO], [Z], and [O2] as explained before. In Section 7 we establish the theorem of algebraization concerning the p-divisible groups in question. In Section 8 we construct the desired example as explained before. In the last section we do a computation of the Hecke correspondence. Using this, we describes the singularities of the supersingular locus near the superspecial point constructed in Section 8.

The problems addressed in this paper are all local, hence we use the convenient language of algebraic stacks. The reader is free to replace the word “algebraic stack(s)” by “scheme(s)” in this paper by adding an auxiliary level structure.

Acknowledgments. This paper grows out from a letter to Chai responding his questions in March 2001. I am very grateful to him for his stimulating questions and encouragements. I would like to thank J. Achter for the simplified proof of Lemma 3.2. The paper is benefited from a conversation with M. Rapoport. It is my pleasure to thank him. Special thanks are due to G. Kings for useful comments on an earlier manuscript. The manuscript was completed during my stay at the Max-Planck-Institut für Mathematik in Bonn in the summer of 2001. I thank the Institute for the kind hospitality and the excellent working environment.

§2. Dieudonné modules

(2.1) Let F be a totally real field of degree g and O_{F} be its ring of integers. To simplify notation, we will assume that there is only one prime of O_{F} lying over p in Section 2–6. Let e be the ramification index and f be the residue degree of this prime v, thus g = ef. Denote by F_{v} the completion of F at v and O the ring of integers in F_{v}. Let O^{ur} denote the ring of integers in the maximal étale extension of ℤ_{p} in O. The ring O^{ur} is isomorphic to W (𝔽_{p^{f}}). Let π be a uniformizer of O, chosen from O_{F}. Let P (T) be the monic irreducible polynomial of π over O^{ur}.

(2.2) Let (A, λ, ι) be a polarized O_{F} -abelian variety of dimension g over a perfect field k containing 𝔽_{p^{f}}, and let M be its covariant Dieudonné module. The Dieudonné module M is a free O ⊗_{ℤ_{p}} W (k)-module of rank two equipped with a non-degenerate alternating pairing

M × M → W (k)

such that ⟨ax, y⟩ = ⟨x, ay⟩ for all x, y ∈ M and a ∈ O and on which the Frobenius F and Verschiebung V commute with the action of O. We call it briefly a quasi-polarized O-Dieudonné module. The ring O ⊗_{ℤ_{p}} W (k) is isomorphic to

⨁_{i∈ℤ/fℤ}W (k)[T]/(σ_{i}(P (T)),

where σ_{i}, i ∈ ℤ/fℤ, are embeddings of O^{ur} into W (k), arranged in a way that σσ_{i} = σ_{i+1}. Set W^{i} := W (k)[T]/(σ_{i}(P (T)) and denote again by π the image of T in W^{i}. The action of the Frobenius map σ on O ⊗_{ℤ_{p}} W (k) through the right factor gives a map σ : W^{i} → W^{i+1} which sends a ↦ a^{σ} for a ∈ W (k) and σ(π) = π. We also have W^{i} ⊗_{W} k = k[π]/(π^{e}) and O ⊗_{ℤ_{p}} k = ⊕_{i∈ℤ/fℤ}k[π]/(π^{e}). Let

M^{i} := {x ∈ M | ax = σ_{i}(a)x, ∀ a ∈ O^{ur} }

be the σ_{i}-component of M, which is a free W^{i}-module of rank two. We have the decomposition

M = M^{0} ⊕ M^{1} ⊕ ⋯ ⊕ M^{f−1}

in which F : M^{i} → M^{i+1}, V : M^{i+1} → M^{i}. The summands M^{i}, M^{j} are orthogonal with respect to the pair ⟨ , ⟩ for i ≠ j. Conversely, a Dieudonné module together with such a decomposition and these properties is a quasi-polarized O-Dieudonné module.

(2.3) The Lie algebra Lie(A) of A also admits a decomposition

Lie(A) = ⨁_{i∈ℤ/fℤ}Lie(A)^{i}

and Lie(A)^{i} = M^{i}/V M^{i+1} is a k[π]/(π^{e})-module generated by at most two elements. Let

Lie(A)^{i} = k[π]/(π^{e}) · x_{1}^{i} ⊕ k[π]/(π^{e}) · x_{2}^{i} ≃ k[π]/(π^{e_{1}^{i}}) ⊕ k[π]/(π^{e_{2}^{i}}).

Hence we obtain an invariant ({e_{1}^{i}, e_{2}^{i}})_{i}, which we call the Lie type of A and denote it by lie(A).

(2.4) We say an O_{F} -abelian variety satisfies the Deligne-Pappas condition if it admits a separable O_{F} -linear polarization. It is easy to see that an O-Dieudonné module can be separably quasi-polarized if and only if e_{1}^{i} + e_{2}^{i} = e for all i ∈ ℤ/fℤ. The latter is equivalent to the condition that the k-dimensions of Lie(A)^{i} are all the same. This is equivalent to the Kottwitz determinant condition [K2, Sec. 5]. It follows that the Deligne-Pappas condition implies the Kottwitz determinant condition. We will see (3.10) that any O_{F} -abelian variety admits an O_{F} -linear polarization λ. The Kottwitz determinant condition then implies that the polarization λ is a power of π times a separable polarization. Therefore, these two conditions are equivalent.

(2.5) The invariant lie(A) = ({e_{1}^{i}, e_{2}^{i}})_{i} has the property that e_{j}^{i} ≤ e, i ∈ ℤ/fℤ, j = 1, 2, ∑_{i} e_{1}^{i} + e_{2}^{i} = g, and the invariant e_{1}^{i} + e_{2}^{i} is a locally constant function in a family. The latter follows from the fact that the σ_{i}-component Lie(A)^{i} of Lie(A) is a locally free sheaf. One can see from this that the Kottwitz determinant condition is an open condition for the reduction of Hilbert-Blumenthal varieties.

In [DP], Deligne and Pappas showed that the stratum of each Lie type ({e_{1}^{i}, e_{2}^{i}})_{i}, in the Deligne-Pappas space, is a regular locally closed subset, and has dimension g − 2 ∑_{i} min{e_{1}^{i}, e_{2}^{i}} provided the stratum is non-empty. However, non-emptiness follows directly from the algebraization theorem (7.5).

(2.6) Let Def[A, ι] denote the deformation functor of the O_{F} -abelian variety (A, ι) over k. It follows from crystalline theory that Def[A, ι](k[ϵ]) = Hom_{k⊗O_{F}} (V M/pM, M/V M). From dim_{k} Hom_{k[π]/π^{e}}(k[π]/(π^{a}), k[π]/(π^{b})) = min{a, b}, we compute that

dim_{k} Def[A, ι](k[ϵ]) = ∑_{i∈ℤ/fℤ} ∑_{1≤j,k,≤2} min{e_{j}^{i}, e − e_{k}^{i}}.  (1)

If (A, ι) satisfies the Deligne-Pappas condition, then

dim_{k} Def[A, ι](k[ϵ]) = ef + 2 ∑_{i∈ℤ/fℤ}min{e_{1}^{i}, e_{2}^{i}}.  (2)

(2.7) Assume that (A, λ, ι) is a separably polarized O_{F} -abelian variety, that is, it lies in the Deligne-Pappas space. Let Def[A, λ, ι] denote the deformation functor of (A, λ, ι). We can choose a k[π]/(π^{e})-basis {x_{1}^{i}, x_{2}^{i}} of (M/pM)^{i} for each i ∈ ℤ/fℤ such that (V M/pM)^{i} is generated by {π^{e_{1}^{i}}x_{1}^{i}, π^{e_{2}^{i}}x_{2}^{i}} and ⟨π^{e−1}x_{1}^{i}, x_{2}^{i}⟩ = 1, ∀ i ∈ ℤ/fℤ. The first order universal deformation (over the deformation ring R) of the O_{F} -abelian variety (A, ι) is given by the following data:

F̃ il ⊂ H_{1}^{cris}(A/R), F̃ il = ⊕F̃ il^{i}, F̃ il^{i} =< X_{1}^{i}, X_{2}^{i} >_{R[π]/(π^{e})},

where

X_{1}^{i} = π^{e_{1}^{i}}x_{1}^{i} + ∑_{j=0}^{e_{1}^{i}−1} a_{j}^{i}π^{j}x_{1}^{i} + ∑_{j=0}^{e_{2}^{i}−1} b_{j}^{i}π^{j}x_{2}^{i}, X_{2}^{i} = π^{e_{2}^{i}}x_{2}^{i} + ∑_{j=0}^{e_{1}^{i}−1} c_{j}^{i}π^{j}x_{1}^{i} + ∑_{j=e_{2}^{i}−e_{1}^{i}}^{e_{2}^{i}−1} d_{j}^{i}π^{j}x_{2}^{i}.

Here we assume that e_{1}^{i} ≤ e_{2}^{i} for simplicity. From ⟨X_{1}^{i}, X_{2}^{i}⟩ = 0, the vector space Def[A, λ, ι](k[ϵ]) is cut down by the equations d_{e_{2}^{i}−1}^{i} + a_{e_{1}^{i}−1}^{i} − b_{e_{1}^{i}−1}^{i} = 0, which is empty if min{e_{1}^{i}, e_{2}^{i}} = 0. Hence we conclude that

dim_{k} Def[A, λ, ι](k[ϵ]) = ef + ∑_{i∈ℤ/fℤ}(2 min{e_{1}^{i}, e_{2}^{i}} − min{e_{1}^{i}, e_{2}^{i}, 1}) .  (3)

This shows that dim_{k} Def[A, λ, ι](k[ϵ]) = g if and only if (A, λ, ι) satisfies the Rapoport condition.

(2.8) Proposition The Rapoport locus is the smooth locus in the Deligne-Pappas space.

Proof. It follows from the dimension statement of [DP, Thm. 2.2] and (3).

(2.9) Remark It is well known that the forgetful morphism Def[A, λ, ι] → Def[A, ι] is formally étale if (A, λ, ι) satisfies the Rapoport condition, and that the Rapoport locus is smooth. In [DP, Thm 2.2], Deligne and Pappas concluded that the singular locus had codimension two. However, they actually showed that the complement of the Rapoport locus had codimension two. Proposition 2.8 fills the harmless gap of their assertion on the dimension of the singular locus. From (2) and (3), one can see that the forgetful morphism Def[A, λ, ι] → Def[A, ι] is not formally étale any more when (A, λ, ι) does not satisfy the Rapoport condition.

(2.10) Let a(A) denote the a-module of A, which is defined to be the cokernel of the Frobenius map F on Lie(A) = M/V M:

M/V M →^{F} M/V M → a(A) → 0.

Note that in the covariant theory, the Frobenius map F is induced from the Verschiebung morphism V : A^{(p)} → A via the covariant functor.

Let each σ_{i}-component of a(A) be

a(A)^{i} ≃ k[π]/(π^{a_{1}^{i}}) ⊕ k[π]/(π^{a_{2}^{i}})

for some {a_{1}^{i}, a_{2}^{i}}. We define the a-number a(A) of A to be the invariant a(A) = ({a_{1}^{i}, a_{2}^{i}})_{i}. Let |a(A)| denote the usual a-number of A, which is the dimension of the k-vector space Hom_{k}(α_{p}, A^{t}). If the O_{F} -abelian variety (A, ι) satisfies the Rapoport condition, we write a(A) = (a^{i})_{i} for a(A) = ({0, a^{i}})_{i}.

(2.11) Lemma The a-module a(A) is canonically isomorphic to the k-linear dual Hom_{k}(α_{p}, A^{t})^{∗} of Hom_{k}(α_{p}, A^{t}) as k ⊗_{ℤ} O_{F} -modules.

Proof. Let M^{∗} denote the contravariant Dieudonné functor. Then we have

Hom_{k}(α_{p}, A^{t}) = Hom_{W [F,V]}(M^{∗}(A^{t}), k) ≃ Hom_{W [F,V]}(M(A), k)
= Hom_{k}(M(A)/(F, V)M(A), k) = a(A)^{∗}.

(2.12) Note that A is ordinary if and only if |a(A)| = 0. It is easy to see that the latter condition implies that {e_{1}^{i}, e_{2}^{i}} = {0, e} for all i ∈ ℤ/fℤ. Therefore, ordinary points satisfy the Rapoport condition automatically. As non-ordinary points do not specialize to the boundary of Deligne-Pappas space, the complement of the Rapoport locus is in the interior of the space. Therefore, the compactification constructed by Rapoport provides that of the Deligne-Pappas space [DP, Introduction].

(2.13) Let a(A) = ({a_{1}^{i}, a_{2}^{i}})_{i}, a_{1}^{i} ≤ a_{2}^{i}, be the a-number of A and lie(A) = ({e_{1}^{i}, e_{2}^{i}})_{i} be the Lie type. It follows from elementary divisors that there are two W^{i}-bases {x_{1}^{i}, x_{2}^{i}}, {y_{1}^{i}, y_{2}^{i}} of M^{i} such that

F M^{i−1} =< π^{e−e_{1}^{i−1}}x_{1}^{i}, π^{e−e_{2}^{i−1}}x_{2}^{i} >, and V M^{i+1} =< π^{e_{1}^{i}}y_{1}^{i}, π^{e_{2}^{i}}y_{2}^{i} > .

If e_{1}^{i} = min{e_{1}^{i}, e_{2}^{i}, e − e_{1}^{i−1}, e − e_{2}^{i−1}}, write y_{1}^{i} = ax_{1}^{i} + bx_{2}^{i}, y_{2}^{i} = cx_{1}^{i} + dx_{2}^{i}, then

F M^{i−1} + V M^{i+1} =< π^{e_{1}^{i}}x_{1}^{i}, π^{e_{2}^{i}}x_{2}^{i}, π^{e−e_{1}^{i−1}}bx_{2}^{i}, π ^{e^{i−1}i_{2}}dx_{2}^{i} > .

We obtain a bound for a-numbers: a_{1}^{i} = e_{1}^{i} and min{e_{2}^{i}, e − e_{1}^{i−1}, e − e_{2}^{i−1}} ≤ a_{2}^{i} ≤ e_{2}^{i} when e_{1}^{i} = min{e_{1}^{i}, e_{2}^{i}, e − e_{1}^{i−1}, e − e_{2}^{i−1}}.

(2.14) Lemma Let a(A) = ({a_{1}^{i}, a_{2}^{i}})_{i} and lie(A) = ({e_{1}^{i}, e_{2}^{i}})_{i}. Then lie(A^{t}) = ({e − e_{1}^{i}, e − e_{2}^{i}})_{i} and the a-number ({b_{1}^{i}, b_{2}^{i}})_{i} of the dual abelian variety A^{t} is given as follows: b_{1}^{i} = min{e − e_{1}^{i}, e − e_{2}^{i}, e_{1}^{i−1}, e_{2}^{i−1}} and b_{1}^{i} + b_{2}^{i} = (a_{1}^{i} + a_{2}^{i}) + (e_{1}^{i−1} + e_{2}^{i−1}) − (e_{1}^{i} + e_{2}^{i}).

Proof. Write \overline{M} = M/pM and \overline{M^{t}} = M^{t}/pM^{t}, where M^{t} is the Dieudonné module of A^{t}. From the definition of lie(A) = (\{e_{1}^{i}, e_{2}^{i}\})_{i}, one concludes that V \overline{M^{i+1}} is isomorphic to k[π]/(π^{e−e_{1}^{i}}) ⊕ k[π]/(π^{e−e_{2}^{i}}). The perfect pairing \overline{M} × \overline{M^{t}} → k induces a perfect pairing

V \overline{M^{i+1}} × \overline{M^{t,i}}/V \overline{M^{t,i+1}} → k,

thus lie(A^{t}) = ({e − e_{1}^{i}, e − e_{2}^{i}})_{i}.

Let W^{i} := F \overline{M^{i−1}} ∩ V \overline{M^{i+1}} in \overline{M^{i}}. We have dim_{k} W^{i} = dim_{k} F \overline{M^{i−1}} + dim_{k} V \overline{M^{i+1}} − dim_{k}(F \overline{M^{i−1}}+V \overline{M^{i+1}}) = (a_{1}^{i}+a_{2}^{i})+(e_{1}^{i−1}+e_{2}^{i−1})−(e_{1}^{i}+e_{2}^{i}). The perfect pairing \overline{M} ×\overline{M^{t}} → k induces a perfect pairing

W^{i} × (M^{t}/(F, V)M^{t})^{i} → k,

thus b_{1}^{i}+b_{2}^{i} = (a_{1}^{i}+a_{2}^{i})+(e_{1}^{i−1}+e_{2}^{i−1})−(e_{1}^{i}+e_{2}^{i}). The assertion b_{1}^{i} = min{e−e_{1}^{i}, e−e_{2}^{i}, e_{1}^{i−1}, e_{2}^{i−1}} is obtained from (2.13) and lie(A^{t}) = ({e − e_{1}^{i}, e − e_{2}^{i}})_{i}.

(2.15) Let D^{−1} = (π^{−d}) be the inverse of the different of O over ℤ_{p}. There is a unique W ⊗ O-bilinear pair ( ,) : M × M → W ⊗ D^{−1} such that ⟨x, y⟩ = Tr_{W ⊗O/W} (x, y). For each W^{i}-basis x_{1}^{i}, x_{2}^{i} of M^{i}, the π-adic valuation ord_{π}(x_{1}^{i}, x_{2}^{i}) is independent of the choice of basis, and the degree of the quasi-polarization is p^{R}, where

R = 2 ∑_{i∈ℤ/fℤ}ord_{p} Norm_{W^{i}/W} (π^{d}(x_{1}^{i}, x_{2}^{i}))).

We can choose two W^{i}-bases {x_{1}^{i}, x_{2}^{i}}, {y_{1}^{i}, y_{2}^{i}} of M^{i} for each i ∈ ℤ/fℤ such that

V y_{1}^{i+1} = π^{e_{1}^{i}}x_{1}^{i}, V y_{2}^{i+1} = π^{e_{2}^{i}}x_{2}^{i}.

It follows from (V x, V y) = p(x, y)^{σ^{−1}} that we get

ord_{π}(y_{1}^{i+1}, y_{2}^{i+1}) = ord_{π}(x_{1}^{i}, x_{2}^{i}) + (e_{1}^{i} + e_{2}^{i} − e).

If the min_{i∈ℤ/fℤ} ord_{π}(x_{1}^{i}, x_{2}^{i}) = −d, say that i = 0 achieves the minimum, then

ord_{π}(x_{1}^{i}, x_{2}^{i}) = −d + ∑_{k=0}^{i−1}(e_{1}^{k} + e_{2}^{k} − e)

and

R = 2 ∑_{i=1}^{f−1} ∑_{k=0}^{i−1}(e_{1}^{k} + e_{2}^{k} − e).

We can divide by a power of π such that min_{i∈ℤ/fℤ} ord_{π}(x_{1}^{i}, x_{2}^{i}) = −d.

(2.16) Proposition There exists a number N, depending on g, such that ord_{p}(deg λ) ≤ N for any polarized O_{F} -abelian variety (A, λ, ι) of dimension g such that the polarization λ is not an integral multiple of another polarization.

Proof. Take any number greater than R as above, e.g. N = ef(f − 1).

§3. Formal isogeny classes

In the rest of this paper we will assume that k is an algebraically closed field of characteristic p > 0.

(3.1) In this section we address the following question: For every isogeny class of polarized O_{F} -abelian varieties over k, is there one which satisfies the Rapoport condition? Two polarized O_{F} -abelian varieties (A_{i}, λ_{i}, ι_{i}), i = 1, 2, are isogenous if there exists an O_{F} -linear isogeny φ : A_{1} → A_{2} such that φ^{∗}λ_{2} = aλ_{1} for some a ∈ ℤ. We answer this question in Corollary 3.9.

(3.2) Lemma Let M be a quasi-polarized O-Dieudonné module. Then the set slope(M) of slopes of M is either \{\frac{i}{g}, …, \frac{i}{g}, \frac{g−i}{g}, …, \frac{g−i}{g}\} for some 0 ≤ i ≤ g/2, or \{\frac{1}{2}, …, \frac{1}{2}\}.

Proof. Suppose that M is not supersingular. The F -isocrystal M ⊗ B(k) contains M_{a,b}^{r} ⊕ M_{b,a}^{r}, where M_{a,b} is the simple F -isocrystal of slope a/(a + b). We want to show that 2(a + b)r = 2g. We first have an embedding F_{v} → End(M_{a,b}^{r}) = M_{r}(D_{a/(a+b)}) and any maximal commutative subalgebra of the latter has ℚ_{p}-dimension r(a + b), so g| r(a + b). On the other hand, we have 2g ≥ 2r(a + b), from M ⊗ B(k) ⊃ M_{a,b}^{r} ⊕ M_{b,a}^{r}. Hence M ⊗ B(k) = M_{a,b}^{r} ⊕ M_{b,a}^{r}. This completes the proof.

(3.3) Let S(g) denote the subset of ℚ which parameterizes possible slopes arising from O_{F} -abelian varieties

S(g) := \{i ∈ ℤ; 0 ≤ i ≤ \frac{g}{2}\} ∪ \{\frac{g}{2}\}.

For each i ∈ S(g), we denote by s(i) the set of slopes \{\frac{i}{g}, …, \frac{i}{g}, \frac{g−i}{g}, …, \frac{g−i}{g}\}. The order on S(g) induced from ℚ is compatible with the Grothendieck’s specialization theorem.

(3.4) Example. Let M = M^{0} ⊕ ⋯ ⊕ M^{f−1} be an O-Dieudonné module, where M^{i} is a free W^{i}-module of rank two generated by x_{1}^{i}, x_{2}^{i} with x_{2}^{i} ∈ V M^{i+1}. Let

\begin{aligned}F x_{1}^{i} & = a_{11}^{i+1}x_{1}^{i+1} + a_{12}^{i+1}x_{2}^{i+1} \\ F x_{2}^{i} & = pa_{21}^{i+1}x_{1}^{i+1} + pa_{22}^{i+1}x_{2}^{i+1} \end{aligned}  (4)

be the action of the Frobenius F. Then

\begin{aligned}F^{f}x_{1}^{0} & = αx_{1}^{0} + βx_{2}^{0} \\ F^{f}x_{2}^{0} & = γx_{1}^{0} + δx_{2}^{0}, \end{aligned}  (5)

where

(\begin{matrix}α & β \\ γ & δ \end{matrix}) = A_{1}^{(f−1)} ⋯ A_{f−1}^{(1)}A_{f}, A_{i} := (\begin{matrix}a_{11}^{i} & a_{12}^{i} \\ pa_{21}^{i} & pa_{22}^{i} \end{matrix}) , A_{f} = A_{0}.

Here we write A^{(n)} for A^{σ^{n}}. Note that M satisfies the Rapoport condition.

Let a, b ∈ ℤ, a + b = g, 0 ≤ a ≤ b. Write a = de + r, 0 ≤ r < e. Take

A_{i} = (\begin{matrix}0 & 1 \\ −p & 0 \end{matrix}) for 1 ≤ i ≤ 2d, A_{i} = (\begin{matrix}1 & 1 \\ −p & 0 \end{matrix}) for 2d < i < f, and A_{f} = (\begin{matrix}π^{r} & 1 \\ −p & 0 \end{matrix}) .

Then we have

(\begin{matrix}α & β \\ γ & δ \end{matrix}) = p^{d} (\begin{matrix}α′ & β′ \\ γ′ & δ′ \end{matrix}) , (\begin{matrix}α′ & β′ \\ γ′ & δ′ \end{matrix}) = ∏_{i=2d+1}^{f} A_{i} ≡ (\begin{matrix}π^{r} & 1 \\ 0 & 0 \end{matrix}) mod p

Note that A_{i}^{σ} = A_{i} for all i ∈ ℤ/ℤ/fℤ. The characteristic polynomial of F^{f} is X^{2} − (α + β)X + p^{f} and ord_{p}(α + β) = \frac{a}{e}. Therefore slope(M) = \{\frac{a}{g}, …, \frac{a}{g}, \frac{b}{g}, …, \frac{b}{g}\}.

(3.5) Remark The example constructed above is non-trivial, due to the following constrain. Let M = M_{a,b} ⊕ M_{b,a}, a + b = g, a ≠ b be an O-Dieudonné module which satisfies the Rapoport condition, where M_{a,b} (resp. M_{b,a}) is a Dieudonné submodule of slope \frac{b}{a+b} (resp. \frac{a}{a+b}). Then a is a multiple of e. Therefore, one can not construct a special Dieudonné module, in the sense of Manin, with an O-action which satisfies the Rapoport condition and has arbitrary possible slopes in Lemma 3.2. The proof of this fact is as follows.

Let M_{a,b} = M^{0} ⊕ ⋯ ⊕ M^{f−1} as in (2.2), where M^{i} is a free W^{i}-module of rank one. We can choose a basis x_{i} for M^{i} such that V x_{i+1} = π^{n_{i}}x_{i} for 1 ≤ i ≤ f − 1 and V x_{1} = u_{0}π^{n_{0}}x_{0}, u_{0} ∈ (W^{0})^{×}, where n_{i} = 0 or e as it satisfies the Rapoport condition. As the slope of M_{a,b} is \frac{b}{a+b}, we have V^{f}x_{0} = π^{a}ux_{0}, u ∈ (W^{0})^{×}, hence a = n_{0} + ⋯ + n_{f−1}.

(3.6) Let V be a 2-dimensional F_{v}-vector space with a non-degenerate alternating form ψ on V with values in ℚ_{p} such that ψ(ax, y) = ψ(x, ay) for a ∈ F_{v}, x, y ∈ V. Let G be the algebraic group of F_{v}-linear similitudes over ℚ_{p} and let G_{1} be the derived group of G. The algebraic group G_{1} is simply-connected. Denote by c the multiplier map G → 𝔾_{m}.

Let M be a quasi-polarized Dieudonné module. We choose an isomorphism between M_{0} ⊗_{W} B(k) and V ⊗_{ℚ_{p}} B(k) for skew-symmetric F_{v}-modules over B(k). Let b ∈ G(B(k)) be the element obtained by the transport of structure of the Frobenius F of M, and let ν be associated slope homomorphism from the pro-torus D into G, [K1, Sec. 4]. Let J_{b} be the algebraic group over ℚ_{p} which represents the group functor ([RZ, Prop. 1.12])

R ↦ {g ∈ G(R ⊗_{ℚ_{p}} B(k)); g(bσ) = (bσ)g}.

(3.7) Lemma H^{1}(ℚ_{p}, J_{b}) = 0.

Proof. As k algebraically closed, we may assume that b satisfies the decent equation ([RZ, 1.8]): (bσ)^{s} = sν(p)σ^{s} for some s ∈ ℕ . If M is supersingular, then J_{b} is an inner form of G [RZ, Remark 1.15]. Therefore, H^{1}(ℚ_{p}, J_{b}) = H^{1}(ℚ_{p}, G) = 0. If M is not supersingular, then J_{b} is an inner form of the centralizer G_{sν(p)}, which is Res_{Q_{p^{s}}/ℚ_{p}}(F_{v} ⊗ ℚ_{p^{s}})^{×} × ℚ_{p}^{×}. Therefore, H^{1}(ℚ_{p}, J_{b}) = 0.

(3.8) Corollary Let M_{1} and M_{2} are two quasi-polarized O-Dieudonné modules. If slope(M_{1}) = slope(M_{2}), then M_{1}⊗_{W} B(k) and M_{2}⊗_{W} B(k) are isomorphic as quasi-polarized F -isocrystals with the O-action.

Proof. This follows from [RR, Prop. 1.17] and Lemma 3.7.

(3.9) Corollary Any polarized O_{F} -abelian variety over k is isogenous to one which satisfies the Rapoport condition.

Proof. It follows from (3.4), Corollary 3.8, and surjectivity of the multiplier map c : G(ℚ_{p}) → ℚ_{p}^{×} that the statement holds for the associated p-divisible group. Then the assertion follows from a theorem of Tate.

(3.10) Corollary Any O_{F} -abelian variety over k admits an O_{F} -linear polarization.

Proof. This follows from Corollary 3.9 and [R, Prop. 1.10].

§4. Normal forms

(4.1) Let M be a non-ordinary separably quasi-polarized O-Dieudonné module over k which satisfies the Rapoport condition. Let a(M) = (a^{i})_{i} be the a-number of M. Denote by τ(M) the a-index of M, which is defined to be the subset of ℤ/fℤ:

{i ∈ ℤ/fℤ; a^{i} ≠ 0}.

Write τ for τ(M).

Denote by \overline{M} the quotient of M module πM. The module \overline{M} is a 2f-dimensional vector space over k together with a k-linear action by O/π = O^{ur}/p that commutes with the action of F and V. The perfect pairing ⟨ , ⟩ on M induces an perfect pairing ⟨ , ⟩ on \overline{M} that satisfies ⟨F x, y⟩ = ⟨x, V y⟩^{p} and ⟨ax, y⟩ = ⟨x, ay⟩ for all x, y ∈ \overline{M}, a ∈ O^{ur}/p. The decomposition of \overline{M} (2.2) into σ_{i}-eigenspaces

\overline{M} = ⊕_{i∈ℤ/fℤ}\overline{M}^{i}

respects the perfect pairing as before (2.2).

As M satisfies the Rapoport condition, V \overline{M} and F \overline{M} are free k ⊗ O^{ur}-module of rank one, and we have that F \overline{M} = ker V and V \overline{M} = ker F. It follows that \overline{M} is isomorphic to N/pN for a separably quasi-polarized O^{ur}-Dieudonné module N over k of rank 2f with the same a-index as M.

(4.2) Proposition There exists a k-basis \{x_{i}, y_{i}\} of \overline{M}^{i} for each i ∈ ℤ/fℤ such that

• ⟨x_{i}, y_{i}⟩ = 1 and y_{i} ∈ V \overline{M} for all i ∈ ℤ/fℤ,

• F x_{i} = \begin{cases}x_{i+1} & if i + 1 ∉ τ \\ −y_{i+1} & if i + 1 ∈ τ \end{cases} V y_{i} = \begin{cases}y_{i−1} & if i ∉ τ \\ 0 & if i ∈ τ \end{cases} V x_{i} = \begin{cases}y_{i−1} & if i ∈ τ \\ 0 & if i ∉ τ \end{cases}

Proof. It follows from [Y, Prop. 4.1] or [GO, Thm. 3.2.8].

Note that this proposition gives the classification of the π-torsion subgroup scheme A[π] of separably polarized O_{F} -abelian varieties over k, classified by the a-indices.

The following results generalize [Y, Prop. 4.1, Prop 4.2, Lemma 4.3], the proofs are the same.

(4.3) Proposition There exists a W^{i}-basis {X_{i}, Y_{i}} of M for each i ∈ ℤ/fℤ such that

(i) ⟨X_{i}, Y_{i}⟩ = 1 and Y_{i} ∈ (V M)^{i}, ∀ i ∈ ℤ/fℤ,

(ii) F X_{i} = \begin{cases}X_{i+1} & if i + 1 ∉ τ \\ −Y_{i+1} + c_{i+1}πX_{i+1} & if i + 1 ∈ τ \end{cases} F Y_{i} = \begin{cases}pY_{i+1} & if i + 1 ∉ τ \\ pX_{i+1} & if i + 1 ∈ τ \end{cases}

for some c_{i+1} ∈ W^{i+1}.

(4.4) Proposition There exists a W^{i}-basis {X_{i}, Y_{i}} of M^{i} for each i ∈ ℤ/fℤ such that

(i) ⟨X_{i}, Y_{i}⟩ ∈ (W^{i})^{×} and Y_{i} ∈ (V M)^{i}, ∀ i ∈ ℤ/fℤ,

(ii) F X_{i} = \begin{cases}X_{i+1} & if i + 1 ∉ τ \\ Y_{i+1} + c_{i+1}πX_{i+1} & if i + 1 ∈ τ \end{cases} F Y_{i} = \begin{cases}π^{e}Y_{i+1} & if i + 1 ∉ τ \\ π^{e}X_{i+1} & if i + 1 ∈ τ, \end{cases}

for some c_{i+1} ∈ W^{i+1}

Note that the i-th component a^{i} of a(M) is min{e, ord_{π}(c_{i}π)} for i ∈ τ.

(4.5) Lemma If M is superspecial, i.e. |a(M)| = g. Then (1) There exists a W^{i}-basis {X_{i}, Y_{i}} of M^{i} for each i ∈ ℤ/fℤ such that

• Y_{i} ∈ (V M)^{i} and ⟨X_{i}, Y_{i}⟩ = 1,

• F X_{i} = −Y_{i+1}, F Y_{i} = pX_{i+1},

for all i ∈ ℤ/fℤ.

(2) There exists a W^{i}-basis {X_{i}, Y_{i}} of M^{i} for each i ∈ ℤ/fℤ such that

• Y_{i} ∈ (V M)^{i} and ⟨X_{i}, Y_{i}⟩ = u_{i} ∈ (W^{i})^{×} with u_{i}^{σ^{r}} = u_{i}, where r = lcm(2, f),

• F X_{i} = Y_{i+1}, F Y_{i} = pX_{i+1},

for all i ∈ ℤ/fℤ.

(4.6) Let M be a non-ordinary separably quasi-polarized O-Dieudonné module satisfying the Rapoport condition. Let a(M) = (a^{i}) be the a-number of M and τ = τ(M) = {n_{1}, n_{2}, … n_{t}} be the a-index of M, where t = |a(M/πM)|, the reduced usual a-number of M. We assume that 0 ≤ n_{1} < n_{2} < ⋯ < n_{t} < f and let n_{0} := n_{t} − f and n_{t+1} = n_{1}. Denote by A(e, f) the set of a-numbers

A(e, f) := {(a^{i}); i ∈ ℤ/fℤ, a^{i} ∈ ℤ, 0 ≤ a^{i} ≤ e},

with the partial order that (a^{i}) ≤ (b^{i}) if a^{i} ≤ b^{i}, ∀ i ∈ ℤ/fℤ. An a-number (a^{i}) ∈ A(e, f) is called spaced if a^{i}a^{i+1} = 0, ∀i ∈ ℤ/fℤ, cf. [GO, Sec. 1, p.112].

By Proposition 4.4, we can choose a W^{i}-basis {X_{i}, Y_{i}} of M^{i} for each i ∈ ℤ/fℤ such that

(\begin{matrix}F X_{i−1} \\ F Y_{i−1} \end{matrix}) = A_{i} (\begin{matrix}X_{i} \\ Y_{i} \end{matrix}) ,

where

A_{i} = (\begin{matrix}1 & 0 \\ 0 & π^{e} \end{matrix}) if i ∉ τ; A_{i} = (\begin{matrix}π^{a^{i}}c_{i} & 1 \\ π^{e} & 0 \end{matrix}) if i ∈ τ,

for some c_{i} ∈ W^{i} for i ∈ τ and c_{i} is a unit if a^{i} < e.

Let ℓ_{i} := n_{i} − n_{i−1} for 1 ≤ i ≤ t. We have

(\begin{matrix}F^{ℓ_{i}}X_{n_{i−1}} \\ F^{ℓ_{i}}Y_{n_{i−1}} \end{matrix}) = (\begin{matrix}π^{a^{i}}c_{n_{i}} & 1 \\ π^{eℓ_{i}} & 0 \end{matrix}) (\begin{matrix}X_{n_{i}} \\ Y_{n_{i}} \end{matrix}) , ∀1 ≤ i ≤ t.

For 1 ≤ s ≤ t, suppose that

(\begin{matrix}F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}X_{n_{s−1}} \\ F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}Y_{n_{s−1}} \end{matrix}) = (\begin{matrix}α_{s} & β_{s} \\ γ_{s} & δ_{s} \end{matrix}) (\begin{matrix}X_{n_{t}} \\ Y_{n_{t}} \end{matrix}) ,

for some coefficients α_{s}, β_{s}, γ_{s}, δ_{s}. It follows from

\begin{aligned}(\begin{matrix}F^{ℓ_{s−1}+ℓ_{s}+···+ℓ_{t}}X_{n_{s−2}} \\ F^{ℓ_{s−1}+ℓ_{s}+···+ℓ_{t}}Y_{n_{s−2}} \end{matrix}) & = (\begin{matrix}F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}(F^{ℓ_{s−1}}X_{n_{s−2}}) \\ F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}(F^{ℓ_{s−1}}Y_{n_{s−2}}) \end{matrix}) \\ & = (\begin{matrix}π^{a^{n_{s−1}}}c_{n_{s−1}}^{(ℓ_{s}+ℓ_{s+1}+···+ℓ_{t})} & 1 \\ π^{eℓ_{s−1}} & 0 \end{matrix}) (\begin{matrix}F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}X_{n_{s−1}} \\ F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}Y_{n_{s−1}} \end{matrix}) \end{aligned}  (6)

that

(\begin{matrix}α_{s−1} & β_{s−1} \\ γ_{s−1} & δ_{s−1} \end{matrix}) = (\begin{matrix}u_{s−1} & 1 \\ π^{eℓ_{s−1}} & 0 \end{matrix}) (\begin{matrix}α_{s} & β_{s} \\ γ_{s} & δ_{s} \end{matrix}) , (\begin{matrix}α_{t} & β_{t} \\ γ_{t} & δ_{t} \end{matrix}) = (\begin{matrix}π^{a^{n_{t}}}c_{n_{t}} & 1 \\ π^{eℓ_{t}} & 0 \end{matrix}) ,  (7)

where u_{s−1} = π^{a^{n_{s−1}}}c_{n_{s−1}}^{(ℓ_{s}+ℓ_{s+1}+···+ℓ_{t})}. Recall that we write a^{(n)} for a^{σ^{n}} (3.4). Therefore,

(\begin{matrix}α & β \\ γ & δ \end{matrix}) := (\begin{matrix}α_{1} & β_{1} \\ γ_{1} & δ_{1} \end{matrix}) = (\begin{matrix}u_{1} & 1 \\ π^{eℓ_{1}} & 0 \end{matrix}) (\begin{matrix}u_{2} & 1 \\ π^{eℓ_{1}} & 0 \end{matrix}) ⋯ (\begin{matrix}u_{t} & 1 \\ π^{eℓ_{t}} & 0 \end{matrix}) .  (8)

(4.7) When c_{i} = 0, ∀ i ∈ τ, if t is even, t = 2d, we have

(\begin{matrix}α & β \\ γ & δ \end{matrix}) = (\begin{matrix}π^{e(ℓ_{2}+ℓ_{4}+···+ℓ_{2d})} & 0 \\ 0 & π^{e(ℓ_{1}+ℓ_{3}+···+ℓ_{2d−1})} \end{matrix})

and slope(M) = s(i) (3.3), where i = min{e(∑_{1≤j≤d} ℓ_{2j}), e(∑_{1≤j≤d} ℓ_{2j−1})}. If t is odd, t = 2d + 1, we have

(\begin{matrix}α & β \\ γ & δ \end{matrix})^{2} = (\begin{matrix}π^{g} & 0 \\ 0 & π^{g} \end{matrix})

and M is supersingular.

(4.8) Proposition Let M be a separably quasi-polarized O-Dieudonné module over k. If a(M) is spaced, i.e. ℓ_{i} ≥ 2 ∀ 1 ≤ i ≤ t, then slope(M) ≥ s(|a(M)|) (3.3).

Proof. We may assume that M is non-ordinary. It follows from (8) that

(\begin{matrix}F^{f}π^{e}X_{n_{t}} \\ F^{f}Y_{n_{t}} \end{matrix}) = A (\begin{matrix}π^{e}X_{n_{t}} \\ Y_{n_{t}} \end{matrix}) , A = ∏_{i=1}^{t} A_{i} = π^{|a(M)|} ∏_{i=1}^{t} (π^{−a^{n_{i}}}A_{i}) , A_{i} = (\begin{matrix}u_{t} & π^{e} \\ π^{e(ℓ_{t}−1)} & 0 \end{matrix}) .

Let N be the O-Dieudonné submodule of M generated by π^{e}X_{n_{t}}, Y_{n_{t}}. As F^{f}(N) ⊂ π^{|a(M)|}N, we get slope(M) ≥ s(|a(M)|).

(4.9) When t = |τ| = 1, say τ = {0}, we have

(\begin{matrix}F^{f}X_{0} \\ F^{f}Y_{0} \end{matrix}) = (\begin{matrix}c_{0} & 1 \\ π^{g} & 0 \end{matrix}) (\begin{matrix}X_{0} \\ Y_{0} \end{matrix}) .

for some c_{0} ∈ π^{a^{0}}W^{0}. It is easy to see that M satisfies a Cayley-Hamilton equation ([O2],[Y]) F^{2f}X_{0} − c_{0}^{(f)}F^{f}X_{0} − π^{g}X_{0} = 0, thus slope(M) = s(i), where i = min\{\frac{g}{2}, ord_{π}(c_{0})\}.

(4.10) When t = |τ| = 2, say τ = {0, ℓ_{2}} and ℓ_{2} ≤ ℓ_{1}, we have

(\begin{matrix}F^{f}X_{ℓ_{1}} \\ F^{f}Y_{ℓ_{1}} \end{matrix}) = (\begin{matrix}u_{1} & 1 \\ π^{eℓ_{1}} & 0 \end{matrix}) (\begin{matrix}u_{2} & 1 \\ π^{eℓ_{2}} & 0 \end{matrix}) (\begin{matrix}X_{ℓ_{1}} \\ Y_{ℓ_{1}} \end{matrix}) ,

for some coefficients u_{1}, u_{2} (u_{1} = c_{0}^{(ℓ_{2})}, u_{2} = c_{ℓ_{2}}). Then

(\begin{matrix}α & β \\ γ & δ \end{matrix}) = (\begin{matrix}u_{1}u_{2} + π^{eℓ_{2}} & u_{1} \\ π^{eℓ_{2}}u_{2} & π^{eℓ_{2}} \end{matrix})  (9)

If u_{1} = 0, then the matrix is (\begin{matrix}π^{eℓ_{2}} & 0 \\ π^{eℓ_{1}}u_{2} & π^{eℓ_{1}} \end{matrix}), hence slope(M) = s(eℓ_{2}). If u_{1} ≠ 0, then

we have the Cayley-Hamilton equation

βF^{2f}X_{ℓ_{1}} − (β^{(f)}δ + βα^{(f)})F^{f}X_{ℓ_{1}} + β^{(f)}(αδ − βδ)X_{ℓ_{1}} = 0.

As ord_{π}(δ) = eℓ_{1} ≥ \frac{g}{2}, we see that slope(M) = s(i), where i = min\{\frac{g}{2}, ord_{π}(a^{(f)}\}. In both cases we have that slope(M) = s(i), where i = min\{\frac{g}{2}, ord_{π}(u_{1}u_{2} + π^{eℓ_{2}})\}.

§5. a-number stratification

(5.1) Let p be a fixed prime number. Let ℳ^{DP} denote the moduli stack over Specℤ_{(p)} of polarized O_{F} -abelian varieties (A, λ, ι) of dimension g = [F : ℚ] with the polarization λ of prime-to-p degree. It is a separated Deligne-Mumford algebraic stack over Specℤ_{(p)} locally of finite type. In [DP], Deligne and Pappas showed that the algebraic stack ℳ^{DP} is flat and a locally complete intersection over Specℤ_{(p)} of relative dimension g, and the closed fibre ℳ^{DP} ⊗ 𝔽_{p} is geometrically normal and has singularities of codimension at least two. It follows from Deligne-Pappas’ results that the irreducible components of geometric special fibre ℳ^{DP}⊗\overline{𝔽_{p}} are in bijection correspondence with those of geometric generic fibre ℳ^{DP}⊗\overline{ℚ}. Those are parameterized by the isomorphism classes of non-degenerate skew-symmetric O_{F} - modules H_{1}(A(ℂ), ℤ) for all (A, λ, ι) ∈ ℳ^{DP}(ℂ).

Let ℳ^{R} denote the Rapoport locus of ℳ^{DP}. It is the smooth locus of ℳ^{DP} (2.8). Let ℳ denote the reduction ℳ^{R} ⊗_{ℤ}k(v) of ℳ^{R} modulo v, where k(v) is the residue field of O_{F} at v. We will define the stratification on ℳ by a-numbers scheme-theoretically, (cf. [Y, Sec. 3]).

(5.2) Let S be a locally noetherian scheme over Speck(v) and π_{A} : (A, λ, ι) → S ∈ ℳ(S) a polarized O_{F} -abelian scheme over S. The sheaf R^{1}(π_{A})_{∗}(O_{A}) is a locally free rank one O_{F} ⊗ O_{S}-module. It admits a decomposition (2.2)

R^{1}(π_{A})_{∗}(O_{A}) = ⨁_{i∈ℤ/fℤ}R^{1}(π_{A})_{∗}(O_{A})^{i}

with respect to the action by O_{F}. Each component R^{1}(π_{A})_{∗}(O_{A})^{i} is a locally free O_{S}[π]/(π^{e})- module of rank one and it admits a filtration of locally free O_{S}-modules

R^{1}(π_{A})_{∗}(O_{A})^{i} = ℱ^{i,0} ⊃ ℱ^{i,1} ⊃ ⋯ ⊃ ℱ^{i,e} = 0, ℱ^{i,j} := ker π^{e−j}.

Let F_{A/S} : A → A^{(p)} be relative Frobenius morphism over S, where A^{(p)} := A ×_{S,F_{abs}} S and F_{abs} is the absolute Frobenius morphism on S. It induces an O_{S} ⊗ O_{F} -linear morphism F_{A/S}^{∗} : R^{1}(π_{A})_{∗}(O_{A})^{(p)} → R^{1}(π_{A})_{∗}(O_{A}).

Let \underline{a} = (a^{i}) ∈ A(e, f) (4.6) be an a-number. Let ℳ_{≥\underline{a}} denote the substack of ℳ whose objects π_{A} : (A, λ, ι) → S satisfying the following condition

F_{A/S}^{∗} (R^{1}(π_{A})_{∗}(O_{A})^{(p),i}) ⊂ ℱ^{i,a^{i}}, ∀ i ∈ ℤ/fℤ.  (10)

It is clear that the condition (10) is closed and locally for Zariski topology ℳ_{≥\underline{a}} is defined by finitely many equations. Therefore ℳ_{≥\underline{a}} is a closed algebraic substack of ℳ.

It is clear that ℳ_{≥\underline{a}} is non-empty as it contains E ⊗ O_{F} with a prime-to-p polarization, where E is a supersingular elliptic curve.

(5.3) Lemma Let (A, λ, ι) ∈ ℳ(k) and a(A) = (a^{i}) be the a-number of A (2.10). Then coker F_{A/k}^{∗} = ⊕_{i∈ℤ/fℤ}k[π]/(π^{a^{i}}).

Proof. Let M^{∗} denote the contravariant Dieudonné module of A. We identify M^{∗}/pM^{∗} = H_{DP}^{1} (A/k) and have H^{1}(A, O_{A}) = M^{∗}/V M^{∗} and coker F_{A/k}^{∗} = M^{∗}/(F, V)M^{∗}. On the other hand, the quasi-polarization gives an isomorphism M ≃ M^{∗} of O-Dieudonné modules, which induces an isomorphism M/(F, V)M ≃ M^{∗}/(F, V)M^{∗} of O_{F} ⊗ k-modules.

(5.4) Theorem The algebraic stack ℳ_{≥\underline{a}} is smooth over Speck(v) of pure dimension g−| \underline{a} |.

Proof. Let x : Speck → ℳ_{≥\underline{a}} be a geometric point. Then there is an affine open neighborhood U of x and a polarized O_{F} -abelian scheme (A, λ, ι) ∈ ℳ(U) such that R^{1}(π_{A})_{∗}(O_{A})^{(p)} and R^{1}(π_{A})_{∗}(O_{A}) are free O_{U}⊗O_{F} -modules. Let x_{i}^{(p)} and x_{i} be O_{U}[π]/(π^{e})-bases of R^{1}(π_{A})_{∗}(O_{A})^{(p),i} and R^{1}(π_{A})_{∗}(O_{A})^{i} respectively for each i ∈ ℤ/fℤ. Let

F_{A/S}^{∗}(x_{i}^{(p)}) = ∑_{0≤j<e} f_{i,j}π^{j}x_{i}, f_{i,j} ∈ O_{U}.

Then locally ℳ_{≥\underline{a}} is defined by the equations f_{i,j}, where (i, j) ∈ I = \{(i, j); i ∈ ℤ/fℤ, 0 ≤ j < a^{i}\}. Therefore, dim_{x} ℳ_{≥\underline{a}} ≥ g − |I| = g − | \underline{a} |.

Let \overline{M} := H_{cris}^{1}(A_{x}/k), \overline{M}^{(p)} := H_{cris}^{1}(A_{x}^{(p)}/k) = \overline{M} ⊗_{k,σ} k. We have the Hodge filtrations:

0 ⟶ F il^{(p)} ⟶ \overline{M}^{(p)} ⟶ Q^{(p)} := H^{1}(A_{x}, O_{A_{x}})^{(p)} ⟶ 0
↓^{F^{∗}}  ↓^{F^{∗}}
0 ⟶ F il ⟶ \overline{M} ⟶ Q := H^{1}(A_{x}, O_{A_{x}}) ⟶ 0

By Proposition 4.3 and Lemma 5.3, we can choose a k[π]/(π^{e})-basis \{x_{i}, y_{i}\} be of (\overline{M})^{i} for each i ∈ ℤ/fℤ such that y_{i} ∈ F il^{i} and F^{∗}(x_{i} ⊗ 1) = −y_{i} + c_{i}x_{i} for some c_{i} ∈ π^{a^{i}}k[π]/(π^{e}). For any PD-extension R of k, the differential forms x_{i}, y_{i} have unique horizontal liftings in M̃ := H_{cris}^{1}(A/R) with respect to the Gauss-Manin connection, which we will denote the liftings by x_{i}, y_{i} again.

Let R := k[[\underline{t}]]/(\underline{t})^{2} be the first order universal deformation ring of (A_{x}, λ_{x}, ι_{x}), where \underline{t} = (t_{i,j}), i ∈ ℤ/fℤ, 0 ≤ j < e. The first order universal deformation (Ã, λ̃, ι̃) gives rise to

0 ⟶ F̃ il^{(p)} ⟶ M̃^{(p)} ⟶ Q̃^{(p)} ⟶ 0
↓^{F^{∗}}  ↓^{F^{∗}}
0 ⟶ F̃ il ⟶ M̃ ⟶ Q̃ ⟶ 0

where F̃ il = span⟨y_{i} +∑_{0≤j<e} t_{i,j}π^{j}x_{i}⟩, F̃ il^{(p)} = span⟨(y_{i} +∑_{0≤j<e} t_{i,j}π^{j}x_{i})⊗1⟩ = span⟨y_{i} ⊗ 1⟩, Q̃ = span⟨[x_{i}]⟩, and Q̃^{(p)} = span⟨[x_{i} ⊗ 1]⟩ for all i ∈ ℤ/fℤ. It follows from

F^{∗}([x_{i} ⊗ 1]) = [−y_{i}] = ∑_{0≤j<e} t_{i,j}[π^{j}x_{i}]

that R/(t_{i,j})_{(i,j)∈I} is the first order deformation ring of ℳ_{≥\underline{a}} at x and the tangent space has dimension g − | \underline{a} | ≤ dim_{x} ℳ_{≥\underline{a}}. The assertion follows.

(5.5) Corollary The ordinary points are dense in ℳ^{DP} ⊗ 𝔽_{p}.

Proof. It follows form Theorem 5.4 and density of the Rapoport locus.

(5.6) Let ℳ\underline{a} denote the subset of ℳ whose points have a-number \underline{a}. It is a locally closed subset of ℳ, hence as a locally closed algebraic substack of ℳ with the reduced induced structure. It follows from Lemma 5.3 that (A, λ, ι) ∈ ℳ_{≥\underline{a}}(k) if and only if a(A) ≥ \underline{a}. It follows from Theorem 5.4 that ℳ\underline{a} is an dense open substack of ℳ_{≥\underline{a}} and ℳ_{≥\underline{a}} is the scheme-theoretic closure of ℳ\underline{a} in ℳ. This justifies our notation.

§6. Deformations of Dieudonné modules

(6.1) We follow the convenient setting of [N, Sec. 0] and [CN, Sec. 2, p. 1011]. As we will only deal with smooth functors, the deformation theory developed by P. Norman [N] and Norman-Oort [NO] is enough for our purpose. We refer the reader to [Z2] for the generalized theory of displays over more general base ring.

Let R be a commutative ring of characteristic p. Let W (R) denote the ring of Witt vectors over R, equipped with the Verschiebung τ and Frobenius σ:

\begin{aligned}(a_{0}, a_{1}, …)^{τ} & = (0, a_{0}, a_{1}, …) \\ (a_{0}, a_{1}, …)^{σ} & = (a_{0}^{p}, a_{1}^{p}, …). \end{aligned}

Let Cart_{p}(R) denote the Cartier ring W (R)[F][[V]] modulo the relations

• F V = p and V aF = a^{τ},

• F a = a^{σ}F and V a^{τ} = aV, ∀ a ∈ W (R).

A left Cart_{p}(R)-module is uniform if it is complete and separated in the V -adic topology. A uniform Cart_{p}(R)-module M is reduced if V is injective on M and M/V M is a free R-module. A Dieudonné module over R is a finitely generated reduced uniform Cart_{p}(R)- module.

There is an equivalence of categories between the category of finite dimensional commutative formal group over R and the category of Dieudonné module over R. We denote this functor by D_{∗}. The tangent space of a formal group G is canonically isomorphic to D_{∗}(G)/V D_{∗}(G).

(6.2) Let \underline{a} = (a^{i}) be an a-number and (A_{0}, λ_{0}, ι_{0}) ∈ ℳ_{≥\underline{a}}(k) be a non-ordinary polarized O_{F} -abelian variety. By Lemma 3.2, the associated p-divisible group G_{0} = A_{0}[p^{∞}] is connected, hence is a smooth formal group. As the forgetful functor Def[A_{0}, λ_{0}, ι_{0}] → Def[A_{0}, ι_{0}] induces an equivalence of deformation functors, we will consider deformations of O_{F} -abelian varieties and their associated formal groups.

Let I be the set {(i, j); i ∈ ℤ/fℤ, a^{i} ≤ j < e − 1} and let R := k[[t_{i,j}]]_{(i,j)∈I}. We have

O ⊗_{ℤ_{p}} W (R) = ⨁_{i∈ℤ/fℤ}W (R)[T]/(σ_{i}(P (T))).

Set W_{R}^{i} := W (R)[T]/(σ_{i}(P (T))) and denote again by π the image of T in W_{R}^{i}.

Let M_{0} be the covariant Dieudonné module of A_{0}. Let \underline{b} = (b^{i}) be the a-number of A_{0} and τ be the a-index of M_{0}. By Proposition 4.4, we can choose a W^{i}-basis of M_{0}^{i} for each i ∈ ℤ/fℤ such that

F X_{i−1} = \begin{cases}X_{i} & if i ∉ τ \\ c_{i}X_{i} + Y_{i} & if i ∈ τ \end{cases} F Y_{i−1} = \begin{cases}π^{e}Y_{i} & if i ∉ τ \\ π^{e}X_{i} & if i ∈ τ, \end{cases}  (11)

for some c_{i} ∈ π^{b^{i}}W^{i}.

By [NO, Lemma 0.2], we construct a Dieudonné module M_{R} over R with a Cart_{p}(R)- linear action by O as follows. It is the Cart_{p}(R)-module generated by {π^{j}X_{i}, π^{j}Y_{i}}_{i∈ℤ/fℤ,0≤j<e} with the relations

F X_{i−1} = \begin{cases}X_{i} & if i ∉ τ \\ c_{i}X_{i} + (Y_{i} + ∑_{a^{i}≤j<e} T_{i,j}π^{j}X_{i}) & if i ∈ τ \end{cases}  (12)

Y_{i−1} = \begin{cases}V u(Y_{i} + ∑_{a^{i}≤j<e} T_{i,j}π^{j}X_{i}) & if i ∉ τ \\ V uX_{i} & if i ∈ τ, \end{cases}  (13)

where T_{i,j} is the Teichmüller lift of t_{i,j}, ∀ (i, j) ∈ I, u = p^{−1}π^{e}. We impose the natural W_{R}^{i}- module structure on the free W (R)-submodule generated by {π^{j}X_{i}, π^{j}Y_{i}}_{0≤j<e}, and impose the trivial W_{R}^{k}-module structure on this submodule for k ≠ i. The action of O on M_{R} comes from the natural embedding O ↪ O ⊗_{ℤ_{p}} W (R).

(6.3) Lemma The O-Dieudonné module M_{R} over R is isomorphic to the universal deformation of M_{0} for ℳ_{≥\underline{a}}.

Proof. It is clear that Cart_{p}(k) ⊗_{Cart_{p}(R)} M_{R} = M_{0}. On M_{R}/V M_{R}, we have

F [X_{i−1}] = c_{i}[X_{i}] + ∑_{a^{i}≤j<e} t_{i,j}π^{j}[X_{i}], ∀ i ∈ τ.

By the Serre-Tate theorem, we obtain a morphism Spf R → ℳ_{≥\underline{a},x}^{∧}. By [N, Thm. 1], this construction induces an injection of tangent spaces. By Theorem 5.4, ℳ_{≥\underline{a}} is smooth and dim ℳ_{≥\underline{a}} = dim R. Hence the morphism is an isomorphism.

(6.4) Remark Another construction of M_{R} is using tensor products modulo relations. Let P_{R} is a free O ⊗_{ℤ_{p}} W (R)-module of rank two, with a W_{R}^{i}-basis {X_{i}, Y_{i}} for each component P_{R}^{i}. Then we construct M_{R} to be the quotient of Cart_{p}(R) ⊗_{W (R)} P_{R} modulo the relations (12) and (13).

(6.5) Let M_{R} be as in (6.2). Let

T_{i} := \begin{cases}∑_{a^{i}≤j<e} T_{i,j}π^{j} & if i ∉ τ \\ c_{i} + ∑_{a^{i}≤j<e} T_{i,j}π^{j} & if i ∈ τ. \end{cases}  (14)

Then we have

(\begin{matrix}F X_{i−1} \\ F Y_{i−1} \end{matrix}) = A_{i} (\begin{matrix}X_{i} \\ Y_{i} \end{matrix}) ,

where

A_{i} = (\begin{matrix}1 & 0 \\ T_{i}π^{e} & π^{e} \end{matrix}) if i ∉ τ; A_{i} = (\begin{matrix}T_{i} & 1 \\ π^{e} & 0 \end{matrix}) if i ∈ τ.

(6.6) Lemma The non-ordinary locus of ℳ_{x}^{∧} is defined by ∏_{i∈τ} t_{i,0} = 0.

Proof. Take \underline{a} = (0, 0, …, 0). On M_{R}/(V M_{R} + πM_{R}), we have

F [X_{i−1}] = [X_{i}] if i ∉ τ; F [X_{i−1}] = t_{i,0}[X_{i}] if i ∈ τ.

As each slope stratum is reduced, it induces a closed reduced subscheme, if not empty, of ℳ_{x}^{∧}. A point 𝔭 ∈ SpecR is in the non-ordinary locus if and only if any of t_{i,0}, i ∈ τ, vanishes on R/𝔭. Therefore the defining equation is ∏_{i∈τ} t_{i,0} = 0.

(6.7) We continue with (6.5). Let t = |a(M_{0}/πM_{0})| and τ = {n_{1}, n_{2}, … n_{t}}. We assume that 0 ≤ n_{1} < n_{2} < ⋯ < n_{t} < f and let n_{0} := n_{t} − f and n_{t+1} = n_{1}. Let ℓ_{i} := n_{i} − n_{i−1} for 1 ≤ i ≤ t. (4.6)

From (6.5), we have

\begin{aligned}(\begin{matrix}F^{ℓ_{i}}X_{n_{i−1}} \\ F^{ℓ_{i}}Y_{n_{i−1}} \end{matrix}) & = (\begin{matrix}1 & 0 \\ T_{n_{i−1}+1}^{(ℓ_{i}−1)}π^{e} & π^{e} \end{matrix}) … (\begin{matrix}1 & 0 \\ T_{n_{i−1}+ℓ_{i}−1}^{(1)}π^{e} & π^{e} \end{matrix}) (\begin{matrix}T_{n_{i}} & 1 \\ π^{e} & 0 \end{matrix}) (\begin{matrix}X_{n_{i}} \\ Y_{n_{i}} \end{matrix}) \\ & = (\begin{matrix}1 & 0 \\ V′_{i} & π^{e(ℓ_{i}−1)} \end{matrix}) (\begin{matrix}U′_{i} & 1 \\ π^{e} & 0 \end{matrix}) (\begin{matrix}X_{n_{i}} \\ Y_{n_{i}} \end{matrix}) \\ & = (\begin{matrix}U′_{i} & 1 \\ U′_{i}V′_{i} + π^{eℓ_{i}} & V′_{i} \end{matrix}) (\begin{matrix}X_{n_{i}} \\ Y_{n_{i}} \end{matrix}) \end{aligned}  (15)

where

U′_{i} := T_{n_{i}}, and V′_{i} := ∑_{j=1}^{ℓ_{i}−1} T_{n_{i−1}+j}^{(ℓ_{i}−j)}π^{ej}.

Recall that we write T^{(n)} for T^{σ^{n}} (3.4) (4.6). For 1 ≤ s ≤ t, suppose that

(\begin{matrix}F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}X_{n_{s−1}} \\ F^{ℓ_{s}+ℓ_{s+1}+···+ℓ_{t}}Y_{n_{s−1}} \end{matrix}) = (\begin{matrix}α_{s} & β_{s} \\ γ_{s} & δ_{s} \end{matrix}) (\begin{matrix}X_{n_{t}} \\ Y_{n_{t}} \end{matrix}) ,

for some coefficients α_{s}, β_{s}, γ_{s}, δ_{s} in W_{R}^{n_{t}}. It follows from the same computation (6) of (4.6) that

(\begin{matrix}α_{s−1} & β_{s−1} \\ γ_{s−1} & δ_{s−1} \end{matrix}) = (\begin{matrix}U_{s−1} & 1 \\ U_{s−1}V_{s−1} + π^{eℓ_{s−1}} & V_{s−1} \end{matrix}) (\begin{matrix}α_{s} & β_{s} \\ γ_{s} & δ_{s} \end{matrix}) ,  (16)
(\begin{matrix}α_{t} & β_{t} \\ γ_{t} & δ_{t} \end{matrix}) = (\begin{matrix}U_{t} & 1 \\ U_{t}V_{t} + π^{eℓ_{t}} & V_{t} \end{matrix}) ,  (17)

where

U_{s} := T_{n_{s}}^{(ℓ_{s+1}+ℓ_{s+2}+···+ℓ_{t})} and V_{s} := ∑_{i=1}^{ℓ_{s}−1} T_{n_{s−1}+i}^{(ℓ_{s}+···+ℓ_{t}−i)}π^{ei}.

Therefore,

(\begin{matrix}α & β \\ γ & δ \end{matrix}) := (\begin{matrix}α_{1} & β_{1} \\ γ_{1} & δ_{1} \end{matrix}) = ∏_{i=1}^{t} (\begin{matrix}U_{i} & 1 \\ U_{i}V_{i} + π^{eℓ_{i}} & V_{i} \end{matrix})  (18)

(6.8) Theorem If \underline{a} = (a^{i}) is spaced, then the points in ℳ_{≥\underline{a}} of slope s(|\underline{a}|) are dense in ℳ_{≥\underline{a}}.

Proof. We will show that for each point x = (A_{0}, λ_{0}, ι_{0}) ∈ ℳ\underline{a}(k), there is a deformation in ℳ_{≥\underline{a}} whose generic point has slope s(|\underline{a}|). Let M_{0} the Dieudonné module of A_{0}. Let R = k[[t_{i,a^{i}}]], i ∈ τ(M_{0}). We construct a deformation M_{R} by (6.2) with T_{i,j} = 0 except for i ∈ τ(M_{0}) and j = a^{i}. Note that V_{i} = 0 in (18). We have (cf. 4.8)

(\begin{matrix}F^{f}π^{e}X_{n_{t}} \\ F^{f}Y_{n_{t}} \end{matrix}) = π^{|\underline{a}|}A (\begin{matrix}π^{e}X_{n_{t}} \\ Y_{n_{t}} \end{matrix}) , A = ∏_{i=1}^{t} (\begin{matrix}U_{i} & π^{e−a^{n_{i}}} \\ π^{e(ℓ_{i}−1)−a^{n_{i}}} & 0 \end{matrix}) ,

where U_{i} = (c_{n_{i}} +T_{n_{i},a^{n_{i}}})^{(ℓ_{i+1}+ℓ_{i+2}+···+ℓ_{t})} for some c_{n_{i}} in W^{n_{i}}. We may assume that e(ℓ_{i} −1) > a^{n_{i}} for some i, otherwise |\underline{a}| = \frac{g}{2} and there is nothing to prove. Assume that e(ℓ_{1} − 1) > a^{n_{1}}, we have

\begin{aligned}A & ≡ (\begin{matrix}U_{1} & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}U_{2} & ∗ \\ ∗ & 0 \end{matrix}) … (\begin{matrix}U_{t} & ∗ \\ ∗ & 0 \end{matrix}) mod π \\ & ≡ (\begin{matrix}P & ∗ \\ 0 & 0 \end{matrix}) mod π \end{aligned}  (19)

where P is a nonzero polynomial in U_{i}’s. Hence P is a unit in W (K)[π], where K is the perfection of Frac(R). Therefore M_{K} = Cart_{p}(K) ⊗_{Cart_{p}(R)} M_{R} has slope s(|\underline{a}|).

(6.9) Corollary The generic point of each irreducible component of ℳ_{≥\underline{a}} has slope ≥ s(λ(\underline{a})), where λ(\underline{a}) := max\{|\underline{b}|; \underline{b} ⊂ \underline{a}, \underline{b} is spaced\} (cf. [GO, p.112]).

Proof. This follows from Proposition 4.8 and Grothendieck’s specialization theorem.

(6.10) Remark Goren and Oort showed [GO, Thm. 5.4.11] that when p is inert in F, the inequality ≥ in Corollary 6.9 can be replaced by equation =. However, the equality does not hold in general. I do not have the sharp formula of the slopes of the generic points of each a-number stratum. When the a-numbers are spaced, the equality can be achieved as it stands in Theorem 6.8. Those are the key cases of [GO, Thm. 5.4.11]. Our proof is different from theirs.

(6.11) Let x = (A_{0}, λ_{0}, ι_{0}) and M_{0} be as in (6.2). Suppose that the reduced (usual) a-number t := |τ| of x is one, where τ is the a-index of M_{0}. We assume that τ = {0} for simplicity. By Proposition 4.4, we choose a basis for M_{0} as in (11) of (6.2).

Let ℓ = ord_{π}(c_{0}). We have a^{0} = min\{e, ℓ\} and slope(M_{0}) = s(min\{\frac{g}{2}, ℓ\}) by (4.9).

Take \underline{a} = (0, 0, … 0) and construct the universal deformation M_{R} of M_{0} for ℳ by (6.2). It is the Dieudonné module of the formal group attached to the universal formal deformation (Ã, λ̃, ι̃) over ℳ_{x}^{∧} = Spf R. For each m ∈ S(g), m ≤ ℓ, let ℳ_{x}^{≥s(m)} denote the reduced closed subscheme (in ℳ_{x}^{∧}) of slope ≥ s(m). We will find the defining equations of the subscheme ℳ_{x}^{≥s(m)} and show that the generic point of ℳ_{x}^{≥s(m)} has slope s(m).

From (18), we have

(\begin{matrix}F^{f}X_{0} \\ F^{f}Y_{0} \end{matrix}) = (\begin{matrix}U_{1} & 1 \\ U_{1}V_{1} + π^{g} & V_{1} \end{matrix}) (\begin{matrix}X_{0} \\ Y_{0} \end{matrix}) ,

where

U_{1} = T_{0} and V_{1} = ∑_{i=1}^{f−1} T_{i}^{(f−i)}π^{ei}.

Therefore we have the Cayley-Hamilton equation F^{2f}X_{0} − (U_{1}^{(f)} + V_{1})F^{f}X_{0} − π^{g} = 0. The subscheme ℳ_{x}^{≥s(m)} is defined by the equations obtained from “ord_{π}(U_{1}^{(f)} + V_{1}) ≥ m”.

From (14) of (6.5), write

\begin{aligned}U_{1}^{(f)} + V_{1} & = c_{0}^{(f)} + ∑_{i=0}^{f−1} ∑_{j=0}^{e−1} T_{i,j}^{(f−i)}π^{ei+j} \\ & = c_{0}^{(f)} + ∑_{i=0}^{g−1} T_{k}π^{k} \end{aligned}  (20)

where T_{k} := T_{i,j}^{(f−i)}, t_{k} = t_{i,j} for k = ei + j, 0 ≤ j ≤ e − 1. We can see that the defining equations for ℳ_{x}^{≥s(m)} are t_{0} = t_{1} = … t_{m−1} = 0. Let K_{m} be the perfection of the generic residue field of ℳ_{x}^{≥s(m)}. The element T_{m} is a unit in W (K_{m}), therefore the generic point of ℳ_{x}^{≥s(m)} has slope s(m).

(6.12) Theorem Let x : Speck → ℳ be a geometric point of reduced a-number one. Then each closed subscheme ℳ_{x}^{≥s(m)}, for m ∈ S(g), s(m) ≤ slope(x) (3.3), is formally smooth of codimension ⌈m⌉ and its generic point has slope s(m). Here ⌈m⌉ denotes the smallest integer not less than m.

(6.13) Corollary The strong Grothendieck conjecture (see [GO. Sec. 5.1]) holds on the open subset of ℳ of reduced a-number ≤ 1.

(6.14) Corollary The strong Grothendieck conjecture for ℳ holds when p is totally ramified in F.

(6.15) Corollary The weak Grothendieck conjecture (see [GO. Sec. 5.1]) for ℳ holds.

Proof. It follows from Lemma 7.3 that there is a supersingular point of reduced a-number one. Then the assertion follows from Theorem 6.12.

(6.16) In the rest of this section we assume that p is totally ramified in F. Denote by ℳ_{v}^{DP} the reduction ℳ^{DP} ⊗_{ℤ} k(v) of ℳ^{DP} modulo v. Let x = (A_{0}, λ_{0}, ι_{0}) ∈ ℳ_{v}^{DP}(k) be a geometric point and let lie(A_{0}) = {e_{1}, e_{2}} and a(A_{0}) = {a_{1}, a_{2}}. We assume that e_{1} ≤ e_{2} and a_{1} ≤ a_{2}. Let M_{0} be the covariant Dieudonné module of A_{0}. We can choose two W [π] = O ⊗ W -bases {X_{1}, X_{2}}, {Y_{1}, Y_{2}} of M_{0} such that

V uY_{1} = π^{e_{2}}X_{1} and V uY_{2} = π^{e_{1}}X_{2},

where pu = π^{e}. We have

F X_{1} = π^{e_{1}}Y_{1} and F X_{2} = π^{e_{2}}Y_{2}.

Write Y_{1} = αX_{1} + βX_{2} and Y_{1} = γX_{1} + δX_{2}. Then we have

(\begin{matrix}F X_{1} \\ F X_{2} \end{matrix}) = (\begin{matrix}π^{e_{1}}α & π^{e_{1}}β \\ π^{e_{2}}γ & π^{e_{2}}δ \end{matrix}) (\begin{matrix}X_{1} \\ X_{2} \end{matrix}) and (F, V)M =< π^{e_{1}}X_{2}, π^{e_{2}}X_{1}, π^{e_{1}}αX_{1} >_{W [π]} .

It follows that a_{1} = e_{1} and a_{2} = min\{e_{2}, e_{1} + ord_{π}(α)\}. As e_{2} ≥ \frac{e}{2}, by the argument in (4.10) we have slope(A_{0}) = s(i), where i = min\{\frac{e}{2}, e_{1} + ord_{π}(α)\}. Note that the a-number a(A_{0}) determines the invariant lie(A_{0}) and slope(A_{0}): e_{1} = a_{1} and slope(A_{0}) = s(min\{\frac{e}{2}, a_{2}\}).

(6.17) In [DP], Deligne and Pappas have defined a closed algebraic substack 𝒩_{i} of ℳ_{v}^{DP} which classifies the objects of Lie type {e_{1}, e_{2}} with e_{1} ≥ i. They have shown that the complement 𝒩_{i}^{o} of 𝒩_{i+1} in 𝒩_{i} is a smooth algebraic stack of dimension e − 2i if it is nonempty. It follows from their results and Theorem 7.5 that points of Lie type {e_{1}, e_{2}} are dense in 𝒩_{e_{1}}. It follows from (6.16) that any point in 𝒩_{e_{1}} has slope ≥ s(e_{1}). The following lemma confirms the density of points of slope s(e_{1}) in 𝒩_{e_{1}}.

(6.18) Lemma If a_{2} > a_{1}, then there is a deformation (Ã, λ̃, ι̃) over k[[t]] of (A_{0}, λ_{0}, ι_{0}) whose generic point has a-number (a_{1}, a_{2} − 1).

Proof. As the forgetful map Def[A_{0}, λ_{0}, ι_{0}] → Def[A_{0}, ι_{0}] induces an equivalence of deformation functors in 𝒩_{e_{1}}, we will construct a deformation of O_{F} -abelian varieties in 𝒩_{e_{1}}. By the reduction step in [DP, 4.3] and the construction of (6.2), we can construct an O-Dieudonné module M_{R} over R := k[[t]] of M_{0} such that

\begin{aligned}F X_{1} & = π^{e_{1}}αX_{1} + π^{e_{1}}β(X_{2} + T π^{a_{2}−e_{1}−1}X_{1}) \\ F X_{2} & = π^{e_{2}}γX_{1} + π^{e_{2}}δ(X_{2} + T π^{a_{2}−e_{1}−1}X_{1}), \end{aligned}  (21)

where T is the Teichmülller lift of t. Note that β is a unit in W [π]: if β ≡ 0 mod π, then α is a unit and a_{2} = a_{1}. By base change to K := k((t))^{perf}, we have

(F, V)M_{K} =< π^{e_{1}}X_{2}, π^{e_{2}}X_{1}, (π^{e_{1}}α + π^{a_{2}−1}βT)X_{1} >_{W (K)[π]},

thus a(M_{K}) = (a_{1}, a_{2} − 1). This completes the proof.

(6.19) Theorem The strong Grothendieck conjecture for ℳ_{v}^{DP} holds when p is totally ramified in F.

Proof. Let x ∈ ℳ_{v}^{DP}(k) be a geometric point with a-number (a_{1}, a_{2}) and slope s(i), where i = min\{\frac{e}{2}, a_{2}\}. It suffices to deform the point x to a point of slope s(i − 1). If a_{2} > a_{1}, then by Lemma 6.18 we can deform to a point of slope s(i − 1). Suppose that a_{2} = a_{1} = e_{1}, we have slope(x) = s(e_{1}). By (6.17), we can deform to a point of Lie type {e_{1} − 1, e_{2} + 1}. As points of slope s(e_{1} − 1) are dense in 𝒩_{e_{1}−1}, we can further deform to a point of slope s(e_{1} − 1).

(6.20) Corollary Each Newton stratum of ℳ_{v}^{DP} of slope s(m), m ∈ S(g) (3.3), has pure dimension g − ⌈m⌉ when p is totally ramified in F.

Proof. This follows from the purity of Newton strata [dJO] and Theorem 6.19.

(6.21) Remark Corollary 6.20 confirms a question of Chai [C, Question 7.6, p.984] in the Hilbert-Blumenthal cases when p is totally ramified.

§7. An algebraization theorem

(7.1) In this section we do not assume that there is only one prime of the totally real field F lying over p. Let v_{1}, …, v_{s} be primes of F over p, F_{p} := F ⊗ ℚ_{p} = F_{v_{1}} ⊕ ⋯ ⊕ F_{v_{s}}, and O := O_{F} ⊗ℤ_{p} = O_{v_{1}} ⊕⋯⊕O_{v_{s}}. The possible slopes are parameterized by S(g_{1})×⋯×S(g_{s}) (3.3), where g_{i} = [F_{v_{i}} : ℚ_{p}]. For each O_{F} -abelian variety A, the associated p-divisible group A(p) := A[p^{∞}] has a decomposition A(p) = A(p)_{v_{1}} ⊕ ⋯ ⊕ A(p)_{v_{s}}.

Let k(p) denote the smallest finite field containing all residue fields k(v_{i}). Denote by ℳ_{p}^{DP} the reduction ℳ^{DP} ⊗_{Z} k(p) of ℳ^{DP} and ℳ the reduction ℳ^{R} ⊗_{Z} k(p) of the Rapoport locus ℳ^{R}.

(7.2) A quasi-polarized O-p-divisible group over k is a tuple (H, λ, ι), where H is a pdivisible group, λ : H → H^{t} an isogeny such that λ^{t} = −λ, and an algebra homomorphism ι : O ↪ End_{k}(H) such that λι(a) = ι(a)^{t}λ for all a ∈ O. Here H^{t} stands for the Serre dual of H, equipped with a natural action by O. We identify (H^{t})^{t} with H by the natural isomorphism H ≃ (H^{t})^{t}. We call such a tuple (H, λ, ι) over k algebrisable if it is attached to a polarized O_{F} -abelian variety (A, λ_{A}, ι_{A}) over k.

(7.3) Lemma Any supersingular quasi-polarized O-p-divisible group (H, λ, ι) over k is algebrisable.

Proof. Let E be a supersingular elliptic curve over k, and A′ := E ⊗ O_{F}. It is clear that A′ satisfies the Rapoport condition. By [R, Prop. 1.10], there exists a separable O_{F} -linear polarization λ′ on A′. Let (H_{1}, λ_{1}, ι_{1}) be the p-divisible group attached to (A′, λ′, ι′). It follows from Corollary 3.8 that (H_{1}, λ_{1}, ι_{1}) is isogenous to (H, λ, ι). By a theorem of Tate, there is a polarized O_{F} -abelian variety (A, λ_{A}, ι_{A}) whose p-divisible group is isomorphic to (H, λ, ι).

(7.4) Theorem (1) The weak Grothendieck conjecture for ℳ holds.

(2) The strong Grothendieck conjecture for ℳ_{p}^{DP} holds when all residue degrees f_{i} are one.

Proof. (1) It follows from Lemma 7.3 that there is a supersingular polarized O_{F} -abelian variety A such that each component A(p)_{v_{i}} of the associated p-divisible group A(p) has reduced a-number one. Then the assertion follows from the theorem of Serre-Tate and Theorem 6.12.

(2) This follows from the theorem of Serre-Tate and Theorem 6.19.

(7.5) Theorem Any quasi-polarized O-p-divisible group (H, λ, ι) over k is algebrisable.

Proof. It follows from Theorem 7.4 (1) that any slope in S(g_{1})×⋯×S(g_{s}) can be realized by a point in ℳ. Then the theorem follows from Corollary 3.8.

§8. An example

In this section we construct a separably polarized O_{F} -abelian scheme A over a complete DVR R whose close fibre A_{k} does not satisfy the Rapoport condition.

Let g = 2 and p > 3 be a ramified prime in the totally quadratic real field F. We have O_{F} ⊗ W (k) = W (k)[π], π^{2} = p. Let M be a free W (k)[π]-module generated by e_{1}, e_{2} with the Verschiebung action

V e_{1} = πe_{2}, V e_{2} = πe_{1}.

and with the alternating form determined by

⟨e_{1}, e′_{2}⟩ = ⟨e′_{1}, e_{2}⟩ = 1

and other pairing are 0 for the W -basis e_{1}, e′_{1}, e_{2}, e′_{2}, where e′_{1} := πe_{1}, e′_{2} := πe_{2}. By the algebraization theorem, there is a polarized O_{F} -abelian variety A_{0} over k with the prescribed Dieudonné module M.

We have the Hodge filtration of M/pM = H_{DP}^{1} (A_{0}/k)^{∗}

0 → V M/pM = kē′_{1} ⊕ kē′_{2} → M/pM → M/V M = kē_{1} ⊕ ē_{2} → 0.

Clearly, A_{0} does not satisfy the Rapoport condition, as π = 0 on M/V M.

Let R := W (k)[\sqrt{p}]. By Grothendieck-Messing’s theory, Serre-Tate’s Theorem and Grothendieck’s Existence Theorem, we can lift the abelian variety A_{0} with the additional structure over R by lifting the Hodge filtration with respect to the addition structure. Let N be the R-submodule of M ⊗_{W} R generated by e′_{1} + \sqrt{p}e_{1} and e′_{2} − \sqrt{p}e_{2}. It is easy to check that N is stable by O_{F} -action, N ⊗_{R} k = V M/pM and ⟨N, N⟩ = 0. Thus, we get a desired polarized O_{F} -abelian scheme over R.

§9. A computation of the Hecke correspondence

(9.1) Let M_{0} be the quasi-polarized O-Dieudonné module of the polarized abelian variety A_{0} in the previous section. Let N be the quasi-polarized Dieudonné module containing M_{0} with V N = M and ⟨ , ⟩_{N} = \frac{1}{p}⟨ , ⟩. We have

\begin{matrix}V e_{1} = πe_{2} = e′_{2} & F e′_{2} = pe_{1} \\ V e_{2} = πe_{1} = e′_{1} & F e′_{1} = pe_{2} \\ V e′_{1} = πV e_{1} = pe_{2} & F e_{2} = e′_{1} \\ V e′_{2} = πV e_{2} = pe_{1} & F e_{1} = e′_{2} \end{matrix}

and ⟨e_{1}, e′_{2}⟩ = 1, ⟨e′_{1}, e_{2}⟩ = 1. From N = \frac{1}{p}F M, we have

N =< e_{1}, e_{2}, \frac{1}{p}e′_{1}, \frac{1}{p}e′_{2} >_{W} .

Write X_{1} = e_{1}, X_{2} = e_{2}, X′_{1} = \frac{1}{p}e′_{1}, X′_{2} = \frac{1}{p}e′_{2}, we have

M_{0} =< X_{1}, X_{2}, pX′_{1}, pX′_{2} >_{W} .

and ⟨X_{1}, X′_{2}⟩_{N} = ⟨X′_{1}, X′_{2}⟩_{N} = 1. Denote by N̄ the quotient of N modulo pN and M̄_{0} the image of M_{0} in N̄. Write x_{i}, x′_{i} the image of X_{i}, X′_{i} in N̄.

(9.2) Let X be space of π-invariant maximal isotropic “Dieudonné ” subspaces of N̄ over k

X (k) = {0 ⊂ M̄ ⊂ N̄; dim M̄ = 2, πM̄ ⊂ M̄, F M̄ ⊂ M̄, V M̄ ⊂ M̄, ⟨M̄, M̄⟩_{N} = 0}

We regard X as a reduced subscheme over k. It is a closed subscheme of the Lagrangian Grassmanian LG(2, 4), hence a projective variety.

There is a finite morphism pr : X → ℳ_{p}^{DP} (cf. [Y. Sec. 14]) sending M̄_{0} ↦ (A_{0}, λ_{0}, ι_{o}) and the morphism pr factors through the supersingular locus 𝒮^{DP} of ℳ_{p}^{DP}.

We have M̄_{0} =< x_{1}, x_{2} >_{k}∈ X (k). Let M̄_{t} ∈ X be the k-subspace of N̄ generated by

x̃_{1} = x_{1} + t_{11}x′_{1} + t_{12}x′_{2}, x̃_{2} = x_{2} + t_{21}x′_{1} + t_{22}x′_{2}.

The points M̄_{t} form a Zariski open neighborhood of M̄_{0} in X, which we denote by U. We will show that U ≃ Speck[t_{1}, t_{2}, t_{3}]/(t_{1}^{p+1} − t_{2}^{p+1}, t_{1}^{2} + t_{2}t_{3}).

(9.3) From ⟨M̄_{t}, M̄_{t}⟩_{N} = 0, we get t_{11} + t_{22} = 0. From X′_{i} = \frac{1}{p}e′_{i} = \frac{1}{π}e_{i} = \frac{1}{π}X_{i}, we have

πx′_{i} = x_{i}, πx_{i} = 0, i = 1, 2.

One computes

πx̃_{1} = t_{11}x_{1} + t_{12}x_{2} = t_{11}x̃_{1} + t_{12}x̃_{2} − (t_{11}^{2} + t_{12}t_{21})x′_{1} − (t_{11}t_{12} + t_{12}t_{22})x′_{2}

and concludes from πx̃_{1} ∈ M̄_{t} that

t_{11}^{2} + t_{12}t_{21} = 0, and t_{12}(t_{11} + t_{22}) = 0.  (22)

Similarly from πx̃_{2} ∈ M̄_{t}, one gets

t_{22}^{2} + t_{12}t_{21} = 0, and t_{21}(t_{11} + t_{22}) = 0.  (23)

For the stability of M̄_{t} by F and V, one computes

F x̃_{1} = t_{11}^{p}x_{2} + t_{12}^{p}x_{1} = t_{11}^{p}x̃_{2} + t_{12}^{p}x̃_{1} − (t_{11}^{p}t_{21} + t_{12}^{p}t_{11})x′_{1} − (t_{11}^{p}t_{22} + t_{12}^{p+1})x′_{2}.

and obtains

t_{11}^{p}t_{21} + t_{12}^{p}t_{11} = 0  (24)
t_{11}^{p}t_{22} + t_{12}^{p+1} = 0  (25)

Similarly from F x̃_{2} ∈ M̄_{t}, one obtains

t_{21}^{p+1} + t_{22}^{p}t_{11} = 0  (26)
t_{21}^{p}t_{22} + t_{22}^{p}t_{12} = 0  (27)

Applying V M̄_{t} ⊂ M̄_{t}, one does not get new equations but (24)–(27). From (25), one has t_{12} = αt_{11}, α^{p+1} = 1. From (26), one has t_{21} = βt_{22} = −βt_{11}, β^{p+1} = 1. From (22), one has 1 = αβ. These parameters satisfy the the equations (24) and (27).

We computed that U = \{(t_{11}, t_{12}, t_{21}, t_{22}) = (t, αt, −\frac{1}{α}t, −t); α^{p+1} = 1\}, hence U ≃ Speck[t_{1}, t_{2}, t_{3}]/(t_{1}^{p+1} − t_{2}^{p+1}, t_{1}^{2} + t_{2}t_{3}). Compared with a result of [BG, p. 476, 3.], the morphism pr maps X onto the (p + 1) irreducible components of 𝒮^{DP} containing (A_{0}, λ_{0}, ι_{0}).

(9.4) Proposition There is a Zariski open neighborhood 𝒱 of 0 in U such that pr : 𝒱 → 𝒮^{DP} is an étale neighborhood of 𝒮^{DP} at (A_{0}, λ_{0}, ι_{0}).

Proof. Choose a finite étale cover ℳ_{p,n}^{DP} → ℳ_{p}^{DP} by adding a prime-to-p level structure with n ≥ 3. Choose a lift (A_{0}, λ_{0}, ι_{0}, η_{0}) of (A_{0}, λ_{0}, ι_{0}). Then there is a lift p̃r : X → ℳ_{p,n}^{DP} which sends M̄_{0} to (A_{0}, λ_{0}, ι_{0}, η_{0}). The morphism p̃r becomes a closed immersion as the automorphisms invloved are trivial. It again factors through the supersingular locus 𝒮_{n}^{DP}. As X and 𝒮_{n}^{DP} are reduced schemes, X is ismorphic to its imagae. The imgage is the union of the p + 1 irreducible components of 𝒮^{DP} containing (A_{0}, λ_{0}, ι_{0}, η_{0}). Then there is a Zariski open 𝒱 of U such that the 𝒱 is an étale neighborhood of 𝒮^{DP} at (A_{0}, λ_{0}, ι_{0}).

§10. Superspecial points

(10.1) Shimura’s exact mass formula. Let B be a quaternion division algebra over a totally real number field F, and let σ denothe standard involution of B. Let V be a left B-module of rank m, and let φ be a quaternion Hermitian form on V. That is,

φ : V × V → B

such that φ(x, y) = φ(y, x)^{σ} and φ(ax, by) = aφ(x, y)b^{σ} for all x, y ∈ V and a, b ∈ B. Let G^{φ} denote the unitary group of φ. It is a reductive group over F and we will regard it as a reductive algebraic group over ℚ via the Weil restriction of scalars from F to ℚ. Assume that G^{φ}(ℝ) is compact. For any open compact subgroup K of G^{φ}(𝔸_{f}), the mass of K is defined as follows. Let {c_{1}, c_{2}, …, c_{h}} be a set of (complete) representatives of the double coset space G^{φ}(ℚ)\G^{φ}(𝔸_{f})/K, and let Γ_{i} := G^{φ}(ℚ) ∩ c_{i}Kc_{i}^{−1}. Note that each Γ_{i} is finite by the assumption of G^{φ}(ℝ). The mass of K is defined to be

mass(K) := ∑_{i=1}^{h} \frac{1}{#Γ_{i}} .

For the general defintion of the mass of K (without the compactness assumption), we refer to [S, Introduction, p. 67]. It follows easily from the definition that for two open compact subgroups K_{1} ⊂ K_{2}, one has mass(K_{1}) = [K_{2} : K_{1}]mass(K_{2}).

Choose a maximal order O_{B} of B. Let L be an O_{B}-lattice in V which is maximal along the lattices on which φ takes its value in O_{B}. Let K_{0} be the maximal compact subgroup of G^{φ}(𝔸_{f}) which stabilizes the adele lattice L ⊗_{ℤ} ℤ̂. In [S, Introduction, p. 68] Shimura gives an explicit formula

mass(K_{0}) = |D_{F} |^{m^{2}} · ∏_{k=1}^{m} {D_{F}^{1/2} [(2k − 1)!(2π)^{−2k}]^{d} ζ_{F} (2k)} ∏_{i=1}^{s} ∏_{k=1}^{m} {(N(𝔭_{i})^{k} + (−1)^{k}} ,(28)

where D_{F} is the discriminant of F over ℚ, d is the degree of F, and 𝔭_{i} are the primes of O_{F} at which B is ramified.

Note that the maximality of L is a local property. That is, L is maximal if and only if each L_{𝔭} is maximal for all primes 𝔭 of F.

(10.2) One can express the exact formula in terms of special values of zeta functions at negative integers. Let F be a totally real field of degree d. Set

Λ_{F} (s) := A^{s}Γ(s/2)^{d}ζ_{F} (s), A := D_{F}^{1/2}π^{−d/2}.

Then one has the functional equation for ζ_{F} (s):

Λ_{F} (1 − s) = Λ_{F} (s).

It gives

ζ_{F} (s) = D_{F}^{1/2} · D_{F}^{−s}π^{− \frac{d}{2} +ds} \frac{Γ(\frac{1−s}{2})^{d}}{Γ(\frac{s}{2})^{d}} · ζ_{F} (1 − s)

Let s = 2k, one gets

ζ_{F} (2k) = D_{F}^{1/2} · D_{F}^{−2k}π^{− \frac{d}{2} +2dk} \frac{Γ(\frac{1−2k}{2})^{d}}{Γ(k)^{d}} · ζ_{F} (1 − 2k).

On the other hand, one has

\frac{Γ(\frac{1−2k}{2})}{Γ(k)} = \frac{(−1)^{k} · \frac{2^{2k−1}(k−1)!}{(2k−1)!} · Γ(1/2)}{(k − 1)!} = \frac{(−1)^{k} · 2^{2k−1} · \sqrt{π}}{(2k − 1)!} .

The term in (28) can be expressed

\begin{matrix}D_{F}^{1/2} [(2k − 1)!(2π)^{−2k}]^{d} ζ_{F} (2k) \\ = & D_{F}^{1−2k} [(2k − 1)!(2π)^{−2k}]^{d} · [\frac{(−1)^{k} · 2^{2k−1} · \sqrt{π}}{(2k − 1)!} ]^{d} · π^{− \frac{d}{2} +2dk} · ζ_{F} (1 − 2k) \\ = & D_{F}^{1−2k} [\frac{(−1)^{k}}{2} ]^{d} · ζ_{F} (1 − 2k). \end{matrix}  (29)

Hence we get

mass(K_{0}) = ∏_{k=1}^{m} \{[\frac{(−1)^{k}}{2} ]^{d} · ζ_{F} (1 − 2k)\} · ∏_{i=1}^{s} ∏_{k=1}^{m} \{(N(𝔭_{i})^{k} + (−1)^{k}\}  (30)

(10.3) If F = ℚ and B is the quaternion algbera which is ramified at {∞, p}, then we have

mass(K_{0}) = \frac{(−1)^{m(m+1)/2}}{2^{m}} \{∏_{k=1}^{m} ζ(1 − 2k)\} · ∏_{k=1}^{m} \{(p^{k} + (−1)^{k}\}  (31)

If m = 1 and B is ramified only at the primes over p and infinite places, then we have

mass(K_{0}) = [\frac{−1}{2}]^{d} · ζ_{F} (−1) · ∏_{i=1}^{s} \{(N(𝔭_{i}) + (−1)\}  (32)

(10.4) Let x = (A_{0}, λ_{0}, ι_{0}) be a supersingular polarized O_{F} -abelian variety over k of dimension g = md. Let G_{x} denote the group scheme over Specℤ whose group of R-points, for each commutative ring R, is

G_{x}(R) = {ϕ ∈ (End_{O_{F}} (A_{0}) ⊗ R)^{×}; ϕ′ϕ = 1},

where the map ϕ ↦ ϕ′ is the Rasoti involution induced by λ_{0}.

Let Λ_{x} denote the set of the isomorphism classes of supersingular polarized O_{F} -abelian varieties (A, λ, ι) of dimension g over k such that (i) the Dieudonné module M(A) is isomorphic to M(A_{0}) as quasi-polarized O-Dieudonné

modules, and

(ii) the Tate module T_{ℓ}(A) is isomorphic to T_{ℓ}(A_{0}) as non-degenerate alternating O_{F} ⊗ ℤ_{ℓ}- modules for all ℓ ≠ p.

Let Λ′_{x} be the subset of Λ_{x} consists of objects (A, λ, ι) such that

(iii) there exists an element ϕ ∈ Hom_{O_{F}} (A_{0}, A) ⊗ ℚ such that ϕ^{∗}λ = λ_{0}.

(10.5) Theorem There is a natural bijection of pointed sets between Λ′_{x} and G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(ℤ̂).

Proof. For any (A, λ, ι) ∈ Λ′_{x}, there is an ℚ-polarized O_{F} -linear isogeny ϕ : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι), by definition. Then there is an ϕ_{ℓ} ∈ G_{x}(ℚ_{ℓ}) for all primes ℓ such that ϕϕ_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)). The element ϕ_{ℓ} lies in G_{x}(ℤ_{ℓ}) for almost all ℓ and it is determined by ϕ up to a right translation of G_{x}(ℤ_{ℓ}). Therefore, ϕ gives an element (ϕ_{ℓ}) ∈ G_{x}(𝔸_{f})/G_{x}(ℤ̂). Let ϕ′ be another ℚ-polarized O_{F} -linear isogeny from A_{0} to A. Then we have ϕ′g = ϕ for some g ∈ G_{x}(ℚ) and (ϕ′_{ℓ})G_{x}(ℤ̂) = (gϕ_{ℓ})G_{x}(ℤ̂). Therefore, we get a well-defined map σ from Λ′_{x} to G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(ℤ̂).

Proof of injectivity. Let α : (A_{0}, λ_{0}, ι_{0}) → (A, λ_{A}, ι_{A}) and β : (A_{0}, λ_{0}, ι_{0}) → (B, λ_{B}, ι_{B}) be two ℚ-polarized O_{F} -linear isogenies such that σ(α) = σ(β). Then there exist ϕ ∈ G_{x}(ℚ) and (ϕ_{ℓ}) ∈ G_{x}(𝔸_{f}) such that αϕ_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)) and βϕϕ_{ℓ} ∈ Isom(A_{0}(ℓ), B(ℓ)) for all ℓ. The polarized O_{F} -quasi-isogeny βϕα^{−1} = βϕϕ_{ℓ}(αϕ_{ℓ})^{−1} : A → B is an isomorphism at every place, hence it is an isomorphism.

Proof of surjectivity. Given an element [(ϕ_{ℓ})] ∈ G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(ℤ̂), then there is a positive integer N such that Nϕ_{ℓ} : A_{0}(ℓ) → A_{0}(ℓ) is an isogeny for all ℓ. Let H be the product of all finite group ker(Nϕ_{ℓ}). Since each Nϕ_{ℓ} is O_{F} -linear, H is stable under the O_{F} - action. Take A := A_{0}/H, then there is a natural O_{F} -action, ι, on A such that the projection π : A_{0} → A is an O_{F} -linear morphism. The degree of π is N^{2g}. The polarization Nλ_{0} descends to a fractional polarization λ on A. Note that we have the induced isomorphism A(ℓ) → A_{0}(ℓ) of quasi-polarized ℓ-divisible groups. Therefore the polarization λ is ℓ-adic integral for all ℓ, hence it is a polarization on A. Therefore, N^{−1}π : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι) gives an element in Λ′_{x} and it induces the element [(ϕ_{ℓ})].

(10.6) Proposition (J.-K. Yu) Let (A_{i}, λ_{i}, ι_{i}), i = 1, 2 be two supersingular polarized O_{F} - abelian varieties over k. Then there exists φ ∈ Hom_{O_{F}} (A_{1}, A_{2}) ⊗ ℚ such that φ^{∗}λ_{2} = λ_{1}. That is, the condition (iii) of (10.4) is satisfied automatically.

Proof. Choose an isogeny φ : A_{1} → A_{2}. By Noether-Skolem’s thoerem, φ can be chosen to be O_{F} -linear. Consider the O_{F} -linear polarizations λ_{1} and λ′_{1} := φ^{∗}λ_{2} on A_{1}. Those induce two positive involutions ∗_{1} and ∗′_{1} on the semi-simple algebra B := End(A_{1}) ⊗ ℚ. By [Z3, Satz 1.1], there exists a positive element c ∈ B^{×}, c = c^{∗_{1}}, such x^{∗′_{1}} = c^{−1}x^{∗_{1}}c, ∀ x ∈ B. Let x ∈ F then we get c ∈ C := End_{O_{F}} (A_{1})⊗ℚ. Consider the algebraic variety P over ℚ defined by {X ∈ C; X^{∗_{1}}X = c}. The algebraic group G_{1} := {g ∈ C^{×}; g^{∗_{1}}g = 1} over ℚ acts on P on the left. The variety P is a G_{1}-torsor, hence associates an element ξ in H^{1}(ℚ, G_{1}). By [K2, Lemma 2.11], P (ℝ) ≠ ∅. Since G_{1} is semi-simple and simply-connected, H^{1}(ℚ_{p}, G_{1}) = 0. Therefore ξ is locally trivial everywhere. By Hasse principle for G_{1}, ξ is trivial. Then there exists g ∈ G_{1}(ℚ) such that g^{∗_{1}}g = c. Replacing φ by φg, we have λ′_{1} = qλ_{1} for some q ∈ ℚ^{×}. Note that q is positive. It follows from the fact that the polarizations lie in the positive cone of the Néron-Severi group NS(A_{1}) ⊗ ℝ. Therefore q is a norm and we we can find a φ ∈ Hom_{O_{F}} (A_{1}, A_{2}) ⊗ ℚ such that φ^{∗}λ_{2} = λ_{1}. This completes the proof.

(10.7) Corollary Λ′_{x} = Λ_{x}.

(10.8) Lemma Let (A, λ, ι) ∈ Λ_{x} and [c] be the correspoding double coset. Then Aut(A, λ, ι) ≃ Γ_{c}, where Γ_{c} := G_{x}(ℚ) ∩ cG_{x}(ℤ̂)c^{−1}.

Proof. Let G′ be the group scheme over Specℤ that represents the group functor

R ↦ {α ∈ (End_{O_{F}} (A) ⊗ R); α′α = 1},

where R is a commutative ring and the map α ↦ α′ is the Rosati involution induced by λ.

Choose a map ϕ ∈ Hom_{O_{F}} (A_{0}, A) ⊗ ℚ such that ϕ^{∗}λ = λ_{0}. Then the element c ∈ G_{x}(𝔸_{f}) has the property (∗): ϕc_{ℓ} ∈ Isom((A_{0}(ℓ), λ_{0}, ι_{0}), (A(ℓ), λ, ι)), ∀ ℓ. Note that α ∈ Aut(A, λ, ι) if and only if α ∈ G′(ℚ) and α_{ℓ} ∈ Aut(A(ℓ), λ, ι), ∀ ℓ.

The map ϕ gives an isomorphism G_{x}(ℚ) → G′(ℚ) which sends β to ϕβϕ^{−1} =: α. From (∗), we have α ∈ G′(ℤ̂) if and only if (ϕc)^{−1}α(ϕc) ∈ G_{x}(ℤ̂). Hence, α ∈ G′(ℤ̂) if and only if c^{−1}βc ∈ G_{x}(ℤ̂). This completes the proof.

(10.9) Corollary

mass(Λ_{x}) := ∑_{(A,λ,ι)∈Λ_{x}} \frac{1}{Aut(A, λ, ι)} = mass(G_{x}(ℤ̂)).

(10.10) The semi-simple algebra C_{x} := End_{O_{F}} (A_{0})⊗ℚ is the centralizer of F into the simple algebra M_{g}(End^{(} E)), where E is a supersingular elliptic curve. We have C_{x} ≃ M_{m}(B), where B is a quaternion algebra over the totally real field F ramified at all real places and unramified at all primes not dividing p. From local invariants, one has B ≃ F ⊗ End^{0}(E). Therefore,

\begin{matrix}C_{x} ⊗ ℝ ≃ M_{m}(ℍ) \\ C_{x} ⊗ ℚ_{ℓ} ≃ M_{2m}(F ⊗ ℚ_{ℓ}), ℓ ≠ p \\ C_{x} ⊗ ℚ_{p} ≃ ∏_{𝔭|p, f(𝔭):odd}M_{m}(B_{𝔭}) × ∏_{𝔭|p, f(𝔭):even}M_{2m}(F_{𝔭}). \end{matrix}  (33)

The Rosati involution induces the standard involution on M_{m}(ℍ) and M_{m}(B_{𝔭}), which we will denote by ∗. It also induces the symplectic involution on M_{2m}(F ⊗ ℚ_{ℓ}) and M_{2m}(F_{𝔭}).

Therefore, we have

G_{x}(ℝ) ≃ {g ∈ M_{m}(ℍ); g^{∗}g = 1}, G_{x}(ℚ_{ℓ}) ≃ Sp_{2m}(F ⊗ ℚ_{ℓ}), ℓ ≠ p

G_{x}(ℚ_{p}) ≃ ∏_{𝔭|p, f(𝔭):odd}G_{𝔭} × ∏_{𝔭|p, f(𝔭):even}Sp_{2m}(F_{𝔭}),

(34)

where G_{𝔭} = {g ∈ M_{m}(B_{𝔭}); g^{∗}g = 1}. We will choose such isomorphisms and replace ≃ by =.

(10.11) Lemma There exists a left B-module V, a quaternion Hermittian from φ on V, and a maximal lattice L in the sense of Shimura (10.1) such that G^{φ} ≃ G_{x} over ℚ and K_{0} defined in (10.1) is

K_{0} = ∏_{ℓ≠p} Sp_{2m}(O_{F} ⊗ ℚ_{ℓ}) × ∏_{𝔭|p, f(𝔭):odd}K_{𝔭} × ∏_{𝔭|p, f(𝔭):even}Sp_{2m}(O_{𝔭}),  (35)

where O_{𝔭} is the ring of integers in F_{𝔭} and K_{0,𝔭} = {g ∈ M_{m}(O_{B_{𝔭}}); g^{∗}g = 1}.

Proof. Let V = B^{⊕g} with φ(\underline{x}, \underline{y}) = ∑ x_{i}ȳ_{i}^{σ} and let L = O_{B}^{⊕g}.

(10.12) The method to compute mass(G_{x}(ℤ̂)) is first to relate with the mass of the standard one K_{0} and then to apply Shimura’s mass formula. For any two open compact subgroups K_{1}, K_{2} of G_{x}(𝔸_{f}), of G_{x}(ℚ_{p}), or of G_{x}(ℚ_{ℓ}), we denote μ_{•}(K_{2}/K_{1}) := [K_{1} : K_{1} ∩ K_{2}]^{−1}[K_{2} : K_{a} ∩ K_{2}], for • = f, p or ℓ.

We have the following properties

• μ_{ℓ}(K_{2}/K_{1}) = 1 for almost all ℓ.

• μ_{•}(K_{2}/K_{1}) = [K_{2} : K_{1}] if K_{1} ⊂ K_{2}.

• If K_{1} = ∏_{ℓ} K_{1,ℓ} and K_{2} = ∏_{ℓ} K_{2,ℓ}, where ℓ runs over all primes of ℚ, then μ_{f}(K_{2}/K_{1}) = ∏_{ℓ} μ_{ℓ}(K_{2,ℓ}/K_{1,ℓ}).

• mass(K_{1}) = μ_{f}(K_{2}/K_{1}) · mass(K_{2}).

(10.13) Lemma Let K_{1}, K_{2} be two maximal open compact subgroups of Sp_{2m}(F_{v}), where F_{v} is a non-archimedean local field. Then μ_{v}(K_{2}/K_{1}) = 1.

Proof. This is obvious.

(10.14) Theorem The mass mass(Λ_{x}) denpends only on the Dieudonné module M_{0} of x. Let K_{M_{0}} denote the group of automorphisms of the quasi-polarized O-Dieudonné module M_{0}. Then

mass(Λ_{x}) = μ_{p}(K_{0,p}/K_{M_{0}}) · mass(K_{0}).

(10.15) In the rest of this section we will concentrate the most interesting cases when M_{0} is superspecial. There are two main reasons to do this. One is that there are only finitely many isomorphism classes of quasi-polarized superspecial O-Dieudonné modules. It is possible to list all of them and calculate the groups of automorphisms. The other one, which is more important, is that the irreducible components of the supersingular locus is ralated to certain superspecial points. This connection has been achieved by canonical isogenies. Therefore, the computation of certain superspecial points will give the number of the irreducible components.

What we will do is to classify the Dieudonné modules in question and compute the groups of automorphisms. Some special cases of them has been done by Li and Oort, and the author. We start with these resuts.

In (10.16)-(10.21), we consider the case m = g and d = 1, hence B = End^{0}(E).

(10.16) Proposition (LO, Proposition 6.1) Let M be a W -rank 2g quasi-polarized superspecial Dieudonné module over k. Then one can decompose

M ≃ M_{1} ⊕ M_{2} ⊕ ⋯ ⊕ M_{d}, (⟨M_{i}, M_{j}⟩ = 0 if i ≠ j)

where each M_{i} is either of the following types:

(i) M_{i} is a W -rank two superspecial Dieudonné submodule generated by some x such that ⟨x, F x⟩ = p^{r}ϵ for some r ∈ ℤ and ϵ ∈ W^{×} with ϵ^{σ} = −ϵ; or

(ii) M_{i} is a W -rank four superspecial Dieudonné submodule generated by some x, y such that ⟨x, y⟩ = p^{r} for some r ∈ ℤ, and ⟨x, F x⟩ = ⟨y, F y⟩ = ⟨x, F y⟩ = ⟨y, F x⟩ = 0.

From the proof of the proposition above, we can deduce the following result.

(10.17) Proposition (E, Proposition 5.1) Let M be a W -rank 2g separably quasi-polarized superspecial Dieudonné module over k. Then M ≃ M_{1}^{⊕g} with M_{1} of type (i) above.

(10.18) Corollary Let M be as in Propostion 10.17. Then the group K_{M} of the automorphisms of M is

K_{M} = {α ∈ M_{g}(O_{B_{p}}); α^{∗}α = 1},

where B_{p} is the quaternion division algebra over ℚ_{p}.

(10.19) Corollary Let x = (A_{0}, λ_{0}) be a separably polarized superspecial abelian variety over k and M_{0} its Dieudonné module. Then K_{M_{0}} = K_{0} and

mass(Λ_{x}) := ∑_{(A,λ)∈Λ_{x}} \frac{1}{Aut(A, λ)} = \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{k=1}^{g} ζ(1 − 2k)\} · ∏_{k=1}^{g} \{(p^{k} + (−1)^{k}\} .

In this case, Λ_{x} is the set of the isomorpshism classes of separably polarized superspecial abelian varieties over k such that the condition (ii) of (10.4) is satisfied.

(10.20) Corollary

∑_{(A,λ)∈Λ} \frac{1}{Aut(A, λ)} = \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{k=1}^{g} ζ(1 − 2k)\} · ∏_{k=1}^{g} \{(p^{k} + (−1)^{k}\} ,  (36)

where Λ is the set of the isomorpshism classes of principally polarized superspecial abelian varieties over k of dimension g.

Proof. Take x = (A_{0}, λ_{0}) to be a principally polarized superspecial abelian variety over k of dimension g. Then we have Λ = Λ_{x}.

(10.21) In [LO, Sec. 4.9], the number of irreducible components of the supersingular locus of the A_{g,1} ⊗ k is #Λ. Here Λ is the set of the isomorphism classes of polarized superspecial abelian varieties (A, λ) of dimension g over k such that ker λ = A[F^{g−1}]. They also show that

#Λ = \begin{cases}H_{g}(p, 1) & if g is odd, \\ H_{g}(1, g) & if g is even. \end{cases}

Here H_{g}(p, 1) (resp. H_{g}(1, g)) is the principal genus (resp. the non-principal genus) of the Hermitian space B^{g}.

If g is odd, then we can divide the polarization by a power of p such that it is principal. This computation has been done. When g is even, by again dividing by a power of p, we can consider the case that the polarization λ satisfies the property ker λ = A[F]. All we need to compute is the group K_{M} of automorphisms and compute μ_{p}(K_{0,p}/K_{M}).

From the previous results, it seems saying that we have K_{M} = Aut(N_{g}), where

N_{g} = O_{B_{p}}^{⊕g} · γ, γ = (\begin{matrix}I_{g/2} & 0 \\ 0 & πI_{g/2} \end{matrix}) · ξ ∈ GL_{g}(B_{p}), ξξ̄^{t} = anti-diag\{1, 1, …, 1\}.

(10.22) Lemma μ_{p}(K_{0,p}/ Aut(N_{g})) = 1

Proof. One can choose a left and right Haar measure μ on G_{x}(ℚ_{p}). Then μ_{p}(K_{0,p}/ Aut(N_{g})) = μ(K_{0,p})/μ(Aut(N_{g})). On the other hand, N_{g} = γ^{−1}K_{0,p}γ and μ(γ^{−1}K_{0,p}γ) = ||g^{−1}|| · ||g||μ(K_{0,p}) = μ(K_{0,p}).

I am not sure if I have the computation wrong. The computation shows that one have one only mass formula not depending on the parity of g.

Let A_{g,1,n}, n ≥ 3, be the moduli scheme of principally polarized abelian varieties with full symplectice n-level structure. Let 𝒮_{g,1,n} be the supersingular locus of A_{g,1,n} ⊗ \overline{𝔽_{p}}.

(10.23) Proposition The number of irreducible components of the supersingular locus 𝒮_{g,1,n} is

Sp(ℤ/nℤ) · \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{k=1}^{g} ζ(1 − 2k)\} · ∏_{k=1}^{g} \{(p^{k} + (−1)^{k}\} .

Proof. This is a reformulation of a result of Li and Oort by the mass formula.

(10.24) We turn to the focus on the Hilbert-Blumenthal cases. That is, the case that m = 1 and d = g. Let (A_{0}, λ_{0}, ι_{0}) be a superspecial point satisfying the Rapoport condition, and let M_{0} be its covariant Dieudonné module. We have M_{0} = ⊕_{𝔭|p}M_{𝔭} and K_{M_{0}} = ∏_{𝔭|p} K_{M_{𝔭}}.

(10.25) Lemma (1) If the residue degree f(𝔭) is odd, then the a-number a(M_{𝔭}) = (e, e, …, e) and K_{M_{𝔭}} = K_{0,𝔭}.

(2) If the residue degree f(𝔭) is even, then there are at least two possibilities:

(i) the a-number a(M_{𝔭}) is either (2e, 0, 2e, 0, …, 2e, 0) or (2e, 0, 2e, 0, …, 2e, 0). In either case,K_{M_{𝔭}} = K_{0,𝔭};

(ii) the a-number a(M_{𝔭}) = (e, e, …, e) and

K_{M_{𝔭}} = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(O_{F_{𝔭}}); c ≡ 0 mod π \} .

In this case we have K_{0,𝔭}/K_{M_{𝔭}} ≃ P^{1}(k(𝔭)) and

μ_{𝔭}(K_{0,𝔭}/K_{M_{𝔭}}) = #P^{1}(k(𝔭)) = N(𝔭) + 1 = p^{f(𝔭)} + (−1)^{f(𝔭)}.

We will give a proof. In fact, we will use (i) and (ii) of (2).

(10.26) Corollary Let x = (A_{0}, λ_{0}, ι_{0}) be a separably polarized superspecial O_{F} -abelian variety over k in the Rapoport locus, and let M_{0} be its Dieudonné module. Then

mass(Λ_{x}) := ∑_{(A,λ,ι)∈Λ_{x}} \frac{1}{Aut(A, λ, ι)} = [\frac{−1}{2}]^{g} · ζ_{F} (−1) · ∏_{𝔭|p}\{p^{f(𝔭)} + (−1)^{f(𝔭)}\}  (37)

In this case, Λ_{x} is the set of the isomorpshism classes of separably polarized superspecial O_{F} -abelian varieties over k in the Rapoport locus such that the condition (ii) of (10.4) is satisfied.

(10.27) Corollary

∑_{(A,λ,ι)∈Λ} \frac{1}{Aut(A, λ, ι)} = [\frac{−1}{2}]^{g} · ζ_{F} (−1) · ∏_{𝔭|p}\{p^{f(𝔭)} + (−1)^{f(𝔭)}\}  (38)

where Λ is the set of the isomorpshism classes of principally polarized superspecial O_{F} -abelian varieties over k of dimension g which satisfy the Rapoport condition.

Proof. Take x = (A_{0}, λ_{0}, ι_{0}) to be a principally polarized superspecial O_{F} -abelian variety over k of dimension g in the Rapoport locus. Then we have Λ = Λ_{x}.

References

[BG] E. Bachmat and E. Z. Goren, On the non-ordinary locus in Hilbert-Blumenthal surfaces. Math. Ann. 313 (1999), 475–506.

[C] C.-L. Chai, Newton polygons as lattice points. Amer. J. Math. 122 (2000), 967–990.

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

[dJO] A.J. de Jong and F. Oort, Purity of the stratification by Newton polygons. J. Amer. Math. Soc. 13 (2000), 209–241.

[DP] P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositio Math. 90 (1994), 59–79.

[E] T. Ekedahl, On supersingular curves and supersingular abelian varieties. Math. Scand. 60 (1987), 151–178.

[GO] E.Z. Goren and F. Oort, Stratifications of Hilbert modular varieties in positive characteristic. J. Algebraic Geom. 9 (2000), 111–154.

[Gr] A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné. Les Presses de l’Université de Montréal, 1974.

[K1] R. E. Kottwitz, Isocrystal with additional structure. Compositio Math. 56 (1985), 201– 220.

[K2] R. E. Kottwitz, Points on some Shimura varieties over finite fields. J. Amer. Math. Soc. 5 (1992), 373–444.

[LO] K.-Z. Li and F. Oort, Moduli of Supersingular Abelian Varieties. Lecture Notes in Math., vol. 1680, Springer-Verlag, 1998.

[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. Oxford University Press, 1974.

[N] P. Norman, An algorithm for computing moduli of abelian varieties. Ann. Math. 101 (1975), 499–509.

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

[O1] F. Oort, Moduli of abelian varieties and Newton polygons. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 385–289.

[O2] F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. Math. 152 (2000), 183–206.

[O3] F. Oort, Newton Polygon strata in the moduli space of abelian varieties. Moduli of Abelian Varieties, 417–440. (ed. by C. Faber, G. van der Geer and F. Oort), Progress in Mathematics 195, Birkhäuser 2001.

[P] G. Pappas, Arithmetic models for Hilbert modular varieties. Compositio Math. 98 (1995), 43–76.

[R] M. Rapoport, Compactifications de l’espaces de modules de Hilbert-Blumenthal. Compositio Math. 36 (1978), 255–335.

[RR] M. Rapoport and M. Richartz, On the classification and specialization of F -isocrystals with additional structure. Compositio Math. 103 (1996), 153–181.

[RZ] M. Rapoport and Th. Zink, Period Spaces for p-divisible groups. Ann. Math. Studies 141, Princeton Univ. Press, 1996.

[Y] C.-F. Yu, On the supersingular locus of Hilbert-Blumenthal 4-folds. MPI Preprint 2000 – 136.

[Z1] T. Zink, Cartiertheorie kommutativer formaler Gruppen. Teubner-Texte Math. Teubner, Leipzip, 1984.

[Z2] T. Zink, The display of a formal p-divisible groups. University of Bielefeld, Preprint 98–017, 1998.

Department of Mathematics Columbia University New York, NY 10027 Email Address: chiafu@math.columbia.edu