The weak Grothendieck conjecture for good reduction of certain Shimura varieties

Chia-Fu Yu^{1}

§1. Introduction

This work was done in the summer of 1997, after the author received F. Oort’s Utrecht preprint no. 995 [O2] in the March from his thesis advisor Ching-Li Chai. For several reasons the author could not write the result in a more formal manner. The main reason was that the author was working on his thesis. Another reason was that the author wished to generalize the results to the cases of good reduction of more general PEL-Shimura varieties. However, the author actually did not seriously work on the cases of type A and D. After four years passed away, the author decided to publish the partial results done in 1997 in the MPI preprint and hoped to come back to complete the rest cases soon.

Before this work, we did not know much about the possible Newton polygons and the expected dimensions of Newton strata of the reduction of PEL-Shimura varieties except the Siegel modular varieties. There were some results obtained by Zink [Z3] on a special case of type C. He used the Bruhat-Tits building to classify the isogeny classes. In [RR] Rapoport and Richartz have given the classification of F -isocrystals with additional structure. The results are very general. However, one could not find the explicit description from their work of possible Newton polygons which are are “expected” to occur in the reduction of the Shimura variety. After this work, we knew the answer in the type C good reduction cases. Now, we have more explicit description by Chai’s work [C]. The description is for general cases too, which possibly applies for certain bad reduction cases as well.

The method of this work is rather classical, which was introduced by F. Oort [O2] called the “Cayley-Hamilton” method. Our main contribution is to classify the Dieudonné modules of (usual) a-number one in question. Our main result is that the weak Grothendieck conjecture for good reduction of type C Shimura varieties holds.

In [K] Kottwitz has defined a moduli problem for Shimura varieties of PEL-type at a good prime p. The formulation is the same as that by Deligne [D] except that he replaces the trace condition in [D] by a determinant condition, now called the Kottwitz determinant condition. Our moduli problem is simpler, which is closer to the formulation by Mumford and Shimura. The relation between Kottwitz’s moduli spaces and the classical moduli spaces we use is: There is an étale morphism from a Kottwitz’s space to our space. For the consideration of geometric properties alone, we can reduce the problems on the Kottwitz’s spaces to those on our spaces.

Acknowledgments. I would like to thank F. Oort for his inspiring work [O2]. This paper was rewritten 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 so that this rewrite is possible.

1This is still a preliminary version.

§2. The moduli spaces

(2.1) Let B be a finite-dimensional division algebra over ℚ with a positive involution ∗. Let (V, ψ) be a ℚ-valued skew-Hermitian finite left B-module with ℚ-dimension 2g. Let G = GL_{B}(V, [ψ]) be the algebraic group of B-linear ψ-similitudes. It is well-known that each factor of the derived group G_{ℂ}^{der} is a symplectic group if and only if (B, ∗) is either of the following cases:

(I) The division algebra B is a totally real field F and the involution is trivial.

(II) The division algebra B is a indefinite quaternion algebra over a totally real field F. Moreover there is an ℝ-algebra isomorphism

B ⊗_{ℚ} ℝ ≃ M_{2}(ℝ) ⊕ ⋯ ⊕ M_{2}(ℝ)

such that the transposition on the right hand side induces the involution ∗.

(2.2) In this work we consider good reduction of type C Shimura varieties. Let B be a division algebra with a positive involution ∗ as (I) or (II) of (2.1). Let O_{B} be an order of B stable under the involution such that O_{B} ⊗ ℤ_{p} is a maximal order. A polarized O_{B}-abelian variety is a datum (A, λ, ι), where

• A is an abelian variety;

• λ : A → A^{t} is a polarization; and

• ι : O_{B} ↪ End(A) is a ring monomorphism such that λι(b^{∗}) = ι(b)^{t}λ.

Fix a prime number p. Let A_{g}^{O_{B},(p)} denote the moduli stack of polarized O_{B}-abelian varieties (A, λ, ι) of dimension g with the polarization λ of prime-to-p degree. It is a standard fact that A_{g}^{O_{B},(p)} is a separated Deligne-Mumford algebraic stack locally of finite type over Specℤ.

(2.3) Lemma The algebraic stack A_{g}^{O_{B},(p)} has good reduction at p if and only if p is unramified in F and B is unramified at every primes of F lying over p.

Proof. This follows from a simple tangent space calculation, which is standard and omitted.

(2.4) In sections 2-4 we assume that p is inert in F and B is unramified at p. Let O := O_{F} ⊗ ℤ_{p} and O_{B,p} := O_{B} ⊗_{ℤ} ℤ_{p}. Note that O_{B,p} ≃ M_{2}(O) if B is of type (II) from our assumption. Let f denote the degree of F and let

r := \begin{cases}\frac{g}{f} & if B is of type (I), \\ \frac{g}{2f} & if B is of type (II). \end{cases}

The moduli space A_{g}^{O_{B},(p)} → Specℤ_{p} is smooth of relative dimension f \frac{r(r+1)}{2}. Let ℳ := A_{g}^{O_{B},(p)} ⊗ 𝔽_{p} denote the reduction of the moduli space A_{g}^{O_{B},(p)} modulo p.

Let k be an algebraically closed field of characteristic p. Let W := W (k) denote the ring of Witt vectors over k and σ denote the Frobenius morphism on W. Note that for each object (A, λ, ι) ∈ ℳ(k), the Lie algebra Lie(A) of the abelian variety A is a free O_{F} ⊗ k-module.

§3. The main theorem

(3.1) Let (A, λ, ι) be a polarized O_{B}-abelian variety of dimension g over k, and let M be its covariant Dieudonné module. The Dieudonné module M is a free W -module of rank 2g together with a σ-linear map F (Frobenius) and a σ^{−1}-linear map V (Verschiebung) such that F V = V F = p. It is also equipped with a W [F, V]-linear action by O_{B,p} and a non-degenerate alternating pairing

M × M → W

such that ⟨bx, y⟩ = ⟨x, b^{∗}y⟩ for all x, y ∈ M. We call it briefly a quasi-polarized O_{B,p}- Dieudonné module.

(3.2) Suppose that B is of type (II). We have O_{B,p} ≃ M_{2}(O) and the involution is the transposition: (a_{ij}) ↦ (a_{ji}). Note that the idempotents

e_{1} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , e_{2} = (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix})

are stable under the involution. The functor M ↦ e_{1}M (Morita equivalence) gives an equivalence between the category of quasi-polarized O_{B,p}-Dieudonné modules to the category of quasi-polarized O-Dieudonné modules. Therefore, we can reduce the problems for case (II) to those for case (I) and treat case (I) alone.

(3.3) Let {σ_{i}}_{i∈ℤ/fℤ} be the set of embeddings from O into W, arranged in a way that σσ_{i} = σ_{i+1}. Let

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

be the σ_{i}-component of M, which is a free W -module of rank 2r. 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 pairing ⟨ , ⟩ for i ≠ j. Conversely, a Dieudonné module together with such a decomposition and these properties is a quasi-polarized O-Dieudonné module.

(3.4) Lemma Any quasi-polarized O-Dieudonné module M has 2r (F -)slopes λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{2r} with each slope multiplicity f. In other words, the Newton polygon of M satisfies the usual symmetric and integral conditions and has the breaking points (x, y) with x divisible by f.

Proof. Consider F^{f} acting on M^{0}. It follows from Katz’s sharp slope estimate [Ka] that the slopes of F^{f} on each M^{i} are the same. Then M has 2r slopes with each slope multiplicity f.

(3.5) Example. Let f = 3 and r = 2. The possible slopes of M are {λ_{1}^{3}, λ_{2}^{3}, λ_{3}^{3}, λ_{4}^{3}} (with each multiplicity 3)listed below.

(λ_{1}, λ_{2}, λ_{3}, λ_{4}) = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}) > (\frac{1}{3}, \frac{1}{2}, \frac{1}{2}, \frac{2}{3}) > (\frac{1}{3}, \frac{1}{3}, \frac{2}{3}, \frac{2}{3}) >

> (0, \frac{1}{2}, \frac{1}{2}, 1) , (\frac{1}{6}, \frac{1}{6}, \frac{5}{6}, \frac{5}{6}) > (0, \frac{1}{3}, \frac{2}{3}, 1) > (0, 0, 1, 1) .

(3.6) The a-number of an O-Dieudonné module M is defined to be a tuple

a(M) = (a_{i})_{i∈ℤ/fℤ}, a_{i} := dim_{k}(M/(F, V)M)^{i}.

The usual a-number is denoted by |a(M)| = ∑_{i∈ℤ/fℤ} a_{i} in this paper. It is the dimension of the k-vector space Hom_{k}(α_{p}, A).

Recall that M is local-local if the operators F and V are nilpotent on M/pM.

If (A, λ, ι) ∈ ℳ(k), then polarization induces a perfect pairing ⟨ , ⟩ on the Dieudonné module M of A. In this case, M is called a separably quasi-polarized O-Dieudonné module.

(3.7) Theorem Let M be a separably quasi-polarized O-Dieudonné module over k. Assume that the a-number a(M) = (10 … 0) and that M is local-local. Then there exists a symplectic basis X_{0}, X_{1}, …, X_{g−1}, Y_{0}, Y_{1}, …, Y_{g−1} such that X_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i} for all i = 0 … g − 1 and the Frobenius map F has the following normal form:

(\begin{matrix}0 & 0 & … & 0 & 0 & 0 & … & 0 & −p \\ 1 & 0 & … & 0 & pa_{2,2} & pa_{2,3} & … & pa_{2,g} & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 & pa_{g,2} & pa_{g,3} & … & pa_{g,g} & 0 \\ 0 & … & 0 & 1 & 0 & 0 & … & … & 0 \\ 0 & … & … & 0 & p & 0 & … & … & 0 \\ 0 & ... & ... & 0 & 0 & p & … & … & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & … & … & 0 & 0 & … & … & p & 0 \end{matrix})

where a_{i,j} = 0 if i ≢ j (mod f) and a_{i,j} = a_{j,i} for all i, j between 2 and g.

Proof. Since the polarization is separable and |a(M)| = 1, it induces a perfect pairing: M/(F, V)M × F^{g}(M/pM) → W/pW ≃ k. Thus there exists X ∈ M^{0} s.t. ⟨X, F^{g}X⟩ ≢ 0 mod p. Since k is algebraically closed , we may adjust X by a constant multiple such that ⟨X, F^{g}X⟩ ≡ 1 mod p, and clearly ⟨X, −V^{g}X⟩ ≡ 1 mod p.

As −V^{g}X̄ ∈ F^{g}M̄ (M̄ := M/pM), we obtain −V^{g}X ≡ F^{g}X mod pM. Define

Y_{0} := F^{g}X, Y_{i} := −V^{g−i}X, X_{i} := F^{i}X ∀ i = 1 … g − 1  (1)

Then we have X_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i} and

⟨X_{i}, Y_{j}⟩ ∈ \begin{cases}pW & if i > j; \\ 1 + pW & if i = j; \\ W & otherwise. \end{cases}  (2)

Set X′ = X +a_{1}Y_{f} +⋯+a_{r−1}Y_{(r−1)f}. Since the (r −1)×(r −1) matrix (⟨Y_{if}, X_{jf}⟩)_{1≤i,j<r} is non-degenerate mod p from (2), we can find a_{i} ∈ W for 1 ≤ i < r such that ⟨X′, X_{jf}⟩ ≡ 0 mod pW for j = 1 … r − 1. Note that F^{i}X′ ≡ F^{i}X mod pM. Replacing X by X′, we have ⟨X, F^{i}X⟩ ≡ 0 mod pW for 0 ≤ i < g. Set X_{0} := X and re-define Y_{i}, X_{i} as (1). Thus we have verified n = 1 for the following induction hypothesis (I_{n}):

∃X ∈ M^{0}, Y_{i} ∈ (V M)^{i}, i = 0 … g − 1 such that

Y_{0} ≡ F^{g}X (modp^{n}W)

Y_{g−1} ≡ −V X (modp^{n}W)

and if we set X_{i} := F^{i}X for i = 0 … g −1, then {X_{i}, Y_{i}} is an symplectic basis modulo p^{n}M.

Suppose that we have a basis {X_{i}, Y_{i}} satisfying (I_{n}), we want to find a new basis {X′_{i}, Y′_{i}} which satisfies (I_{n+1}) and that X′_{i} ≡ X_{i}, Y′_{i} ≡ Y_{i} (mod p^{n}M). Let

X′ := X + a_{f}p^{n}Y_{f} + a_{2f}p^{n}Y_{2f} + ⋯ + a_{(r−1)f}p^{n}Y_{(r−1)f}.

Since F^{i}X′ ≡ F^{i}X mod p^{n+1}M for i > 0, we have ⟨X′, F^{i}X′⟩ ≡ ⟨X′, X_{i}⟩ mod p^{n+1}W. By dividing p^{n}, we get a system of linear equations mod p with coefficients ⟨Y_{j}, X_{i}⟩. Therefore we can find a_{i}, unique modulo p, such that ⟨X′, F^{i}X′⟩ ≡ 0 mod p^{n+1}W for i = 1 … g − 1. Clearly, X′ ≡ X mod p^{n}M.

Suppose ⟨X′, F^{g}X′⟩ ≡ 1 + ap^{n} mod p^{n+1}W. Choose b ∈ W so that b + b^{σ}g + a ≡ 0 mod p. Set X′′ := (1 + bp^{n})X′, then we have ⟨X′′, F^{g}X′′⟩ ≡ 1 mod p^{n+1}W from the choice of b. After replacing X′ by X′′, we find an X′ satisfying

⟨X′, F^{i}X′⟩ ≡ 0 mod p^{n+1}W for 0 < i < g
⟨X′, F^{g}X′⟩ ≡ 1 mod p^{n+1}W

and X′ ≡ X mod p^{n}M.

Set X′_{i} := F^{i}X′, Y′_{g−1} := −V X′, Y′_{0} := F^{g}X′. We can easily adjust the rest of Y_{i} by adding some linear combinations of p^{n}Y_{i} and get Y′_{i} for i = 1 … g − 2 such that {X′_{i}, Y′_{i}} satisfies (I_{n+1}).

By completeness of M, we get a symplectic basis {X_{i}, Y_{i}} with

(∗) : X_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i}, X_{i} = F^{i}X_{0}, Y_{0} = F^{g}X_{0}, and Y_{g−1} = −V X.

It follows from the symplectic basis {X_{i}, Y_{i}} with the relations (∗) that F Y_{j} = pY_{j+1} + ∑_{i=1}^{g−1} pa_{i+1,j+2}X_{i} for j = 0 … g − 2 such that a_{i+1,j+2} = 0 except i + 1 ≢ j + 2 mod f. That is, a_{i,j} ≠ 0 only when i ≡ j (mod f).

§4. Local moduli of Newton strata

(4.1) We briefly review the method of Cayley-Hamilton developed by F. Oort in [O2]. Let M be a Dieudonné module of rank 2g. Suppose that the representative matrix of the Frobenius map on M with respect to a certain basis has the following form

(\begin{matrix}B & C & E \\ 0 & D & 0 \end{matrix}) ,

where B is the g × (g − 1) matrix with b_{i+1,j} = δ_{ij}, D is the g × g matrix diag(1, p, p, …, p), E is the column vector (−p, 0, …, 0) and C is a g × g matrix of the form

(\begin{matrix}a_{1,1} & pa_{1,2} & … & pa_{1,g} \\ a_{2,1} & pa_{2,2} & … & pa_{2,g} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{g,1} & pa_{g,2} & … & pa_{g,g} \end{matrix}) .  (3)

Note that the matrix C is the crucial part of the representative matrix.

The assertion of this method says that the Newton polygon of M is close to the lower convex polygon defined by removing from the following diagram (the Oort diagram) the a_{i,j} if p|a_{i,j}.

\begin{aligned}& & & & & & & −1 (2g, g) \\ & & & a_{g,g} & … & … & a_{1,g} \\ & & · & ⋮ & · & · \\ & · & · & ⋮ & · \\ 1 & a_{1,g} & … & a_{1,1} \\ (0, 0) \end{aligned}  (4)

It is eactly this Newton polygon if each a_{i,j} is either 0 or a unit.

To be more precise, there is a σ-linear version of a Cayley-Hamilton equation relating the action of Frobenius on M. From this Cayley-Hamilton equation, one can show that the Newton polygon of M is the same as that of the following polynomial

X^{2g} + ∑_{k=g−1}^{0} ∑_{i=1}^{g−k} p^{i−1}a_{i+k,i}^{σ^{?}}X^{g+k} + ∑_{k=1}^{g−1} ∑_{i=1}^{g−k} p^{i+k−1}a_{i,i+k}^{σ^{?}}X^{g−k} − p^{g}

We refer the reader to [O2] for details and proofs.

(4.2) We recall the deformation theory of Dieudonné modules developed by Norman [N] and Norman-Oort [NO]. We follow the convenient setting of [N, Sec. 0] and [CN, Sec. 2, p. 1011].

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).

(4.3) Let M_{0} be a local-local separably quasi-polarized O-Dieudonné module with usual a-number |a(M_{0})| = 1. For simplicity, we assume that a(M_{0}) = (10 … 0). Let {X_{i}, Y_{i}} be a symplectic basis as in Theorem 3.7. Set

\begin{aligned}e_{k}^{l} & := X_{(k−1)f+l} for 0 ≤ l ≤ f − 1, 1 ≤ k ≤ r \\ f_{k}^{l} & := Y_{(k−1)f+l} for − 1 ≤ l ≤ f − 2, 1 ≤ k ≤ r \end{aligned}  (5)

with the natural convention that X_{i±g} = X_{i} and Y_{i±g} = Y_{i}. Then {e_{k}^{i}, f_{k}^{i}}_{1≤k≤r} is a symplectic W -basis of M^{i} for each i ∈ ℤ/fℤ. It is convenient to extend the notations f_{k}^{l} and e_{k}^{l} for all integers l, k by the rule (5). We have

F e_{j}^{l} = e_{j}^{l+1} except (l, j) = (f − 1, r)
F e_{r}^{f−1} = f_{1}^{0}
F f_{j}^{l} = pf_{j}^{l+1} + ∑_{i=1}^{r} pa_{i,j}^{l+1}e_{i}^{l+1} except (l, j) = (−1, 1)
F f_{1}^{−1} = −pe_{1}^{0}

(4.4) Let Def[M_{0}, ⟨ , ⟩, ι] be the deformation functor of quasi-polarized O-Dieudonné modules of M_{0}. This functor is smooth. Moreover, for any local artin k-algebra (R, 𝔪) with residue field k there is a bijection [N]

Def[M_{0}, ⟨ , ⟩, ι](R) ≃ Hom_{O}(V M_{0}/pM_{0}, M_{0}/V M_{0})^{sym} ⊗_{k} 𝔪,

where we identify V M_{0}/pM_{0} with the dual (M_{0}/V M_{0})^{∗} by the pairing ⟨ , ⟩. Let (e_{k}^{l})^{∗}, (f_{k}^{l})^{∗} be the dual basis for the basis e_{k}^{l}, f_{k}^{i}. We may identify e^{∗}’s with f’s and f^{∗}’s with e’s. Thus any homomorphism that sends f̄_{j}^{l} ↦ ∑ t_{i,j}^{l}ē_{i}^{l} with t_{i,j}^{l} = t_{j,i}^{l} ∈ 𝔪 gives a deformation of M_{0} over R, and conversely any deformation of M_{0} over R arises from a unique such homomorphism. Hence R_{u} := k[|t_{i,j}^{l}|]/(t_{i,j}^{l} − t_{j,i}^{l}) = k[|t_{i,j}^{l}|]_{0≤l<f, 1≤j≤i≤r} is the universal deformation ring of M_{0} with additional structure.

By [N] or [NO], one can construct a quasi-polarized O-Dieudonné module M_{R_{u}} over R^{u} which parameterizes all deformations of M_{0} with additional structure. The universal Dieudonné module M_{R_{u}} is given by the following generators and relations:

F e_{j}^{l} = e_{j}^{l+1} except (l, j) = (f − 1, r)
F e_{r}^{f−1} = f_{1}^{0} + ∑_{i=1}^{r} T_{i,1}^{0}e_{i}^{0}

f_{j}^{l} = V (f_{j}^{l+1} + ∑_{i=1}^{r}(a_{i,j}^{l+1} + T_{i,j}^{l+1})e_{i}^{l+1}) except (l, j) = (−1, 1)
f_{1}^{−1} = −V e_{1}^{0},

(6)

where T_{i,j}^{l} := (t_{i,j}^{l}, 0, …) is the Teichmüller lifting of t_{i,j}^{l} for each l, i, j.

(4.5) Let x = (A_{0}, λ_{0}, ι_{0}) ∈ ℳ(k) with usual a-number |a(A_{0})| = 1 and p-rank 0, and let M_{0} be the associated covariant Dieudonné module. Let β be an admissible Newton polygon in (3.4) lower than that of M_{0}. Denote by ℳ_{x} the local moduli space of ℳ at x and by ℳ_{x}^{≥β} the reduced closed subscheme (in ℳ_{x}) of Newton polygon ≥ β. By the theory of Cayley-Hamilton, the local moduli of the Newton stratum ℳ_{x}^{≥β} is defined by the equations t_{i,j}^{l} whose “position” in the Oort diagram (4) of (4.1) is strictly below the Newton polygon β.

From (3) of (4.1) and (6) of (4.4), it is not hard to see that the position of t_{k+j,j}^{l} in the Oort diagram is ((r − k)f, (j − 1)f + l). Let

S(β) := { t_{i,j}^{l} ; j ≤ i and the position of t_{i,j}^{l} is higher or on β }.  (7)

Then ℳ_{x}^{≥β} ≃ Spf k[|t|]_{t∈S(β)}.

In the case β = NP (A_{0}) is supersingular, we get the local moduli of the supersingular locus at x and the dimension is

|S(β)| = ∑_{i=1}^{r}[i f/2] = \begin{cases}k \frac{r(r+1)}{2} & if f = 2k; \\ k \frac{r(r+1)}{2} + [r^{2}/4] & if f = 2k + 1. \end{cases}

(4.6) Theorem Let x : Speck → ℳ be a geometric point of usual a-number one and p-rank 0. Then for each admissible Newton polygon β, β ≤ NP (x), the reduced closed subscheme ℳ_{x}^{≥β} of Newton polygon ≥ β in ℳ_{x} is formally smooth of dimension |S(β)| (7)and the generic point has Newton polygon β. In particular, the supersingular locus of ℳ_{x} has dimension

|S(β)| = \begin{cases}k \frac{r(r+1)}{2} & if f = 2k; \\ k \frac{r(r+1)}{2} + [r^{2}/4] & if f = 2k + 1. \end{cases}

(4.7) Corollary The strong Grothendieck conjecture holds on the open subset of ℳ of usual a-number ≤ 1.

(4.8) Example. Let f = 3, r = 2. The crucial part of the representative matrix of the Frobenius on the universal Dieudonné module is

(\begin{matrix}T_{1,1}^{0} & & & p(a_{1,2}^{0} + T_{1,2}^{0}) \\ & p(a_{1,1}^{1} + T_{1,1}^{1}) & & & p(a_{1,2}^{1} + T_{1,2}^{1}) \\ & & p(a_{1,1}^{2} + T_{1,1}^{2}) & & & p(a_{1,2}^{2} + T_{1,2}^{2}) \\ T_{2,1}^{0} & & & p(a_{2,2}^{0} + T_{2,2}^{0}) \\ & p(a_{2,1}^{1} + T_{2,1}^{1}) & & & p(a_{2,2}^{1} + T_{2,2}^{1}) \\ & & p(a_{2,1}^{2} + T_{2,1}^{2}) & & & p(a_{2,2}^{2} + T_{2,2}^{2}). \end{matrix})

The corresponding Oort diagram is The local moduli of the supersingular locus is isomorphic to Spf k[|t_{2,1}^{2}, t_{2,2}^{0}, t_{2,2}^{1}, t_{2,2}^{2}|] and its dimension is 4.

§5. The weak Grothendieck conjecture

(5.1) Let F_{0} be a finite extension of ℚ_{p}. Let V be a 2r-dimensional F_{0}-vector space with a non-degenerate alternating form ψ_{0} on V with values in ℚ_{p} such that ψ_{0}(ax, y) = ψ_{0}(x, ay) for a ∈ F_{0}, x, y ∈ V. Let G_{0} be the algebraic group of F_{0}-linear similitudes over ℚ_{p} and let G_{1} be the derived group of G_{0}. The algebraic group G_{1} is connected and simply-connected. Denote by c the multiplier map G_{0} → 𝔾_{m}. Note that ker c = G_{1} = Res_{F_{0}/ℚ_{p}} Sp_{2r,F_{0}}.

Let b be an element of G_{0}(B(k)) and let ν_{b} be associated slope homomorphism from the pro-torus D into G_{0}, [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}.

By [RR, Prop. 1.17], the fibre of the Newton map over ν_{b} ∈ N (G_{0}) is classified by H^{1}(ℚ_{p}, J_{b}). The following lemma shows that the Newton map from the σ-conjugacy classes B(G_{0}) to the Newton set N (G_{0}) is injective. In other words, the isomorphisms classes of an isocrystal with G_{0}-structure is determined by its Newton polygon.

(5.2) Lemma H^{1}(ℚ_{p}, J_{b}) = 0.

Proof. Consider the F -isocrystal (V ⊗_{ℚ_{p}} B(k), b ⊗ σ) with additional structure. By the decomposition into isotypic components and replacing V by a suitable invariant subspace W, we may assume that either b has two slopes λ and 1 − λ or has only one slope \frac{1}{2}. In the first case, the group J_{b} is an inner form of of the centralizer G_{0,sν(p)} for some s ∈ ℕ, which is Res_{Q_{p^{s}}/ℚ_{p}}(GL_{r}(F_{0} ⊗ ℚ_{p^{s}})) × ℚ_{p}^{×}. Therefore, we have H^{1}(ℚ_{p}, J_{b}) = 0 by Hilbert’s 90.

In the other case (supersingular), the group J_{b} is an inner form of the group G_{0}. Hence H^{1}(ℚ_{p}, J_{b}) = H^{1}(ℚ_{p}, G_{0}) = 0. This completes the proof.

(5.3) Corollary Let M_{i}, i = 1, 2 be two quasi-polarized O_{F_{0}}-Dieudonné modules over k = k̄ (with the trivial involution on F_{0}). If NP (M_{1}) = NP (M_{2}), then M_{1}⊗_{W} B(k) ≃ M_{2}⊗_{W} B(k) as quasi-polarized F -isocrystals with F_{0}-action.

Proof. This follows from [RR, Prop. 1.17], Lemma 5.2 and the surjectivity of the multiplier c : G_{0}(ℚ_{p}) → 𝔾_{m}(ℚ_{p}).

(5.4) We recall briefly basic results on simple algebras over ℚ with involutions. Let D be a finite-dimensional simple algebra with a positive involution ∗. Then (D ⊗_{ℚ} \overline{ℚ}, ∗) ≃ ⊕_{i=1}^{r}(L, ∗_{1}) for some r, where (L, ∗_{1}) is one of the following three cases:

(A) : L = M_{n}(\overline{ℚ}) ⊕ M_{n}(\overline{ℚ}) and (A, B)^{∗_{1}} = (B^{t}, A^{t}).

(C) : L = M_{2n}(\overline{ℚ}) and ∗_{1} is the standard symplectic involution.

(D) : L = M_{n}(\overline{ℚ}) and ∗_{1} is the transposition.

We also have the corresponding cases for (D ⊗_{ℚ}ℝ, ∗). It is isomorphic to ⊕_{i=1}^{r}(N, ∗_{2}) for the same r, where (N, ∗_{2}) is

(A) : N = M_{n}(ℂ) and (a_{ij})^{∗_{2}} = (ā_{ji}).

(C) : N = M_{n}(ℍ) and (a_{ij})^{∗_{2}} = (ā_{ji}).

(D) : N = M_{n}(ℝ) and (a_{ij})^{∗_{2}} = (a_{ji}).

Here a ↦ ā is the standard conjugation.

(5.5) Let ′ be another positive involution on D which has the same type as ∗. Then there is an positive element c ∈ D, c^{∗} = c, such that x′ = cx^{∗}c^{−1} for all x ∈ D^{×}. Let U(D, ∗)_{ℚ} be the unitary group

U(D, ∗)(R) := {x ∈ (D ⊗_{ℚ} R); xx^{∗} = 1},

for each commutative ℚ-algebra R. Consider the algebraic variety X_{ℚ} defined by

X(R) := {x ∈ (D ⊗_{ℚ} R); xx^{∗} = c}.

Note that X(R) ≠ ∅ if and only if (D ⊗ R,′) ≃ (D ⊗ R, ∗). We have X(\overline{ℚ}) ≠ ∅ and U(D, ∗) acts on X from the right. Therefore, X is a right principal homogeneous space under the group U(D, ∗). It associates to an element [c] in H^{1}(ℚ, U(D, ∗)). Note that the image of [c] in H^{1}(ℝ, U(D, ∗)) is trivial.

(5.6) Proposition If U(D, ∗) is semi-simple and simply-connected, then (D,′) ≃ (D, ∗) over ℚ.

(5.7) Corollary If (D, ∗) is of type C, then (D,′) ≃ (D, ∗) over ℚ.

(5.8) Let (D_{1}, ∗_{1}) and (D_{2}, ∗_{2}) be finite-dimensional simple algebras over ℚ with positive involutions. Suppose that the fixed fields of the centers are the same, called F. We define a new simple algebra with a positive involution by

(D, ∗) := (D_{1} ⊗_{F} D_{2}, ∗),

where (d_{1} ⊗ d_{2})^{∗} := d_{1}^{∗_{1}} ⊗ d_{2}^{∗_{2}}. It is clear that ∗ is an involution on D and not hard to show that ∗ is positive. Moreover, we have the following results:

• If one of them is of type A, then (D, ∗) is of type A.

• If both are type C or type D, then (D, ∗) is of type D.

• If one is of type C and the other is of type D, then (D, ∗) is of type C.

(5.9) Let (B, ∗) be as in (I) or (II) of (2.1) and let O_{B} be an order of B as in (2.2). In this section we want to construct a supersingular polarized O_{B}-abelian variety of dimension g. We assume that the Brauer invariant inv_{v}(B) = 0 for all v|p. Let r be the number defined in (2.4). Let E be a supersingular elliptic curve over k and A_{0} := (E ⊗ O_{F})^{g/[F:ℚ]}. One can choose a polarization on A_{0} induced from E and make A_{0} a supersingular polarized O_{F} -abelian variety. We have done for the case of type (I).

Now we assume that B is of type (II). Let (D,′) be the semi-simple algebra End_{O_{F}} (A)⊗ℚ with the Rosati involution. Note that D = M_{2r}(D_{F}), where D_{F} := D_{p} ⊗_{ℚ} F and D_{p} is the quaternion algebra over ℚ ramified at {∞, p}.

We can find a central simple division algebra C over F such that the Brauer invariants of B ⊗_{F} C and of D are the same. We can replace C by M_{n}(C) for suitable n so that B ⊗_{F} C is isomorphic to D. Choose any positive involution on C and make the product positive involution ∗ on B ⊗_{F} C that extends ∗ on B. By Corollary 5.7, there is an isomorphism (B ⊗_{F} C, ∗) ≃ (D,′). Therefore, we can construct a supersingular polarized O_{F} -abelian variety A_{0} equipped with an monomorphism B → End_{O_{F}} (A_{0}) ⊗ ℚ perserving involutions. Furthermore, one may choose a supersingular O_{B}-abelian variety within this isogeny class.

(5.10) Theorem If inv_{v}(B) = 0 for all place v of F over p, then there is a supersingular polarized O_{B}-abelian variety over k.

(5.11) Remark The assumption of the Brauer invariants of B should be removed.

(5.12) Theorem Let (B, ∗) be as (I) or (II) in (2.1). Assume that the moduli space A_{g}^{O_{B},(p)} has good reduction at p. Then the weak Grothendieck conjecture for ℳ := A_{g}^{O_{B},(p)} ⊗𝔽_{p} holds.

Proof. By Theorem 5.10, there is a supersingular polarized O_{B}-abelian variety A_{0}. It follows from Corollary 5.3 that A_{0} can be chosen such that each component G_{v} of the associated p-divisible group A_{0}[p^{∞}] has usual a-number one. By the Serre-Tate theorem and Theorem 4.6, given any sequence of admissible Newton polygons NP (A_{0}) = β_{1} > β_{2} > ⋯ > β_{s}, there is a sequence of smooth irreducible subschemes V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{s} of the local moduli space ℳ_{x} such that the generic point of each subscheme V_{i} has Newton polygon β_{i}. Here x is the polarized O_{B}-abelian variety A_{0}. This completes the proof.

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