Models of Shimura varieties in mixed characteristic

Rutger Noot

1 Introduction

1.1 In the study of integral models of Shimura varieties, results are largely limited to Shimura varieties of PEL-type. These are Shimura varieties serving as moduli spaces of abelian varieties endowed with a polarization, a given ring acting as endomorphisms and a level structure. An example of Mumford, see [Mu2, §4], shows that not all Shimura varieties constructed from subgroups of the symplectic group are obtained in this way. It can however be shown that, over C, all such Shimura varieties parametrize polarized abelian varieties with level structure and given spaces of Hodge classes. In this paper, we will attempt to use this fact to study models in mixed characteristic of these Shimura varieties.

To this end, a “local p-adic analogue” of a Shimura variety is introduced in the following way. We start with a principally polarized abelian scheme X over the ring W of Witt vectors of the algebraic closure \overline{F}_{p} of a finite field. We fix a tensor product T_{DR}(X/W) of DeRham cohomology groups of X and their duals, twisted to make the weight equal to 0, and we choose a class ξ ∈ Fil^{0} T_{DR}(X/W). By the comparison theorem between the crystalline and the DeRham cohomology of X, there is a semi-linear Frobenius endomorphism on T_{DR}(X/W) ⊗ Q. We will assume that ξ is invariant under this endomorphism, or in other words, that ξ is a Tate class.

In Section 2, we will investigate the subspace N of the formal deformation space M of X_{0}, the special fibre of X, parametrizing principally polarized liftings Y/W of X_{0} such that the image of ξ in T_{DR}(Y/W) is a Tate class lying in Fil^{0} T_{DR}(Y/W). We have to make the assumption that the reduction X_{0}/\overline{F}_{p} of X is ordinary to obtain any results. Recall that X_{0} is ordinary if and only if rk_{Z_{p}} T_{p}X_{0} = dim X_{0}. In that case we can show that we can find an N with properties analogous to those of a Shimura variety. In fact, it will be the largest connected formal subscheme of M such that the class ξ can be extended to a horizontal section ξ̃ ∈ Fil^{0} T_{DR}(X_{N} /N) on the formal deformation of X_{0} over N and such that the polarization of X_{0} can be lifted to a polarization of X_{N}. We will prove that N is formally smooth. We can in fact say more about it. Because we assume X_{0} to be ordinary, its deformation space M is canonically endowed with the structure of a toroidal formal group, i. e. a formal group which is isomorphic to Ĝ_{m}^{d} for some integer d. We will find that N is a formal subtorus of M. This implies in particular that, if it has positive dimension, it contains infinitely many torsion points. Because these torsion points correspond to abelian varieties of CM-type, this provides us with another similarity with Shimura varieties, see [Mu2] or [De1].

The results of Section 2 are actually somewhat more general that those described above, because the abelian schemes we consider are defined over the integral closure W̄ of W in the algebraic closure of its fraction field. In this more general case we can still construct N as above, the only difference being that it will no longer be a subtorus of M, but the translate of such a subtorus over a W̄ -valued torsion point of M. The statement about the CM-points remains true. The reader is referred to Theorem 2.8 for the precise statement.

In Section 3, the results of Section 2 will be applied to the study of models in mixed characteristic of Shimura varieties, let us say over the ring W̄ considered above. For the lack of a better way of obtaining such a model of a Shimura variety, we will construct it by taking the Zariski closure of the canonical model of the Shimura variety inside an appropriate moduli space of abelian schemes over W̄. In order to apply the results about Tate classes to this situation, we make use of a theorem of Blasius, which states that if ξ is a Hodge class on an abelian variety over a number field, then it is a Tate class at all primes where the abelian variety has good reduction. As we noted above, a Shimura variety serving as moduli space of principally polarized abelian varieties parametrizes abelian varieties endowed with given spaces of Hodge classes. Therefore, the results of Section 2 can be used to study the completion of our model in a point in characteristic p corresponding to an ordinary abelian variety X_{0}. We find that all components of this completion are translates of formal subtori of the formal deformation space M of X_{0}. The exact result is given in Theorem 3.7.

Acknowledgements. This paper is a generalization of a part of my thesis which was written under the direction of F. Oort. His encouragement and support have been of great value to me. Among the many comments on my thesis and on earlier versions of this paper, I would like to mention those of L. Illusie and of J.-P. Wintenberger. This paper owes much to both of them and I thank them most heartily.

1.2 Notation. Let us fix some notation to be in force for the rest of the paper. We choose an embedding of Q̄ into C and by this embedding we will consider all number fields as subfields of C. Throughout the paper, p will be a prime number, F_{p} will denote the finite field with p elements and \overline{F}_{p} an algebraic closure of F_{p}. The ring of infinite Witt vectors of \overline{F}_{p} will be denoted by W and its fraction field by B. We will write B̄ for an algebraic closure of B and W̄ for the integral closure of W in B̄. The Frobenius mapping on W will be denoted by σ.

We will use the symbol T to denote tensor constructions of cohomology groups, roughly in the way as in [De2]. If X/k is an algebraic variety, then

T_{DR}(X/k) = ⨂_{α∈I} H_{DR}^{n_{α}}(X/k)(m) = ⨂_{α∈I} H_{DR}^{n_{α}}(X/k) ⊗ k(m),

where I is a finite set of indices and n_{α}, m ∈ Z. We will always assume that ∑_{α} n_{α} = 2m. In the above expression, k(m) = k(1)^{⊗m} denotes a Tate twist, which, in the case of DeRham cohomology, means that k(1) = k and

Fil^{i} k(1) = \{ \begin{matrix}k(1) & if & i ≤ −1 \\ 0 & if & i ≥ 0. \end{matrix}

If i < 0, then H_{DR}^{i}(X/k) is defined to be the dual of H_{DR}^{−i}(X/k). If X is an S-scheme for some scheme S, we will write H_{DR}^{i}(X/S) and T_{DR}(X/S) for the relative DeRham cohomology groups and tensor products of such groups respectively. A Tate twist only affects the Hodge filtration, just as in the absolute case.

In the case of crystalline cohomology we write T_{cris}(X_{0}/B) for a tensor product of the crystalline cohomology groups of a smooth algebraic variety X_{0}/\overline{F}_{p} and their duals. In this case, the Tate twist is defined by B(1) = B on which Frobenius acts as p^{−1}σ.

When considering the Betti cohomology of a smooth variety X/C, the cohomology groups will be denoted by H_{B}^{i}(X(C), Q), and T_{B}(X(C), Q) will be a tensor product of these groups and their duals. Here the Tate twist is given by Q(1) = 2πiQ ⊂ C. It only has an effect when T_{B} is compared with the corresponding tensor in the DeRham cohomology. If we have a family

X ⟶^{f} S

over C, we write Tf_{∗}^{an}Q for a tensor product of groups R^{i}f_{∗}^{an}Q.

2 Local moduli of Tate classes

2.1 In this section we will study the formal deformation space of an ordinary abelian variety over a finite field endowed with a number of Tate classes. Let us first recall some well-known facts about deformations of ordinary abelian varieties.

An abelian variety X_{0}/\overline{F}_{p} is called ordinary if the Tate module T_{p}X_{0} is a Z_{p}-module of rank g = dim X_{0}. If X_{0} is ordinary, then the formal deformation space M/W of X_{0} is canonically isomorphic to

Hom(T_{p}X_{0} ⊗ T_{p}X_{0}^{t}, Ĝ_{m}),

see [Ka, 2.1] for the proof. This means in particular that M is canonically endowed with the structure of a formal group. We will refer to this structure as the Serre–Tate group structure on M.

Let X_{M}/M be the universal formal deformation of X_{0}. We write R = W [[t_{1}, …, t_{g^{2}}]] and R_{0} = \overline{F}_{p}[[t_{1}, …, t_{g^{2}}]] = R/pR and choose an isomorphism M ≅ Spf(R). These notations will be fixed for the rest of the section. In the proof of our theorem, we will need a description of the cohomology group

H = H_{DR}^{1}(X_{M}/M) ≅ H_{cris}^{1}(X_{M} ⊗ R_{0}/R).

Because X_{0} is ordinary, this is an “ordinary Hodge-F -crystal on R_{0} of level ≤ 1”. This means that it is provided with the following data:

– A Hodge filtration of length at most 1, denoted by Fil.

– An integrable connection ∇: H ⟶ H ⊗_{R} Ω_{R/W}.

– A φ-linear Frobenius mapping F (φ)φ^{∗} for each lifting φ of the Frobenius of R_{0} to R that is compatible with σ on W.

These data satisfy a number of conditions for which the reader is referred to [DI, 1.4]. All we need is a theorem of Deligne and Illusie, [DI, 1.4.2], which provides us with a detailed description of such a crystal. They prove the following theorem.

2.2 Theorem (Deligne, Illusie). Suppose that H is an ordinary Hodge F -crystal on R_{0} of level ≤ 1. Then the following statements hold.

1. There is a unique U ⊂ H such that for every lifting φ of Frobenius, the map

F (φ)φ^{∗}|_{U}: U → U

is an isomorphism and such that H = U ⊕ Fil^{1} H.

2. There are R-bases a_{1}, …, a_{r} of U and b_{1}, …, b_{s} of Fil^{1} H with the properties that for every φ as above

∇a_{i} = 0 for 1 ≤ i ≤ r,

there exist η_{ij} ∈ Ω_{R/W}^{1} such that ∇b_{i} = ∑_{j=1}^{r} a_{j} ⊗ η_{ij} for 1 ≤ i ≤ s,

F (φ)φ^{∗}a_{i} = a_{i} for 1 ≤ i ≤ r, and

there are u_{ij}(φ) ∈ R such that F (φ)φ^{∗}b_{i} = pb_{i} + p ∑_{j=1}^{r} u_{ij}(φ)a_{j} for 1 ≤ i ≤ s.

Each η_{ij} is a closed form satisfying

φ^{∗}η_{ij} = pη_{ij} + pdu_{ij}(φ).

3. There exist formal power series τ_{ij} ∈ B[[t_{1}, …, t_{g^{2}}]] such that we have η_{ij} = dτ_{ij} for all i and j and

φ^{∗}τ_{ij} − pτ_{ij} − pu_{ij}(φ) = 0

for all i, j and φ. We have τ_{ij}(0) ∈ pW and if p ≠ 2, the series q_{ij} = exp(τ_{ij}) are defined, lie in R and satisfy q_{ij}(0) ≡ 1 mod pW.

4. Suppose that M = Spf(R) is the formal deformation space of an ordinary abelian variety X_{0}, that X_{M}/M is the universal family and that H = H_{DR}^{1}(X_{M}/M). Then r = s = g = dim X_{0} and 2.2.3 also holds for p = 2. Moreover, p and the elements q_{ij} − 1 form a regular system of parameters in R = W [[t_{1}, …, t_{g^{2}}]]. If φ is the lifting of Frobenius for which φ(q_{ij}) = q_{ij}^{p}, we have

F (φ)φ^{∗}b_{i} = pb_{i}

for all 1 ≤ i ≤ g.

Proof. Statement 2.2.1 follows from [DI, 1.3.2]. The statements 2.2.2 and 2.2.3 can be found in [DI, 1.4.2]. In the case that p ≠ 2, statement 2.2.4 follows from [DI, 1.4.7] because the map

gr ∇: T_{R/W} ⟶ Hom_{R}(Fil^{1} H, U)

induced by ∇ is the Kodaira–Spencer mapping, which is bijective because X_{M}/M is the universal family, see [DI, 2.1.3]. The case p = 2 follows from the appendix by Katz to [DI]. □

2.3 The q_{ij} ∈ R provided by the above theorem give “canonical coordinates” on M and hence an isomorphism

M ⟶^{≅} (Ĝ_{m})^{g^{2}},

see [DI, A3], thus inducing a new group structure on M. Note that although the above isomorphism depends on the choice of the a_{i} and b_{i}, the group structure does not. We will call it the Dwork group structure on M. The relation with the Serre–Tate group structure is given by the following theorem, which can be found in [DI, A3.1].

2.4 Theorem (Katz). The Serre–Tate group structure and the Dwork group structure on M are the same.

2.5 We come to the definition of a Tate class. It is shown in [BO, 2.4], that if X is a smooth proper W̄ -scheme, there is a canonical isomorphism

H_{DR}^{i}(X/W̄) ⊗ B̄ ≅ H_{cris}^{i}(X_{0}/W) ⊗ B̄

for each nonnegative integer i. Here X_{0} denotes the special fibre of X/W̄. This isomorphism induces an isomorphism

T_{DR}(X/W̄) ⊗ B̄ ≅ T_{cris}(X_{0}/B) ⊗ B̄

of tensor constructions in the DeRham and crystalline theories. A cohomology class ξ ∈ T_{DR}(X/W̄) ⊗ B̄ is called a Tate class if this isomorphism maps it to a Frobenius invariant element of T_{cris}(X_{0}/B) ⊂ T_{cris}(X_{0}/B) ⊗ B̄.

2.6 We are now ready to describe the main result of this section. It provides an analogue to the construction of Shimura varieties in the following way. Given an abelian variety X/W̄ with ordinary reduction and a Tate class ξ ∈ Fil^{0} T_{DR}(X/W̄), we will construct the largest irreducible formal subscheme N ⊂ M such that ξ can be continued to a horizontal section over N, lying in the middle step of the Hodge filtration everywhere. The proof of this theorem is an explicit calculation using the description 2.2 of the DeRham cohomology of the universal deformation of X. Theorem 2.2 will allow us to give an explicit description of a formal class ξ̃ providing a horizontal continuation of ξ. With formal class we mean to say that the power series expressing this class are allowed to have coefficients in B̄ rather than in W̄. Evidently, this class will not necessarily lie in Fil^{0} at every point of M.

The theorem of Deligne and Illusie provides us with a splitting of the Hodge filtration on H_{DR}^{1}(X_{M}/M) and hence with a decomposition ⊕T^{p,q} of the tensor T_{DR}(X_{M}/M), respecting the Hodge filtration. It is a rather surprising observation that the vanishing of the component in T^{−1,1} of ξ̃ at a point of M implies that the value of ξ̃ at that point lies in Fil^{0}. This leads us to consider the formal subscheme N of M defined by the vanishing of this component. We will see that on N, the class ξ̃ gives a real, and not just a formal, section of T_{DR}. At that point, it will also be evident that the vanishing of the said component of ξ̃ gives rise to a system of linear equations in the log(q_{ij}) with coefficients in W̄. The fact that ξ is a Tate class will be used to show that these coefficients actually lie in Z_{p}. This suffices to show that N is the subscheme we are looking for and that it is the translate of a formal subtorus of M over a torsion point.

We need a lemma before we can state the promised theorem.

2.7 Lemma. Choose an isomorphism M ≅ Spf(W [[t_{1}, …, t_{g^{2}}]]) such that x has coordinates t_{1} = ⋯ = t_{g^{2}} = 0. If η ∈ T_{DR}(X/W̄), then there is a unique element

η̃ ∈ T_{DR}(X_{M}/M) ⊗_{W [[t_{1},...,t_{g^{2}}]]} B̄[[t_{1}, …, t_{g^{2}}]] =: T_{DR}(X_{M}/M) ˆ⊗_{x} B̄

such that ∇η̃ = 0 and η̃_{X} = η. We will refer to η̃ as the formal horizontal continuation of η.

Proof. It follows from 2.2 that H_{DR}^{1}(X_{M}/M) ˆ⊗_{x} B̄ has a basis of horizontal sections, namely the basis consisting of the elements a_{i} and b_{i} − ∑_{j} τ_{ij}a_{j}. Because τ_{ij} = log(q_{ij}) and since the q_{ij} − 1 lie in the maximal ideal of W̄ [[t_{1}, …, t_{g^{2}}]], this basis gives us a horizontal basis β̃ of T_{DR}(X_{M}/M) ˆ⊗_{x} B̄. We can specialize the basis β̃ to obtain a basis β of T_{DR}(X/W̄)⊗B̄. The element η ∈ T_{DR}(X/W̄) can be written in a unique way as a linear combination of the elements of β and the same linear combination of the corresponding elements of β̃ is the promised horizontal continuation of η. The unicity of η̃ follows from the fact that in a power series ring an element with zero derivative is constant. □

2.8 Theorem. Let X be a principally polarized abelian scheme over W̄ with ordinary special fibre X_{0}/\overline{F}_{p}. Suppose that we are given a finite number of tensor constructions T_{DR}^{(k)}(X/W̄) and Tate classes ξ_{k} ∈ T_{DR}^{(k)}(X/W̄). Then there exists a largest formal subscheme N ⊂ MW̄ with the properties that A1- the point x ∈ M(W̄) corresponding to X lies in N (W̄),

A2- N is irreducible,

A3- after replacing the classes ξ_{k} by suitable multiples, the formal horizontal continuations ξ̃_{k} of the classes ξ_{k} restrict to elements χ_{k} ∈ Fil^{0} T_{DR}^{(k)}(X_{N} /N) and

A4- the polarization on X extends to a polarization on X_{N}.

In A3 and A4, X_{N} denotes the pull back of the universal deformation X_{M} → M to N ⊂ M. The formal subscheme N has the properties that

B1- N is the translate of a formal subtorus of M over a torsion point and

B2- the relative dimension of N over W̄ is equal to the dimension of the subvariety of Gr(g, H_{DR}^{1}(XB̄/B̄)) parametrizing the V ⊂ H_{DR}^{1}(XB̄/B̄) with the properties that

– V is totally isotropic for the pairing induced by the polarization and

– for all k, the class ξ_{k} ∈ T_{DR}^{(k)}(XB̄/B̄) lies in the middle step of the filtration induced by V.

2.9 Remark. Condition A3 in the theorem may require some explanation. Recall that we fixed coordinates t_{1}, …, t_{g^{2}} such that x corresponds to the origin. The condition says that the image of ξ̃_{k} in T_{DR}^{(k)}(X_{N} /N) ˆ⊗_{x} B̄ is equal to the image of χ_{k} ∈ T_{DR}^{(k)}(X_{N} /N) in this space. The phrasing of A3 is justified by the fact that the map

T_{DR}^{(k)}(X_{N} /N) ⟶ T_{DR}^{(k)}(X_{N} /N) ⊗ B̄[[t_{1}, …, t_{g^{2}}]] = T_{DR}^{(k)}(X_{N} /N) ˆ⊗_{x} B̄

is injective if N satisfies the conditions of the theorem. To see why this is the case, it suffices to show that the prime ideal I ⊂ W̄ [[t_{1}, …, t_{g^{2}}]] defining N satisfies

B̄[[t_{1}, …, t_{g^{2}}]]I ∩ W̄ [[t_{1}, …, t_{g^{2}}]] = I.

This equality holds because the map W̄ [[t_{1}, …, t_{g^{2}}]] ↪ B̄[[t_{1}, …, t_{g^{2}}]], being flat, satisfies the going down condition and the ideal B̄[[t_{1}, …, t_{g^{2}}]]I is contained in the maximal ideal of B̄[[t_{1}, …, t_{g^{2}}]] since x ∈ N (W̄).

2.10 We will first introduce some notation which will be used in the proof of this theorem.

Let A be a commutative ring with 1 of characteristic 0 and V a free A-module. If we have elements z_{1}, …, z_{g}, w_{1}, …, w_{g} ∈ V, if \underline{i} = (i_{1}, …, i_{n}), \underline{j} = (j_{1}, …, j_{n}) is a pair of multi-indices with 1 ≤ i_{ℓ}, j_{ℓ} ≤ g and if π ⊂ {1, …, 2n} is a subset with |π| = n, we define

\underline{z}^{⊗\underline{i}} := z_{i_{1}} ⊗ ⋯ ⊗ z_{i_{n}} ∈ V^{⊗n}

and

(\underline{z}^{⊗\underline{i}} ⊗ \underline{w}^{⊗\underline{j}})^{π} = (z_{i_{1}} ⊗ ⋯ ⊗ z_{i_{n}} ⊗ w_{j_{1}} ⊗ ⋯ ⊗ w_{j_{n}})^{π} := x_{1} ⊗ ⋯ ⊗ x_{2n} ∈ V^{⊗2n},

where the x_{i} are defined as follows. Let m_{1}, …, m_{n} and m′_{1}, …, m′_{n} be the unique ascending sequences of integers such that we have π = {m_{1}, …, m_{n}} and {1, …, 2n} = {m_{1}, …, m_{n}} ∪ {m′_{1}, …, m′_{n}}. We put

x_{m_{ℓ}} := z_{i_{ℓ}},

x_{m′_{ℓ}} := w_{j_{ℓ}}.

Let R = A[[X_{ij}]]_{i,j=1,...,g} and suppose that {a_{1}, …, a_{g}, b_{1}, …, b_{g}} is a basis of V. We identify V with the submodule 1 ⊗ V ⊂ R ⊗_{A} V and we put

c_{i} = b_{i} − ∑_{j} X_{ij}a_{j} ∈ R ⊗_{A} V.

Suppose that elements γ_{\underline{i},\underline{j}}^{π} ∈ A are given and let

ξ̃ = ∑_{\underline{i},\underline{j},π} γ_{\underline{i},\underline{j}}^{π}(\underline{a}^{⊗\underline{i}} ⊗ \underline{c}^{⊗\underline{j}})^{π} ∈ R ⊗_{A} V^{⊗2n}, (2.10.1)

where the summation runs over all n-tuples \underline{i} and \underline{j} and all π ⊂ \{1, …, 2n\} as above. We expand the right hand side of 2.10.1 and write it as Σ_{0}(\underline{X}) + ⋯ + Σ_{n}(\underline{X}), where Σ_{ℓ} is a linear combination, with coefficients in R, of tensors of ℓ b_{i} and 2n − ℓ a_{i}.

To make this precise, let J = \{1, …, n\} and, for each Δ ⊂ J and every multi index \underline{i}, put

e_{Δ,\underline{i}}(\underline{z}, \underline{w}) = x_{1} ⊗ ⋯ ⊗ x_{n}

where

x_{ℓ} = \{ \begin{matrix}z_{i_{ℓ}} & if & ℓ ∈ Δ \\ w_{i_{ℓ}} & if & ℓ ∉ Δ. \end{matrix}

We then have

Σ_{ℓ}(\underline{X}) = ∑_{{Δ⊂J \atop |Δ|=ℓ}} ∑_{\underline{i},\underline{j},π} γ_{\underline{i},\underline{j}}^{π}(\underline{a}^{⊗\underline{i}} ⊗ e_{Δ,\underline{j}}(\underline{b}, − ∑_{j} X_{ij}a_{j}))^{π}.

2.11 Proof of Theorem 2.8 Since the abelian scheme X is principally polarized, the spaces H_{DR}^{i}(X/W̄) are isomorphic to their duals. Because of condition A4, this isomorphism will extend over all of N if it exists, so we can assume that all the T_{DR}^{(k)}(X/W̄) are tensor products of H_{DR}^{i}(X/W̄)^{⊗j} with i, j > 0. As

H_{DR}^{i}(X/W̄) = ⋀^{i} H_{DR}^{1}(X/W̄) ⟶ H_{DR}^{1}(X/W̄)^{⊗i},

we can even assume that, for every k, T_{DR}^{(k)}(X/W̄) = H_{DR}^{1}(X/W̄)^{⊗2j_{k}}(j_{k}) with j_{k} > 0.

Only for the proof of statement B2 we have to consider all the classes ξ_{k} simultaneously, the rest of the theorem follows as soon as we prove it in the case where there is only one single class. This is what we will do first.

Let ξ be one of the classes ξ_{k} and let T_{DR}(X/W̄) = T_{DR}^{(k)}(X/W̄). We resort to the notations of Deligne and Illusie in [DI], cf. 2.2. Because τ_{ij} = log q_{ij}, the τ_{ij} can be evaluated in the point x ∈ M(W̄), giving τ_{ij}(x) ∈ B̄. It follows that

c_{i} = b_{i} − ∑_{j} τ_{ij}a_{j} ∈ H_{DR}^{1}(X_{M}/M) ˆ⊗_{x} B

and that these elements can be evaluated in x to give elements c_{i}(x) ∈ H_{DR}^{1}(XB̄/B̄). The set {a_{i}(x), c_{i}(x)}_{i=1,...,g} is a basis of H_{DR}^{1}(XB̄/B̄) because {a_{i}(x), b_{i}(x)}_{i=1,...,g} is one.

Suppose that A/W is the canonical lifting of X_{0} and let a ∈ M(W) be the corresponding point. Under the canonical isomorphism

H_{DR}^{1}(XB̄/B̄) ⟶^{∼} H_{cris}^{1}(X_{0}/W) ⊗ B̄ ⟶^{∼} H_{DR}^{1}(AB̄/B̄), (2.11.1)

the class a_{i}(x) is mapped to a_{i}(a) and c_{i}(x) is mapped to c_{i}(a) = b_{i}(a), see [BO, 2.10]. By the definition of a Tate class in T_{DR}(XB̄/B̄), the isomorphism

T_{DR}(XB̄/B̄) ⟶^{∼} T_{DR}(AB̄/B̄)

deduced from 2.11.1 maps Tate classes to Tate classes. The tensors in the a_{i}(a) and the b_{i}(a) form a basis of the space T_{DR}(AB̄/B̄) and the Tate classes in this space are precisely the classes that can be written as a linear combination with coefficients in Q_{p} of tensors consisting of the same numbers of a_{i}(a) and b_{i}(a). It follows that the Tate class ξ can be written as ξ̃(x), where

ξ̃ = ∑_{\underline{i},\underline{j},π} γ_{\underline{i},\underline{j}}^{π}(\underline{a}^{⊗\underline{i}} ⊗ \underline{c}^{⊗\underline{j}})^{π} ∈ T_{DR}(X_{M}/M) ˆ⊗_{x} B, (2.11.2)

with γ_{\underline{i},\underline{j}}^{π} ∈ Q_{p}. Here we use the notations introduced in 2.10. Because the classes a_{i} (i = 1, …, g) and the formal classes c_{i} (i = 1, …, g) are horizontal, it follows that ∇ξ̃ = 0. Therefore, ξ̃ is the formal horizontal continuation of ξ which was introduced in 2.7. By replacing ξ by a multiple, we can make sure that γ_{\underline{i},\underline{j}}^{π} ∈ Z_{p} for each \underline{i}, \underline{j} and π. From now on we will assume that such a replacement has taken place.

We adopt the notations of 2.10 and we write ξ̃ = Σ_{0}(\underline{τ}) + ⋯ + Σ_{n}(\underline{τ}). Recall that the Σ_{ℓ}(\underline{τ}) are elements of T_{DR}(X_{M}/M) ˆ⊗_{x} B̄, in fact even of T_{DR}(X_{M}/M) ˆ⊗_{x} B. For every N with property A3 of the theorem, Σ_{n−1}(\underline{τ}) must vanish on N. In the expression of this element with respect to the basis formed by the tensors in the a_{i} and the b_{j}, the coefficients are homogeneous linear polynomials in the τ_{ij} with coefficients in Z_{p}. Let \underline{Π}_{n−1}(\underline{X}) be the vector of expressions provided by these coefficients, written multiplicatively. This means that log(\underline{Π}_{n−1}(\underline{q})) is the vector of coefficients of Σ_{n−1}(\underline{τ}). The condition that Σ_{n−1}(\underline{τ}) should vanish on the N we are looking for implies that log(\underline{Π}_{n−1}(\underline{q})) should vanish on N and hence that every N satisfying A1–A3 is contained in the formal subscheme of M given by the equations

\underline{Π}_{n−1}(\underline{q}) = \underline{ζ}, (2.11.3)

where \underline{ζ} denotes a sequence of p-power roots of unity. The condition A1, that x should be a point of N, completely determines \underline{ζ}. Gaussian elimination shows that the formal subscheme of M described by the equations in 2.11.3 is the union of translates of a fixed formal subtorus of M over various torsion points. Two such translates are either equal or the sets of their W̄ -valued points are disjoint. Let N′ be the unique component of the formal subscheme described by 2.11.3 containing the point x. It follows from the above that every N ⊂ M satisfying A1–A3 is contained in N′. As we saw, N′ is the translate of a formal subtorus of M and hence it is smooth and irreducible. It is therefore evident that N′ satisfies A1 and A2, and we will show that it also verifies condition A3.

Consider the class

χ = ∑_{\underline{i},\underline{j},π} γ_{\underline{i},\underline{j}}^{π}(\underline{a}^{⊗\underline{i}} ⊗ \underline{b}^{⊗\underline{j}})^{π} ∈ T_{DR}(X_{M}/M),

where the γ_{\underline{i},\underline{j}}^{π} ∈ Z_{p} are the same as in Expression 2.11.2 for ξ̃. With the notations of 2.10, we have that χ = Σ_{n}(\underline{τ}). The structure theorem of Deligne and Illusie shows that

∇χ = −Σ_{n−1}(\underline{η}) ∈ T_{DR}(X_{M}/M) ⊗ Ω_{M/W} .

Since the coefficients of the tensors in the a_{i} and b_{j} in Σ_{n−1}(\underline{X}) are linear polynomials in the X_{ij}, the expression Σ_{n−1}(\underline{η}) has an obvious meaning. As Σ_{n−1}(\underline{τ}) vanishes on N′ and \underline{η} = d\underline{τ}, it follows that Σ_{n−1}(\underline{η}) = 0, so χ is horizontal on N′. The fact that

(Σ_{n}(\underline{τ}) + Σ_{n−1}(\underline{τ}) + … + Σ_{0}(\underline{τ}))(x) = ξ̃(x) ∈ Fil^{0} T_{DR}(XB̄/B̄)

implies that

Σ_{n−1}(\underline{τ})(x) = Σ_{n−2}(\underline{τ})(x) = … = Σ_{0}(\underline{τ})(x) = 0,

and hence that ξ̃(x) = Σ_{n}(\underline{τ})(x) = χ(x). Because N′ is smooth and both classes are horizontal on N′, it follows that the image of ξ̃ in T_{DR}(X_{N} /N) ˆ⊗_{x} B̄ is equal to the image of χ in this module. This shows that N′ satisfies A1–A3, and since we already saw that any formal subscheme satisfying these conditions is contained in it, it follows that N′ is the largest such subscheme.

It is a well-known fact, see for example [dJN, 3.7], that the condition A4 of the theorem determines a formal subtorus N′′ ⊂ M. The intersection N of N′ with N′′ is the translate of a formal subtorus of M over a torsion point and it has the properties A1–A4. By what was said above, it is also clear that N is the largest formal subscheme of M satisfying A1–A4. With the exception of statement B2, the theorem has now been proved.

For the proof of B2, we write H = H_{DR}^{1}(XB̄/B̄), T^{(k)} = T_{DR}^{(k)}(XB̄/B̄), ā_{i} = a_{i}(x) and c̄_{i} = c_{i}(x), for i = 1, …, g. Recall that {ā_{1}, …, ā_{g}, c̄_{1}, …, c̄_{g}} is a basis of H. The g^{2}- dimensional affine space U/B̄ with coordinates X_{ij} (i, j = 1, …, g) can be identified with an open subset of the Grassmann variety Gr(g, H) parametrizing g-dimensional linear subspaces of H by letting the subspace generated by

{c̄_{i} + ∑_{j} X_{ij}ā_{j}}_{i=1,...,g}

correspond to the point (X_{ij}). In analogy with the above we consider the vector bundle H × U → U with constant basis {a_{1}, …, a_{g}, c_{1}, …, c_{g}}. The sections b_{1}, …, b_{g} with b_{i} = c_{i} + ∑_{j} X_{ij}a_{j} generate the subbundle Fil ⊂ H × U which gives a g-dimensional subspace in each fibre. Writing

ξ_{k} = ∑_{\underline{i},\underline{j},π}(γ_{\underline{i},\underline{j}}^{π})_{k}(\underline{a}^{⊗\underline{i}} ⊗ \underline{c}^{⊗\underline{j}})^{π}(x),

we see that the intersection of U with the subvariety of Gr(g, H) determined by the conditions in B2 consists of the points y for which

– Fil_{y} ⊂ H is totally isotropic for the pairing induced by the polarization and

– for each k, the class

∑_{\underline{i},\underline{j},π}(γ_{\underline{i},\underline{j}}^{π})_{k}(\underline{a}^{⊗\underline{i}} ⊗ \underline{c}^{⊗\underline{j}})^{π}(y)

lies in the middle step of the filtration on T^{(k)} induced by Fil_{y} ⊂ H.

Application of the same arguments as we used above shows that this intersection is given by the same linear equations we found for N. □

2.12 Corollary. If Y = X_{y} for some point y ∈ N (W̄), then the image of each of the classes ξ_{k} under the canonical isomorphism

T_{DR}^{(k)}(XB̄/B̄) ⟶^{∼} T_{cris}^{(k)}(X_{0}/B) ⊗ B̄ ⟶^{∼} T_{DR}^{(k)}(YB̄/B̄)

is a Tate class lying in Fil^{0} T_{DR}^{(k)}(YB̄/B̄).

Proof. This follows immediately from Theorem 2.8 and its proof, because the image of ξ_{k} = (ξ̃_{k})_{x} under the above isomorphism is equal to (ξ̃_{k})_{y}, see Equation 2.11.1 and the remarks following it. □

3 Application to Shimura varieties

3.1 In this section, the results of Section 2 will be applied to the study of Shimura varieties. We will first recall some generalities on Shimura varieties and then we will construct a model M̃ in mixed characteristic of a Shimura variety M by embedding it in a moduli space of principally polarized abelian varieties and taking the Zariski closure of the image. Although this is a rather ad hoc method, we will nevertheless be able to investigate the completion of M̃ at a point x_{0} in positive characteristic corresponding to an ordinary abelian variety.

For this investigation we will use Theorem 2.8, which we can apply because a Shimura variety can be considered as the “largest” moduli space of abelian varieties over which a given Hodge class can be continued horizontally inside Fil^{0}. Since a Hodge class is also a Tate class, this interpretation shows that the components of the completion of M̃ at x_{0} are formal subschemes of the formal deformation space of the ordinary abelian variety corresponding to x_{0} to which we can apply Theorem 2.8. This implies that all these components are translates of formal subtori, so in particular that they are smooth. It also shows that the normalization of M̃ is smooth at x_{0}.

3.2 Let Y be a principally polarized abelian variety of dimension g over C and let V = H_{B}^{1}(Y (C), Q) and V_{Z} = H_{B}^{1}(Y (C), Z). Let G ⊂ GL(V) be the Mumford–Tate group of Y and K ⊂ G(A^{f}) a compact open subgroup. The Hodge structure on V is determined by a map h: S := R_{C/R}G_{m} → G_{R}. Let K_{∞} ⊂ G(R) be the centralizer of h. In this situation, we can form the Shimura variety M_{K}(C) = (K_{∞} × K)\G(A)/G(Q) as in [De1]. By loc. cit., M_{K}(C) is the set of complex points of a C-scheme M_{K}, which, again by [De1], can be defined over a number field E. In the sequel, we will always consider weakly canonical models of the Shimura varieties we encounter, see [De1, 3.13] for the definition and [De1, 5.9] for the proof that such a model exists in this case. If we choose K sufficiently small, then M_{K} is smooth and there exists a universal family of abelian varieties f_{K}: X_{K} ⟶ M_{K}. This family can be defined over a number field.

On the abelian variety Y there are various spaces of Hodge classes. These will be denoted C_{B}^{(k)}(Y, Q) ⊂ T_{B}^{(k)}(Y (C), Q) for the Betti cohomology and C_{DR}^{(k)}(Y/C) ⊂ T_{DR}^{(k)}(Y/C) for the DeRham cohomology. The spaces C_{DR}^{(k)} are obtained from the C_{B}^{(k)} by tensoring with C and applying the appropriate comparison theorem. The C_{B}^{(k)} and hence the C_{DR}^{(k)} are determined by the Mumford–Tate group G. Since it follows from [De2, 3.1] that G is the stabilizer of a one-dimensional space of Hodge classes in a finite direct sum ⨁_{k} T_{B}^{(k)}(Y (C), Q), the reader will convince himself that in what follows it suffices to consider a finite number of tensors an a finite number of Hodge classes ξ_{k} ∈ C_{B}^{(k)}(Y (C), Q) ⊂ T_{B}^{(k)}(Y (C), Q). For each k, we will write Ξ^{(k)} ⊂ T_{B}^{(k)}(Y (C), Q) for the subspace generated by ξ_{k}.

If K is chosen sufficiently small, the C_{B}^{(k)}(Y, Q) can be extended to spaces of global sections

C_{B}^{(k)}(X_{K}/M_{K}) ⊂ H^{0}(M_{K}(C), T^{(k)}(f_{K}^{an})_{∗}(Q)),

and the C_{DR}^{(k)}(Y/C) can be extended to spaces of horizontal sections

C_{DR}^{(k)}(X_{K}/M_{K}) ⊂ H^{0}(M_{K}, Fil^{0} T_{DR}^{(k)}(X_{K}/M_{K})).

In fact, M_{K}(C) is locally determined by the second property, or by the equivalent property for the Ξ^{(k)}, cf. [Mu1]. The spaces C_{DR}^{(k)}(X_{K}/M_{K}) can be defined over a number field.

If we consider the algebraic group CSp(V), and if K′ ⊂ CSp(V)(A^{f}) is a compact open subgroup, the Shimura varieties we obtain are the moduli spaces A_{K′}/Q of principally polarized abelian varieties with level K′ structure. If K′ is sufficiently small, A_{K′} is a fine moduli space and there exists a universal family over it. Choosing K′ still smaller, we can also make sure that for all K^{(3)} ⊂ K′′ ⊂ K′ the transition maps A_{K^{(3)}} → A_{K′′} are finite étale. From now on, we will assume that all our compact open subgroups are small enough for this to be the case. For each K′ there is a natural number n such that K′_{n} ⊂ K′, where K′_{n} denotes the congruence subgroup of CSp(V_{Z})(Ẑ) ⊂ CSp(V)(A^{f}) of the elements congruent to 1 modulo n. For such a number n there is a map A_{n} := A_{K′_{n}} ↠ A_{K′}. The Shimura variety A_{n} is the moduli space of principally polarized abelian varieties with symplectic level n structure, so by [MF], Appendix to Chapter 7, A, it admits a canonical model over Q, which in turn has a smooth model A_{n} over Z[1/n]. Given a K ⊂ G(A^{f}) as above and a number n such that K ⊂ K′_{n}, there is a map M_{K} → A_{n}. Because we stick with our convention to use weakly canonical models, this map can be defined over E. In particular, such a map exists if we take K to be the compact open subgroup K_{n} = K′_{n} ∩ G(A^{f}) of G(A^{f}), in which case we will denote it by φ_{n}. From now on, we will write M_{n} for M_{K_{n}} and f_{n}: X_{n} → M_{n} for the universal family X_{K_{n}}.

3.3 Lemma. For each prime number p, there exists a number n, prime to p, such that the map φ_{n}: M_{n} → A_{n} is a closed immersion.

Proof. For the duration of this proof, we will write T^{(k)}(V) := T_{B}^{(k)}(Y (C), Q). Recall that we are given one-dimensional subspaces Ξ^{(k)} ⊂ T^{(k)}(V) for each of a finite number of k. We will need the following description of the points of M_{n}(C) cf. [De1, 1.12].

The points of A_{n}(C) correspond to abelian varieties X over C endowed with a principal polarization and a full symplectic level n structure, i.e. a symplectic isomorphism X[n] ≅ V_{Z}/nV_{Z}. A point of M_{n}(C) corresponds to a principally polarized abelian variety X/C with level n structure and endowed with a G(Q)-class ῑ of isomorphisms V ⟶^{≅} H_{B}^{1}(X(C), Q) with the properties that

– letting h_{X} and h_{Y} be the maps determining the Hodge structures on H_{B}^{1}(X(C), Q) and V respectively, the maps ι^{−1}h_{X}ι and h_{Y}: S → G_{R}, are conjugate by an element of G(R) for any ι ∈ ῑ, and

– the level structure lifts to an isomorphism k: H_{B}^{1}(X(C), A^{f}) → V ⊗ A^{f} such that the induced map

V ⊗ A^{f} ⟶^{ι_{A^{f}}} H_{B}^{1}(X(C), A^{f}) ⟶^{k} V ⊗ A^{f}

belongs to G(A^{f}).

The first condition implies that for any element ι ∈ ῑ, the isomorphism T^{(k)}(ι): T^{(k)}(V) ⟶^{≅} T_{B}^{(k)}(X(C), Q) induced by ι maps Ξ^{(k)} into C_{B}^{(k)}(X, Q). The image only depends on ῑ. It follows that such a class ῑ can be determined by giving a 1-dimensional subspace of C_{B}^{(k)}(X, Q) for each k.

It follows from [De1, 1.15] that the map φ_{n} is finite for n sufficiently large. Let Δ_{n} ⊂ M_{n}×M_{n} be the diagonal and let Γ_{n} ⊂ M_{n}×M_{n} be the graph of the relation φ_{n}(x) = φ_{n}(y). We will consider the first projection R_{n} of \overline{Γ_{n}\Δ_{n}}. For each n, this is a closed subscheme of M_{n} and it follows from the above description of the points of M_{n}(C) that R_{n′} maps into R_{n} if n′ ≥ n. We have to show that, unless R_{n} is empty, we can increase its codimension by increasing n.

Suppose that R is an irreducible component of R_{n} of maximal dimension and let x ∈ R(C) be a generic point. The fact that x ∈ R_{n}(C) implies that, on the corresponding abelian variety X, there are several one dimensional subspaces of Hodge classes in some C_{B}^{(k)}(X(C), Q) endowing X with the structure of a point of M_{n}. Since φ_{n} is finite, there are only finitely many such subspaces. If we choose n′ > n, prime to p, so large that the intersections of all these spaces with T_{B}^{(k)}(X(C), Z) are different modulo n′, then R_{n′} does not have any irreducible components mapping onto R. Repeating this for all components of R_{n}, we find a number n′ such that the codimension of R_{n′} in M_{n′} is strictly larger than that of R_{n}. □

3.4 We intend to study the reduction of our Shimura variety at a prime of which we now fix the residue characteristic p > 0. By the lemma, there is an integer n, prime to p, such that M_{n} admits a closed immersion into the moduli space of principally polarized abelian varieties with symplectic level n structure. We fix such an n and a field E such that M_{n} has a weakly canonical model over Spec(E). We will assume that E is large enough for X_{n}, the level structure and the spaces C_{DR}^{(k)}(X_{n}/M_{n}) to be defined over E. As we noted above, the moduli space A_{n} has a model A_{n} over Spec(Z[1/n]). After the necessary base changes, this provides us with the diagram

M_{n}  ⟶ A_{n}  ⟶  A_{n}
↓  ↓  ↓
Spec(E) = Spec(E) ↪ Spec (O_{E}[\frac{1}{n}]) .

Note that f_{n}: X_{n} → M_{n} is the pull back to M_{n} of the universal family on A_{n}. After replacing E by an algebraic extension and M_{n} by a component, we can assume that M_{n} is geometrically connected. From now on, we will assume this to be the case.

We define M̃ to be the Zariski closure of M_{n} in A_{n}. By pulling back the universal family on A_{n} to M̃, we obtain an abelian scheme

f̃: X̃ ⟶ M̃.

The pull back of X̃ to the generic fibre M_{n} = M̃_{E} is canonically isomorphic to X_{n}.

3.5 Lemma. There are unique subspaces

C̃_{DR}^{(k)} ⊂ H^{0}(M̃, Fil^{0} T_{DR}^{(k)}(X̃/M̃))

restricting to C_{DR}^{(k)}(X_{n}/M_{n}) ∩ T_{DR}^{(k)}(X̃/M̃) on M_{n}.

Proof. Any section of C_{DR}^{(k)}(X_{n}/M_{n}) has a multiple that can be extended to a section of Fil^{0} T_{DR}^{(k)}(X̃/M̃) because T_{DR}^{(k)}(X_{n}/M_{n}), T_{DR}^{(k)}(X̃/M̃), Fil^{0} T_{DR}^{(k)}(X_{n}/M_{n}) and Fil^{0} T_{DR}^{(k)}(X̃/M̃) are locally free and their formation commutes with base change. The space C̃_{DR}^{(k)} is unique for the same reason. □

3.6 We are now in a position to apply Theorem 2.8 to the variety M_{n} and its model M̃. Assume that F ⊃ E is a number field with ring of integers O. Suppose that 𝔭 is a prime ideal of O and that the residue field F_{q} = O/𝔭 has characteristic p. Assume that we are given a point x ∈ M̃(O_{𝔭}) corresponding to an abelian scheme X/O_{𝔭} with ordinary reduction X_{0}/F_{q}. The reduction of x modulo 𝔭 will be denoted by x_{0} ∈ M̃(F_{q}). We will use the notations W, W̄, B and B̄ from 1.2 and choose an embedding of O_{𝔭} in W̄.

3.7 Theorem. Let M̂/W̄ be the base change to W̄ of the completion of M̃ in the point x_{0} ∈ M̃(F_{q}). There is a unique irreducible component

C ⊂ M̂ ⊂ MW̄ ≅ Hom(T_{p}X_{0} ⊗ T_{p}X_{0}^{t}, Ĝ_{m})

containing x. Moreover, C is a translate of a formal subtorus of M over a torsion point.

Proof. Let us first note that M̂ has a finite number of irreducible components. To show this, one has to prove that there is a finite extension R of Ô_{𝔭} such that the irreducible components over R of the completion of M̃ are irreducible over W̄. By the flatness of M̃, it suffices to show this for the generic fibre, and that case follows from [GD, 4.5.10].

For each k in our finite set of indices, let C̃_{DR}^{(k)} ⊂ H^{0}(M̃, T_{DR}^{(k)}(X̃/M̃)) be as in Lemma 3.5. We will first show that every formal subscheme C ⊂ MW̄ satisfying the conditions of the theorem has the properties that

1. X corresponds to a point of C,

2. C is irreducible,

3. dimW̄ C = dim M_{n},

4. the polarization on X extends to a polarization on X_{C} and

5. every class in each C^{(k)} := (C̃_{DR}^{(k)})_{x_{F}} ⊂ T_{DR}^{(k)}(X_{F} /F) has a multiple ξ such that the formal horizontal continuation ξ̃ of ξ is defined on C and lies in Fil^{0} T_{DR}^{(k)}(X_{C}/C).

In 3.7.4 and 3.7.5, X_{C} denotes the pull back to C of the universal formal deformation X_{M}/M of X_{0}. See Remark 2.9 on Property 3.7.5. The above properties will place us in a situation similar to that of theorem 2.8, and we will finish the proof by showing that C is equal to the translated subtorus N constructed there.

It is clear from the construction of C that it satisfies 3.7.1 and 3.7.2. Property 3.7.3 follows from the fact that M̃ is flat over O_{𝔭} and dim M_{n} = dimW̄ M̃W̄ = dimW̄ C. Property 3.7.4 holds because the polarization on X extends to X̃/M̃. This leaves us with point 3.7.5.

Let ξ ∈ C^{(k)}. It is the value at x_{F} of a horizontal section ξ_{n} ∈ Fil^{0} T_{DR}^{(k)}(X_{n}/M_{n}). It follows from Lemma 3.5 that after replacing ξ and ξ_{n} by multiples, there exists a class η ∈ H^{0}(M̃, Fil^{0} T_{DR}^{(k)}(X̃/M̃)) restricting to ξ_{n} on M_{n}. By base change, η gives an element of Fil^{0} T_{DR}^{(k)}(X_{C}/C), which, by abuse of notation, we continue to denote by η. Let ηB̄ be the image of η in T_{DR}^{(k)}(X_{C}/C) ˆ⊗_{x} B̄. We will show that ηB̄ is the restriction to CB̄ of the formal horizontal continuation ξ̃ of ξ, cf. 2.9.

We will write S for the local ring of A_{n} at x_{0} and T = S/I for the local ring of M̃ at this point. We have ŜW̄ ≅ W̄ [[t_{1}, …, t_{g(g+1)/2}]], this ring is a quotient of the coordinate ring W̄ [[t_{1}, …, t_{g^{2}}]] of MW̄. Let SB̄ and TB̄ be the local rings at xB̄ of (A_{n})B̄ and (M_{n})B̄ respectively. We have isomorphisms ŜB̄ ≅ B̄[[t_{1}, …, t_{g(g+1)/2}]] and

T̂B̄ = ŜB̄/ŜB̄I = T̂ ⊗Ŝ ŜB̄ ≅ T̂W̄ ⊗_{W̄ [[t_{1},...,t_{g^{2}}]]} B̄[[t_{1}, …, t_{g^{2}}]] = T̂W̄ ˆ⊗_{x} B̄.

Since M_{n} is smooth, T̂B̄ is isomorphic to B̄[[s_{1}, …, s_{d}]], where d = dim M_{n}.

The class

ηB̄ ∈ T_{DR}^{(k)}(X_{C}/C) ˆ⊗_{x} B̄ = T_{DR}^{(k)}(X̃T̂/T̂) ˆ⊗_{x} B̄ ≅ T_{DR}^{(k)}(X̃_{TB̄}/T̂B̄)

is horizontal. At xB̄, it specializes to ξ, so the fact that T̂B̄ is a power series ring implies that ηB̄ is equal to the image of ξ̃ in T_{DR}^{(k)}(X_{C}/C) ˆ⊗_{x} B̄. This completes the proof of the fact that C possesses the five properties listed above.

Once we show that C is determined by the above properties and that such a C is the translate of a subtorus over a torsion point, we have proved Theorem 3.7. Note that property 3.7.5 holds for every Hodge class on X, but by the remark made in 3.2, it will suffice to consider only the finite set of tensors T^{(k)} and the classes ξ_{k} generating the Ξ^{(k)} ⊂ T^{(k)}.

The classes ξ_{k} ∈ T_{DR}^{(k)}(XW̄ /W̄) are Hodge classes, so by a theorem of Blasius, see [Og, 4.1], they are Tate classes. We can therefore apply Theorem 2.8, which provides us with a translated subtorus N which contains C. Because dimW̄ N = dimW̄ C, and since both N and C are irreducible, it follows that N = C. This proves Theorem 3.7. □

3.8 Corollary. The components over W̄ of the completion of M̃ at a closed point corresponding to an ordinary abelian variety are smooth. The normalization of M̃ at such a point is smooth.

3.9 Corollary. Let S be the local ring of M̃ at x_{0} and let S′ be the normalization of S. The point xW̄ ∈ Spec(S)(W̄) extends to a point x′ ; W̄ ∈ Spec(S′)(W̄). There exists a point y ∈ Spec(S′)(W̄) with the properties that

– the reduction y_{0} ∈ Spec(S′)(\overline{F}_{p}) is equal to the reduction of x′ ; W̄ and

– the abelian variety corresponding to the image of y in M̃(W̄) is of CM-type.

Proofs. According to [GD, 7.8.3(vii)], S′ is a semi-local ring and its completion Ŝ′ is equal to the normalization of Ŝ. This proves Corollary 3.8. There is a one to one correspondence between the closed points of Spec(S′) and the components of Spec(Ŝ). To find the promised point y, it therefore suffices to find a point y ∈ M̂(W̄), on the same component as xW̄, such that the abelian variety corresponding to y is of CM-type.

By Theorem 3.7, the component of M̂ containing x is the translate of a formal subtorus of M over a torsion point, so it contains a torsion point y ∈ M(W̄). By [dJN, 3.5], the corresponding abelian variety is of CM-type. □

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Rutger Noot I. R. M. A. R. Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France

email: noot@univ-rennes1.fr