INTEGRAL MODELS FOR SHIMURA

VARIETIES OF ABELIAN TYPE.^{1}

Mark Kisin

Department of Mathematics, Harvard University (email: kisin@ math.harvard.edu)

Contents

Introduction. 1

1. Reductive groups and p-divisible groups.

1.1 Cocharacters and filtrations. 5 1.2 Review of 𝔖-modules. 7 1.3 Reductive groups and crystalline representations. 9 1.4 Application to p-divisible groups. 12 1.5 Deformation theory. 15

2. Integral canonical models of Hodge type 2.1 Shimura varieties. 21 2.2 Absolute Hodge cycles. 23 2.3 Integral models. 27

3. Integral canonical models of Abelian type 3.1 Twisting abelian varieties. 31 3.2 The action of G^{ad}(ℚ). 33 3.3 Connected Shimura varieties. 36 3.4 Integral models II. 41

References 47

Introduction

The problem of constructing smooth models of Shimura varieties was raised by Langlands in his Jugendtraum paper [La] as a (very small) part of his program to describe the zeta function of a Shimura variety in terms of automorphic Lfunctions. The point of this paper is to show the existence of reasonable models for many Shimura varieties, namely for those which are of abelian type.

To explain some of the properties these integral models ought to satisfy, consider a Shimura datum (G, X) and a Shimura variety Sh_{K}(G, X) attached to a compact open subgroup K ⊂ G(𝔸_{f}). We will assume that K has the form K_{p}K^{p} where K_{p} ⊂ G(ℚ_{p}), K^{p} ⊂ G(𝔸_{f}^{p}), and 𝔸_{f}^{p} denotes the finite adeles with trivial p-component. By results of Shimura, Deligne, Borovoi and Milne it is known that Sh_{K}(G, X) has

Typeset by AMS-TEX

The author was partially supported by NSF grant DMS-0017749000 1This version corrects some mistakes in the published version of the paper.

a model over a number field E = E(G, X) ⊂ ℂ. Let O be the ring of integers of E and v|p a prime of O. Langlands suggested that when K_{p} is hyperspecial [Ti, 3.8], the tower of E-schemes

Sh_{K_{p}}(G, X) = \underleftarrow{lim}_{K^{p}}Sh_{K_{p}K^{p}}(G, X)

ought to have a G(𝔸_{f}^{p})-equivariant extension to a tower of smooth O_{(v)}-schemes 𝒮_{K_{p}}(G, X). The condition on K_{p} means that K_{p} = G_{ℤ_{p}}(ℤ_{p}) for a reductive group G_{ℤ_{p}} over ℤ_{p} with generic fibre G. The example of Siegel modular varieties (G = GSp) already shows that, in general, it is not reasonable to ask for a smooth model if K_{p} is merely maximal compact. Hyperspecial subgroups exist if and only if G is quasi-split at p, and split over an unramified extension of ℚ_{p}.

Without a further condition the existence of 𝒮_{K_{p}}(G, X) is formally vacuous, since one could take 𝒮_{K_{p}}(G, X) = Sh_{K_{p}}(G, X). If the varieties Sh_{K}(G, X) are proper one can ask that 𝒮_{K_{p}}(G, X) be proper. In general, Milne [Mi 1] suggested that 𝒮_{K_{p}}(G, X) should satisfy a certain extension property. One of the consequences of this property is that the pro-scheme 𝒮_{K_{p}}(G, X) satisfies the valuative criterion for properness for discrete valuation rings of mixed characteristic. In those cases when one imagines that 𝒮_{K_{p}}(G, X) should be a moduli space for abelian varieties this is justified by the Néron-Ogg-Shafarevich criterion for good reduction of an abelian variety. In fact the extension property formulated by Milne is stronger, and has the added benefit of characterizing the pro-scheme 𝒮_{K_{p}}(G, X). We refer to §2.3 for a precise formulation. Milne termed a smooth G(𝔸_{f}^{p})-equivariant model of Sh_{K_{p}}(G, X) over O_{(v)}, satisfying the extension property, an integral canonical model.

To state our main theorem we recall that a Shimura datum is called of Hodge type if there is embedding (G, X) ↪ (GSp, S^{±}) into the Shimura datum for the symplectic group. Concretely, this means that Sh_{K}(G, X) has an interpretation as a moduli space for abelian varieties equipped with certain Hodge cycles. A datum (G, X) is called of abelian type if there is a datum of Hodge type (G_{2}, X_{2}) and a central isogeny G_{2}^{der} → G^{der} which induces an isomorphism (G_{2}^{ad}, X_{2}^{ad}) ⟶^{∼} (G^{ad}, X^{ad}). Then we prove the following^{2}

Theorem. Suppose that (G, X) is of abelian type and K_{p} ⊂ G(ℚ_{p}) hyperspecial. Let v|p be a prime of the ring of integers O ⊂ E(G, X). If p > 2 then the proscheme Sh_{K_{p}}(G, X) admits an extension to a G(𝔸_{f}^{p})-equivariant system of smooth O_{(v)}-schemes

𝒮_{K_{p}}(G, X) = \underleftarrow{lim}_{K^{p}}𝒮_{K_{p}K^{p}}(G, X)

which has the extension property.

When p = 2 we can still prove the theorem under some restrictions on (G, X). We refer the reader to the text for a precise statement in this case.

Our construction also shows that 𝒮_{K}(G, X) carries a vector bundle with an integrable connection, equipped with a family of Hodge tensors. Using this one can obtain natural integral models of the automorphic vector bundles on Sh_{K}(G, X) (see [Mi 2]). This ought to be useful in studying the arithmetic properties of algebraic automorphic forms on G.

2Note that the roles of G and G_{2} are reversed in §3.4 below.

An important class of Shimura varieties for applications consists of those of PEL type. For these Shimura varieties integral canonical models were constructed by Kottwitz [Ko], who moreover carried out a large part of Langlands’ program on mod p points in this case. Let us also remark that a proof in the case of abelian type has been claimed by Vasiu. See [Va 1], [Va 2], as well as the more recent [Va 4], [Va5].

We now describe some of the techniques of this paper in more detail. For an overview, the reader may also wish to consult [Ki 4], where we announced these results and sketched some of the arguments. Unsurprisingly, the theorem is first proved for Shimura varieties of Hodge type, and the case of abelian type is deduced from this. (The restrictions when p = 2 all arise in the first step.) Suppose that V is a ℚ-vector space equipped with a symplectic pairing ψ. We will write GSp = GSp(V, ψ). For a Shimura datum of Hodge type a symplectic embedding (G, X) ↪ (GSp, S^{±}) induces an embedding of Shimura varieties

(0.1)  Sh_{K}(G, X) ↪ Sh_{K′}(GSp, S^{±}),

where K′ = K′_{p}K^{′p} ⊂ GSp(𝔸_{f}) is a compact open subgroup which can be chosen so that K′_{p} is maximal compact. The interpretation of Sh_{K′}(GSp, S^{±}) as a moduli space for abelian varieties leads to a natural integral model 𝒮_{K′}(GSp, S^{±}). A natural candidate for 𝒮_{K}(G, X) is the normalization of 𝒮_{K}^{−}(G, X), the closure of Sh_{K}(G, X) in 𝒮_{K′}(GSp, S^{±}). The pro-scheme 𝒮_{K_{p}}(G, X) is then obviously G(𝔸_{f}^{p})- equivariant and the extension property follows from the corresponding property for 𝒮_{K′_{p}}(GSp, S^{±}). It remains to show that these schemes are smooth.

This construction is considered in the papers of both Moonen [Mo] and Vasiu [Va 1], and in fact goes back to Milne’s paper [Mi 1, 2.15]. To try and address the remaining problem of smoothness Moonen and Vasiu propose relating the complete local ring at a closed characteristic p point of 𝒮_{K}(G, X) to a deformation ring for p-divisible groups equipped with a collection of Tate cycles, introduced by Faltings [Fa, §7].^{3}

To apply Faltings’ construction one needs a kind of crystalline realization of the group G. To explain how we obtain this crystalline version of G, let k be a perfect field, K a finite, totally ramified extension of K_{0} = W (k)[1/p] and 𝒢 a p-divisible group over O_{K}. Let L = T_{p}𝒢^{∗} be the Tate module of the Cartier dual 𝒢^{∗} and 𝔻 the Dieudonné module of 𝒢 . Suppose that G_{ℤ_{p}} ⊂ GL(L) is a reductive group defined by a finite collection of Galois invariant tensors (s_{α}) ⊂ L^{⊗} (See (1.3) below for the precise definition of a tensor). The p-adic comparison isomorphism

L ⊗_{ℤ_{p}} B_{cris} ⟶^{∼} 𝔻 ⊗_{W (k)} B_{cris}

allows us to think of the s_{α} as being Frobenius invariant tensors in Fil^{0}𝔻[1/p]^{⊗}. The key which allows us to apply Faltings’ construction is the following Proposition (0.2). Suppose that p > 2 or that 𝒢^{∗} is connected. Then

3Of course Vasiu’s papers claim to carry out this proposal using his theory of well positioned tensors. Using the notation above, let (t_{α}) ∈ L[1/p]^{⊗} be a family of tensors fixed by G, and containing a perfect alternating pairing on L[1/p]. The family is said to be well positioned if for any faithfully flat ℤ_{p}-algebra R, and any finite free R-module M equipped with an isomorphism M[1/p] ⟶^{∼} L[1/p] ⊗_{ℤ_{p}} R such that (t_{α}) ⊂ M^{⊗}, the closure of G ⊗_{ℤ_{p}} R in GL(M) is a reductive group scheme. We make no use of this notion in this paper. Thus, for example, Proposition (0.2)

is a much more specific statement about the behavior of G-structures under the p-adic comparison isomorphism. In particular, it does not imply the existence of a family of well position tensors.

(1) s_{α} ∈ 𝔻^{⊗} (not just after inverting p).

(2) The s_{α} define a reductive subgroup of G_{W} ⊂ GL(𝔻).

The proposition is proved using the theory of our previous paper [Ki 1]. In fact we are able to prove something much more general: If L is a Galois stable lattice in a crystalline representation V, then the theory of [Ki 1] produces a natural, Frobenius stable lattice L_{cris} ⊂ D_{cris}(L). For any closed, reductive subgroup G_{ℤ_{p}} ⊂ GL(L) defined by a collection of Galois invariant tensors, there is a corresponding reductive subgroup G_{W} ⊂ GL(L_{cris}). The restriction in the proposition when p = 2 arises because we do not know whether L_{cris} may be identified with 𝔻 in this case.

With (0.2) in hand we can consider Faltings’ deformation ring. This produces a formally smooth quotient R_{G} of the universal deformation ring R of 𝒢_{0} = 𝒢 ⊗_{W (k)}k. Roughly speaking, R_{G} carries a p-divisible group whose crystal is equipped with Frobenius invariant tensors in degree 0 which deform the s_{α}. Faltings shows that R_{G} is universal for such deformations, at least for maps into formally smooth complete local W (k)-algebras. Using results related to (0.2) above, we are able to show that there is a version of this result for maps into rings of integers in finite extensions of W (k)[1/p].

Finally consider a closed point x ∈ 𝒮_{K}(G, X) in characteristic p, and let A_{x} be the corresponding abelian variety. Applying the above theory to the p-divisible group of A_{x}, with the (s_{α}) a suitable collection of Hodge tensors, allows us to identify the complete local ring at x with R_{G}.

To deduce the case of abelian type from the case of Hodge type we follow Deligne’s construction of canonical models for these varieties [De 3]. Suppose that K_{2,p} ⊂ G_{2}(ℚ_{p}) is hyperspecial, and let Sh_{K_{p}}(G, X)^{+} ⊂ Sh_{K_{p}}(G, X) denote the connected component corresponding to 1 ∈ G(𝔸_{f}). Then Sh_{K_{p}}(G, X)^{+} depends only on the derived group G^{der}, and one can obtain Sh_{K_{p}}(G, X)^{+} from Sh_{K_{2,p}}(G_{2}, X_{2})^{+} by dividing by the action of a profinite group Δ. Combining this with the (not completely trivial) combinatorics which govern the connected components of Sh_{K_{p}}(G, X) one can construct Sh_{K}(G, X) from Sh_{K_{2,p}}(G_{2}, X_{2}).

Moonen [Mo, §3], observed that one can make an analogous construction using the components of 𝒮_{K_{2,p}}(G_{2}, X_{2}). When p ∤ |Δ| he used a general result about quotients by groups of prime to p order to show that, if 𝒮_{K_{2,p}}(G_{2}, X_{2}) consists of smooth schemes, then so does the resulting tower 𝒮_{K}(G, X).

To prove this in general one needs to show that Δ acts freely on 𝒮_{K_{2,p}}(G_{2}, X_{2}). The description of the complex points of a Shimura variety as a double quotient shows that, roughly speaking, the elements of Δ arise from the adelic points of ker (G_{2}^{der} → G^{der}) and from the cokernel of G_{2}^{der}(ℚ) → G^{der}(ℚ). The action of the latter group is related to the action of G_{2}^{ad}(ℚ) on Sh(G_{2}, X_{2}) considered by Deligne. We give a moduli theoretic description of the action of G_{2}^{ad}(ℚ) which can be “reduced mod p”, and using it we are able to show that the action of Δ is always free.

Acknowledgment: It is a pleasure to thank A. Beilinson, O. Bültel, B. Conrad, O. Gabber, R. Kottwitz, D. Li, T. Lovering, K. Madapusi, B. Moonen, M. Nori and G. Pappas, R. Zhou, Y. Zhu for useful comments regarding various versions of this paper. We also thank the referee for a careful and perceptive reading of the paper.

§1 Reductive groups and p-divisible groups

(1.1) Cocharacters and filtrations: A connected reductive group scheme G over S is a flat S-group scheme whose fibres are geometrically connected and reductive. (This is what is called simply a reductive group scheme in [DG]).

By a reductive group over a base S we shall mean a flat S-group G together with a closed, connected reductive subgroup G^{0} ⊂ G such that G/G^{0} is finite étale. We call G^{0} the connected component of the identity of G. We will use this more general notion in §1.3-1.4 below, however the main applications of this paper will be concerned with connected reductive groups.

In this subsection we study the condition that a filtration on a representation of a connected reductive group G corresponds to a parabolic subgroup of G.

Lemma (1.1.1). Let S = Spec A be an affine scheme, and M a finite free Amodule with a decreasing, finite length filtration M^{•}, such that gr^{•}M is finite flat over A.

Let G ⊂ GL(M) be a closed, connected reductive subgroup over S. Denote by P ⊂ G the closed subgroup which respects the filtration M^{•} and U ⊂ P the closed subgroup which acts trivially on gr^{•}M.

Then the following conditions are equivalent.

(1) The filtration M^{•} admits a splitting such that the corresponding cocharacter μ : 𝔾_{m} → GL(M) factors through G.

(2) P ⊂ G is a parabolic subgroup, U ⊂ P is its unipotent radical, and the grading on gr^{•}M is induced by a cocharacter ν : 𝔾_{m} → P/U.

Proof. (1) ⟹ (2): Note that μ factors through P. Hence once we know that P and U are flat then we may take ν to be the composite 𝔾_{m} →^{μ} P → P/U.

Thus it suffices to prove that P is parabolic with unipotent radical U. For this we may work locally in the étale topology of S, and assume that G admits a split maximal torus T ⟶^{∼} 𝔾_{m}^{r} which contains the image of μ. Let 𝔱 and 𝔤 denote the Lie algebras of T and G respectively, and consider the decomposition

𝔤 = 𝔱 ⊕_{α∈Φ} 𝔤^{α}

of 𝔤 into root spaces under the adjoint action of T [DG, I, 4.7.3]. Let Φ′ ⊂ Φ (resp. Φ′′ ⊂ Φ) be the set of roots α such that α∘μ is a non-negative (resp. positive) power of the tautological character of 𝔾_{m}. Set 𝔭 = 𝔱 ⊕_{α∈Φ′} 𝔤^{α} and 𝔲 = ⊕_{α∈Φ′′}𝔤^{α}.

By [DG, XXVI, 1.4] 𝔭 is the Lie algebra of a parabolic subgroup P′ ⊂ G (Φ′ satisfies the condition (iii) of loc. cit). The action of 𝔭 on M respects the filtration so we have P′ ⊂ P. Since P′ is flat over S, to show that P′ is open in P it suffices to check fibre by fibre, so we may assume that S = k an algebraically closed field. Then an element of s ∈ 𝔤 acts on M respecting the filtration if and only if it is contained in 𝔭. Hence Lie P = 𝔭. To see this write s = ∑_{β∈ℤ} s_{β} where μ has weight β on s_{β} ∈ 𝔤. If β_{0} is the smallest β such that s_{β} ≠ 0, choose m ∈ M of some weight j under the action on μ, and such that s_{β_{0}}m ≠ 0. Then

μ(z) · sm = (adμ(z))(s)μ(z)m = ∑_{β∈ℤ} z^{β+j}s_{β}m.

Hence the component of sm in degree β_{0} + j is s_{β_{0}}m ≠ 0 and s preserves the filtration if and only if β_{0} ≥ 0.

It follows that P′ is open in P. As P′ has connected fibres, this implies that P′ is normal in P, and hence that P = P′. Similarly 𝔲 is the Lie algebra of a unipotent subgroup U′ ⊂ G by [DG, XXII 5.11.3] and an argument similar to that above shows that U′ is open in U. However, the fibres of U are clearly unipotent, so we have U = U′.

(2) ⟹ (1). The group P/U is connected reductive, and since S is affine, there exists a Levi subgroup L ⊂ P, which maps isomorphically to P/U [DG, XXVI, 2.3]. Hence we may take μ to be ν composed with the inclusion P/U ⟶^{∼} L ⊂ P. □

(1.1.2) If M and M^{•} are as in (1.1.1) and G ↪ GL(M) a connected reductive subgroup, then we say a filtration M^{•} is G-split if it satisfies the equivalent conditions of (1.1.1). More generally, if G ↪ GL(M) is a reductive subgroup, we say that M^{•} is G-split if it is G^{0}-split, where G^{0} ⊂ G denotes the connected component of the identity.

When the above embedding arises from an A-linear fibre functor on a Tannakian category, the lemma is closely related to [Sa, IV, 2.2.5].

Lemma (1.1.3). With the notation of (1.1.1), suppose that A is a field of characteristic 0. Then M^{•} is G-split if and only if ⟨M⟩^{⊗}, the Tannakian category of G-representations generated by M, admits a filtration which induces the given filtration on M.

Proof. We refer to [Sa, IV §2] for the notion of a filtration on a Tannakian category. If (1.1.1)(1) holds then μ even equips ⟨M⟩^{⊗} with a grading. Conversely, a filtration on ⟨M⟩^{⊗} admits a splitting by a theorem of Deligne [Sa, IV §2.4]. If such a filtration restricts to the given filtration on M, then the splitting gives the required cocharacter μ. □

Proposition (1.1.4). Suppose that A is a discrete valuation ring with field of fractions K, and that G ↪ GL(M) and M^{•} are as above.

Then M^{•} is G-split if and only if the induced filtration on M ⊗_{A} K is G|_{K}-split.

Proof. If M^{•} is G-split then so is the filtration (M ⊗_{A} K)^{•} on M ⊗_{A} K. We prove the converse.

Let P_{K} be the subgroup of G_{K} which respects the filtration on M ⊗_{A} K, and U_{K} ⊂ P_{K} the subgroup which acts trivially on gr^{•}M ⊗_{A} K. If (M ⊗_{A} K)^{•} is G|_{K}- split, then P_{K} is parabolic in G|_{K}, U_{K} ⊂ P_{K} is the unipotent radical of P_{K} and the action of P_{K}/U_{K} on gr^{•}M ⊗_{A} K is given by a cocharacter ν_{K} : 𝔾_{m} → P_{K}/U_{K}.

By [DG, XXVI, 3.5] P_{K} extends uniquely to a parabolic subgroup P′ ⊂ G. The unipotent radical of P′ is a smooth subgroup U′ extending U_{K}. Since P_{K} and U_{K} are dense in P′ and U′ respectively, P′ leaves stable the filtration on M and U′ acts trivially on gr^{•}M.

The cocharacter ν : 𝔾_{m} → GL(gr^{•}M) factors through P′/U′ since this is true for its restriction to K. As in the proof of (1.1.1), we may identify P′/U′ with a Levi subgroup of P′ and lift this cocharacter to a cocharacter μ : 𝔾_{m} → P′ which defines a splitting of the filtration on M^{•}. This shows that M^{•} is G-split (and hence that P′ = P and U′ = U). □

Proposition (1.1.5). Let S = Spec A, be an affine scheme, M a finite free Amodule and G ⊂ GL(M) a closed, connected reductive subgroup. For any A-algebra B let G-Fil(B) denote the set of G_{B} = G ⊗_{A} B-split filtrations on M_{B} = M_{A} ⊗_{A} B. Then the functor B ↦ G-Fil(B) is representable by a smooth proper A-scheme.

Proof. Let Par denote the functor which assigns to an A-algebra B the set of parabolic subgroups P_{B} ⊂ G_{B}. Denote by P̃ar the functor which assigns to B the set of pairs (P_{B}, ν), where P_{B} ⊂ G_{B} is a parabolic subgroup with unipotent radical U_{B}, and ν : 𝔾_{m} → P_{B}/U_{B} is a cocharacter. Then Par is smooth and representable over A [DG, XVI, 3.5] and the morphism P̃ar → Par is smooth and relatively representable by [DG, XI, 4.2]. Hence P̃ar is smooth and representable.

We have a map G-Fil → P̃ar which associates to a filtration M_{B}^{•} the parabolic respecting the filtration, together with the cocharacter giving the grading on gr^{•}M_{B}. We will show that G-Fil is represented by a closed and open subscheme of P̃ar. The properness then follows from (1.1.4).

Consider an A-algebra B, and a parabolic P_{B} of G_{B} with unipotent radical U_{B}, together with a cocharacter ν : 𝔾_{m} → P_{B}/U_{B}. Fix a Levi subgroup L_{B} ⊂ P_{B} mapping isomorphically to P_{B}/U_{B}, and let μ : 𝔾_{m} → L_{B} be the cocharacter induced by ν. Let 𝔲 = Lie U_{B}, 𝔭 = Lie P_{B} and decompose 𝔲 = ⊕_{n∈ℤ}𝔲^{n} and 𝔭 = ⊕_{n∈ℤ}𝔭^{n} under the adjoint action of μ, so that μ has weight n on 𝔲^{n} and 𝔭^{n}. Let G-Fil′(B) ⊂ P̃ar(B) consist of those pairs (P_{B}, ν) such that 𝔲^{n} = 0 for n ⩽ 0 and 𝔭^{n} = 0 for n < 0. Any two choices of L_{B} differ by conjugation by an element of U_{B}, so this condition does not depend on the choice of L_{B} and G-Fil′ is well defined. It is clearly represented by an open and closed subscheme of P̃ar.

We saw in the proof of (1.1.1) that the functor G-Fil → P̃ar factors through G-Fil′. Conversely, given a point (P_{B}, ν) of G-Fil′(B) we may choose a Levi subgroup L_{B} ⊂ P_{B} and consider the filtration given by μ. Let P′_{B} and U′_{B} denote the parabolic and unipotent subgroups defined by μ, and denote their Lie algebras by 𝔭′ and 𝔲′ respectively. To see that P′_{B} = P_{B}, we may work étale locally and assume that G_{B} contains a split maximal torus T_{B} ⊂ P_{B} with Lie algebra 𝔱 such that μ factors through T_{B}. Decompose 𝔤 = 𝔱 ⊕_{α∈Φ} 𝔤^{α} under the adjoint action of T. The proof of (1.1.1) shows that 𝔭 ⊂ 𝔭′ and 𝔲 ⊂ 𝔲′. If 𝔭 ⊊ 𝔭′, there is a root space 𝔤^{α} ⊂ 𝔭′ which is not contained in 𝔭. But then 𝔤^{−α} ⊂ 𝔲 [DG, XXII, 5.11.3] and so 𝔤^{−α} ⊂ 𝔲′ whence 𝔤^{α} ⊈ 𝔭′, a contradiction.

It follows that P′_{B} = P_{B} and U′_{B} = U_{B}. In particular these groups are independent of the choice of L_{B} and the filtration depends only on ν and not on μ. This gives a map G-Fil′ → G-Fil which is a section to the inclusion G-Fil → G-Fil′. Hence G-Fil = G-Fil′ is open and closed in P̃ar. □

(1.1.6) In the situation of (1.1.1) let μ : 𝔾_{m} → G be a cocharacter inducing the given filtration M^{•}. The centralizer L of μ in G is a Levi subgroup with Lie algebra 𝔱 ⊕_{α∈Φ′\Φ′′} 𝔤_{α} [DG, XXVI, 5.11.3]. The parabolic subgroup P′ defined by μ^{−1} satisfies P′ ∩ P = L. We will refer to the unipotent radical U′ of P′ as the opposite unipotent defined by μ. We have Lie G = Lie P ⊕Lie U′ [DG, XXVI, 4.3.2].

(1.2) Review of 𝔖-modules: We now recall some of the results of [Ki 1] regarding the classification of crystalline representations and p-divisible groups.

Let k be a perfect field of characteristic p, W = W (k) its ring of Witt vectors and K_{0} = W (k)[1/p]. Let K be a finite totally ramified extension of K_{0}, and O_{K} its ring of integers. Fix an algebraic closure K̄ of K, and set G_{K} = Gal(K̄/K).

We denote by Rep_{G_{K}}^{cris} the category of crystalline G_{K}-representations, and by Rep_{G_{K}}^{cris∘} the category of G_{K}-stable ℤ_{p}-lattices spanning a representation in Rep_{G_{K}}^{cris}. For V a crystalline representation, recall Fontaine’s functors

D_{cris}(V) = (B_{cris} ⊗_{ℚ_{p}} V)^{G_{K}} and D_{dR}(V) = (B_{dR} ⊗_{ℚ_{p}} V)^{G_{K}} .

Fix a uniformiser π ∈ K, and let E(u) ∈ W (k)[u] be the Eisenstein polynomial for π. We set 𝔖 = W [[u]] equipped with a Frobenius φ which acts as the usual Frobenius on W and sends u to u^{p}.

Let Mod_{/𝔖}^{φ} denote the category of finite free 𝔖-modules 𝔐 equipped with a Frobenius semi-linear isomorphism^{7}

1 ⊗ φ : φ^{∗}(𝔐)[1/E(u)] ⟶^{∼} 𝔐[1/E(u)].

Note that this definition differs slightly from that of [Ki 1], where we insisted that the above map be induced by a map φ^{∗}(𝔐) → 𝔐. This is related to the fact that in loc. cit we considered crystalline representations with Hodge-Tate weights ⩽ 0, whereas here we will allow arbitrary Hodge-Tate weights.

For i ∈ ℤ we set

Fil^{i}φ^{∗}(𝔐) = (1 ⊗ φ)^{−1}(E(u)^{i}𝔐) ∩ φ^{∗}(𝔐).

Let O_{E} denote the p-adic completion of 𝔖_{(p)}, and denote by Mod_{/O_{E}}^{φ} the category of finite free O_{E}-modules M equipped with an isomorphism φ^{∗}(M) ⟶^{∼} M. We have a functor

Mod_{/𝔖}^{φ} → Mod_{/O_{E}}^{φ}; 𝔐 ↦ O_{E} ⊗_{𝔖} 𝔐.

Theorem (1.2.1). There exists a fully faithful tensor functor

𝔐 : Rep_{G_{K}}^{cris∘} → Mod_{/𝔖}^{φ},

which is compatible with formation of symmetric and exterior powers. If L is in Rep_{G_{K}}^{cris∘}, V = L ⊗_{ℤ_{p}} ℚ_{p}, and 𝔐 = 𝔐(L), then

(1) There are canonical isomorphisms

D_{cris}(V) ⟶^{∼} 𝔐/u𝔐[1/p] and D_{dR}(V) ⟶^{∼} φ^{∗}(𝔐) ⊗_{𝔖} K,

where the map 𝔖 → K is given by u ↦ π. The first isomorphism is compatible with Frobenius and the second maps Fil^{i}φ^{∗}(𝔐)⊗_{W} K_{0} onto Fil^{i}D_{dR}(V) for i ∈ ℤ.

(2) There is a canonical isomorphism

OÊ^{ur} ⊗_{ℤ_{p}} L ⟶^{∼} OÊ^{ur} ⊗_{𝔖} 𝔐,

where OÊ^{ur} is a certain faithfully flat, and formally étale O_{E}-algebra.

(3) If k′/k is an algebraic extension of fields, then there exists a canonical φequivariant isomorphism

𝔐(L|_{G_{K′}}) ⟶^{∼} 𝔐(L) ⊗_{𝔖} 𝔖′

where 𝔖′ = W (k′)[[u]] and G_{K′} = Gal(K̄ · W (k′)[1/p]/K · W (k′)[1/p]).

7We will need to consider various rings equipped with a fixed lift of Frobenius. Such a lift will be denoted simply by φ, except if we need to consider different lifts of Frobenius on the same ring, in which case this will of course be made clear in the notation. A Frobenius on a module over such a ring is always semi-linear and will be denoted by the same symbol as the Frobenius lift on the ring.

Proof. This is an exercise in assembling the results of [Ki 1]:

Let Mod_{/𝔖}^{φ,+} denote the full subcategory of Mod_{/𝔖}^{φ} such that the map 1 ⊗ φ induces a map φ^{∗}(𝔐) → 𝔐. Let K_{∞} = K( \sqrt[p^{n}]{π})_{n≥1} be the field obtained from K by adjoining a compatible sequence of p-power roots of π. Write G_{K_{∞}} = Gal(K̄/K_{∞}), and denote by Rep_{G_{K∞}}^{∘} the category of continuous representations of G_{K_{∞}} on finite free ℤ_{p}-modules.

Then we have functors

(1.2.2)  Mod_{/𝔖}^{φ,+} ⟶^{𝔐↦O_{E}⊗_{𝔖}𝔐} Mod_{/O_{E}}^{φ} ⟶^{∼} Rep_{G_{K∞}}^{∘} ,

where the second functor is the equivalence of [Fo, A.1.2.7], and the first functor is fully faithful by [Ki 1, 2.1.12] (see also [Ki 2, E.4]). Denote by Rep_{G_{K∞}}^{fh,∘} the essential image of the composite of (1.2.2).

Let Rep_{cris}^{−,∘} denote the category of G_{K}-stable ℤ_{p}-lattices in crystalline representations with Hodge-Tate weights ⩽ 0. By [Ki 1, 2.1.14] the functor

Rep_{cris}^{−,∘} → Rep_{G_{K∞}}^{∘}; L ↦ L|_{G_{K∞}}

is fully faithful. If L is in Rep_{cris}^{−∘} then [Ki 1, 2.1.5] implies that L[1/p] contains a G_{K_{∞}}-stable lattice in Rep_{G_{K∞}}^{fh,∘} , and hence L|_{G_{K∞}} itself is in Rep_{G_{K∞}}^{fh,∘} by [Ki 1, 2.1.15].

We define 𝔐 as the composite

𝔐 : Rep_{cris}^{−,∘} → Rep_{G_{K∞}}^{fh,∘} ⟶^{∼}_{(1.2.2)} Mod_{/𝔖}^{φ,+}.

To extend this to a functor on Rep_{G_{K}}^{cris∘}, let 𝔖(1) (resp. 𝔖(−1)) in Mod_{/𝔖}^{φ} be the object with underlying 𝔖-module equal to 𝔖 and φ given by sending 1 to E(0)/pE(u) (resp. pE(u)/E(0)). Then 𝔐(ℤ_{p}(−1)) = 𝔖(−1), for example by [Ki 1, 2.1.5], and for any L in Rep_{cris}^{∘} we set 𝔐(L) = 𝔐(L(−m)) ⊗ 𝔖(1)^{⊗m} for m a positive integer such that L(−m)[1/p] has all Hodge-Tate weights ⩽ 0.

(1) now follows from [Ki 1, 2.1.5] and the construction of the quasi-inverse functors D and M in [Ki 1, §1.2]. (2) follows from the construction of the equivalence in [Fo, A.1]. This also implies the compatibility with symmetric tensors: For n ≥ 1, there is a natural map

Sym^{n}(𝔐(L)) → 𝔐(Sym^{n}(L)).

Using (2) we see that this map is an isomorphism after tensoring by ⊗_{𝔖}O_{E}. Hence it is an isomorphism by [Ki 1, 2.1.9]. The argument for exterior powers is similar.

Finally (3) is easily seen directly from the construction of the functor 𝔐. □

(1.3) Reductive groups and crystalline representations: Let L be a G_{K}- stable lattice in a crystalline representation, and G ⊂ GL(L) a reductive group such that the G_{K}-action on L factors through G(ℤ_{p}). We will apply the theory of the previous section to produce a “crystalline realization” of G.

(1.3.1) Let R be a Noetherian ring and M a finite free R-module. We begin by establishing criteria under which a closed R-flat subgroup G ⊂ GL(M) can be described as the scheme theoretic stabilizer of a collection of tensors.

It will be convenient to denote simply by M^{⊗} the direct sum of all the R-modules which can be formed from M using the operations of taking duals, tensor products, symmetric powers and exterior powers. Note that M^{⊗} ⟶^{∼} M^{∗⊗} so that a tensor in the left hand side may be regarded in the right hand side.

If (s_{α}) ⊂ M^{⊗}, and G ⊂ GL(M) is the pointwise stabilizer of the s_{α}, we say G is the group defined by the tensors s_{α}.

Proposition (1.3.2). Suppose that R is a discrete valuation ring of mixed characteristic, and let G ⊂ GL(M) be a closed R-flat subgroup whose generic fiber is reductive. Then G is defined by a finite collection of tensors (s_{α}) ⊂ M^{⊗}.

Proof. The proof is similar to that of [De 2, Prop. 3.1].

Let N be an R-module equipped with an action of GL(M) and N′ ⊂ N a submodule such that N′/N is R-flat. If (s_{α}) ⊂ N′ then the subgroup of GL(M) fixing the s_{α} is the same as the subgroup fixing the s_{α} considered as elements of N.

Let O_{GL} denote the Hopf algebra of GL(M). For any finite projective R-module W with an action of GL(M), let W_{0} denote W with the trivial GL(M) action.

We have the inclusion of R-schemes GL(M) ⊂ End (M), which is fibre by fibre dense. Thus

O_{GL} = \underrightarrow{lim}_{n}Sym^{•}(M ⊗ M_{0}^{∗}) ⊗ (det M)^{−n},

with the transition maps being given by multiplication by det ⊗δ^{−1}, where det ∈ Sym^{•}(M ⊗ M_{0}^{∗}) is the determinant and δ ∈ det M is some fixed basis vector. The transition maps in the direct limit remain injective after any base change R → R′. Hence each term in the limit is a direct summand in the next and it suffices to find tensors (s_{α}) ⊂ O_{GL} defining G.

For any finite projective R-module W with an action of GL(M), the O_{GL}- comodule structure on W gives a GL(M)-equivariant map W → W_{0} ⊗_{R} O_{GL}. This map is injective and its cokernel is a direct summand, a section being induced by the identity section O_{GL} → R. Hence it suffices to find elements defining G in any representation of GL(M) on a finite projective R-module.

Now let I ⊂ O_{GL} denote the ideal of G. Then G is the scheme theoretic stabilizer of I. Let W ⊂ O_{GL} be a finite rank, GL(M)-stable, saturated R-submodule such that W ∩ I contains a set of generators of I. Then G is the stabilizer of W ∩ I ⊂ W. If r = rk_{R}W ∩ I then L = ∧^{r}(W ∩ I) ⊂ ∧^{r}W is a line, and G is the stabilizer of L.

Since G has reductive generic fibre the quotient map (∧^{r}W)^{∗} → L^{∗} has a Gequivariant splitting over the generic point η ∈ Spec R. Hence there exists a G-stable line L̃^{∗} ⊂ (∧^{r}W)^{∗} which maps isomorphically to L^{∗} over η. Now G acts trivially on L ⊗_{R} L̃^{∗} as this is true over η, and the stabilizer of L ⊗_{R} L̃^{∗} ⊂ (∧^{r}W) ⊗_{R} (∧^{r}W)^{∗} is equal to G. □

(1.3.3) Let L be a G_{K}-stable lattice in a crystalline representation and G ⊂ GL(L) a reductive group. Then by (1.3.2) G is defined by a finite collection of tensors (s_{α}) ⊂ L^{⊗}, and the G_{K}-action on L factors through G(ℤ_{p}) if and only if these tensors are G_{K}-invariant.

If this is the case, we may view the tensors s_{α} as morphisms s_{α} : 1 → L^{⊗} in Rep_{G_{K}}^{cris∘}. Applying the functor 𝔐 of the theorem, we obtain morphisms s̃_{α} : 1 → 𝔐(L)^{⊗} in Mod_{/𝔖}^{φ}.

Proposition (1.3.4). Let L be in Rep_{G_{K}}^{cris∘} and G ⊂ GL(L) a reductive ℤ_{p}-subgroup defined by a finite collection of G_{K}-invariant tensors (s_{α}) ⊂ L^{⊗}. If 𝔐 = 𝔐(L), then (s̃_{α}) ⊂ 𝔐^{⊗} defines a reductive subgroup of GL(𝔐).

If k is separably closed, then there is an 𝔖-linear isomorphism. 𝔐 ⟶^{∼} L ⊗_{ℤ_{p}} 𝔖 which takes the tensor s̃_{α} to s_{α}. In particular, the subgroup G_{𝔖} ⊂ GL(𝔐) defined by (s̃_{α}) is isomorphic to G ×_{Spec ℤ_{p}} Spec 𝔖.

Proof. Using the compatibility (1.2.1)(3), it suffices to prove the proposition when k is separably closed. Moreover the second statement implies the first.

Now suppose that k is separably closed, set 𝔐′ = L ⊗_{ℤ_{p}} 𝔖, and let P ⊂ \underline{Hom}_{𝔖}(𝔐, 𝔐′) be the subscheme of isomorphisms between 𝔐 and 𝔐′ which take s̃_{α} to s_{α}. The fibres of P are either empty or a torsor under G. We claim that P is a G-torsor. That is, P is flat over 𝔖 with non-empty fibres. The claim implies the proposition since a torsor under a reductive group is étale locally trivial, while the ring 𝔖 is strictly henselian as k is separably closed, so any G torsor over 𝔖 is trivial.

To prove the claim we proceed in several steps. For R a 𝔖-algebra, we set P_{R} = P ×_{Spec 𝔖} Spec R.

Step 1: P_{𝔖_{(p)}} is a G-torsor. Since OÊ^{ur} is faithfully flat over O_{E} and O_{E} is faithfully flat over 𝔖_{(p)}, it suffices to show that P_{O_{E^{ur}}} is a G-torsor. However the isomorphism in (1.2.1)(2) shows that P_{O_{E^{ur}}} is a trivial G-torsor.

Step 2: P_{K_{0}} is a G-torsor, where we regard K_{0} as a 𝔖-algebra via u ↦ 0. This follows from (1.2.1)(1), which implies the existence of a canonical isomorphism

B_{dR} ⊗_{ℤ_{p}} L ⟶^{∼} B_{dR} ⊗_{W} 𝔐/u𝔐.

Step 3: P_{𝔖[1/pu]} is a G-torsor. Let U ⊂ Spec 𝔖[1/up] denote the maximal open subset over which P is flat with non-empty fibres. By Step 1, we know this subset is non-empty, since it contains the generic point. In particular, the complement of U in Spec 𝔖[1/up] contains finitely many closed points.

Let x ∈ Spec 𝔖[1/up] be a closed point. If x ∉ U, we consider two cases. If |u(x)| < |π|, then since the s_{α} are Frobenius invariant, we have P_{𝔖[1/p]} ⟶^{∼} φ^{∗}(P_{𝔖[1/p]}) in a formal neighbourhood of x. Hence P_{𝔖}[1/p] cannot be a G-torsor at φ(x), since φ is a faithfully flat map on 𝔖. Repeating the argument we find φ(x), φ^{2}(x), ⋯ ∉ U, which gives a contradiction.

Similarly, if |u(x)| ≥ |π| consider a sequence of points x_{0}, x_{1}, … with x_{0} = x, and φ(x_{i+1}) = x_{i}. For i ≥ 1, we have P_{𝔖[1/p]} ⟶^{∼} φ^{∗}(P_{𝔖[1/p]}) in a formal neighbourhood of x_{i}, so we find that x_{i} ∉ U for i ≥ 1.

Step 4: P_{𝔖[1/p]} is a G-torsor. By Step 3, it suffices to show that the restriction of P to K_{0}[[u]] is a G-torsor. For any 𝔑 in Mod_{/𝔖}^{φ} there is a unique, φ-equivariant isomorphism

𝔑 ⊗_{𝔖} K_{0}[[u]] ⟶^{∼} K_{0}[[u]] ⊗_{K_{0}} 𝔑/u𝔑[1/p]

lifting the identity map on 𝔑/u𝔑 ⊗_{O_{K_{0}}} K_{0}, which is functorial in 𝔑 (see, for example, [Ki 1, 1.2.6]). Applying this to 𝔐 and the morphisms s̃_{α} shows that the restriction of P to K_{0}[[u]] is isomorphic to P_{K_{0}} ⊗_{K_{0}} K_{0}[[u]], which is a G-torsor by Step 2.

Step 5: P is a G-torsor. Let U be the complement of the closed point in Spec 𝔖. By Steps 1 and 4 we know that P |_{U} is a G-torsor. By a result of Colliot-Thélène and Sansuc [CS, Thm. 6.13], P extends to a G-torsor over 𝔖 and, as we remarked above, any such torsor is trivial. Hence P |_{U} is trivial, and there is an isomorphism 𝔐|_{U} ⟶^{∼} 𝔐′|_{U} taking s̃_{α} to s_{α}. Since any vector bundle over U has a canonical extension to 𝔖, obtained by taking its global sections, this isomorphism extends to 𝔖. This implies that P is the trivial G-torsor, and completes the proof of the proposition. □

Corollary (1.3.5). With the assumptions of (1.3.4), suppose that G is connected and k is finite. Then there exists an isomorphism 𝔐 ⟶^{∼} L ⊗_{ℤ_{p}} 𝔖 which takes the tensor s̃_{α} to s_{α}. In particular, the subgroup G_{𝔖} ⊂ GL(𝔐) defined by (s̃_{α}) is isomorphic to G ×_{Spec ℤ_{p}} Spec 𝔖.

Proof. As in (1.3.4) we set 𝔐′ = L ⊗_{ℤ_{p}} 𝔐, and we denote by P ⊂ \underline{Hom}_{𝔖}(𝔐, 𝔐′) the subscheme of isomorphisms between 𝔐 and 𝔐′ which take s̃_{α} to s_{α}. Then P is a G-torsor by (1.3.4). Since G is connected and k is finite, any such torsor is trivial [Sp, 4.4], and the corollary follows. □

Corollary (1.3.6). Let L be a G_{K}-stable lattice in a crystalline representation V, 𝔐 = 𝔐(L) and (s_{α}) ⊂ L^{⊗} a collection of G_{K}-invariant tensors which define a reductive subgroup G of GL(L). Then

(1) If we view (s_{α}) ⊂ Fil^{0}D_{cris}(V)^{⊗} via the p-adic comparison isomorphism

B_{cris} ⊗_{ℤ_{p}} L ⟶^{∼} B_{cris} ⊗_{O_{K_{0}}} D_{cris}(V)

then (s_{α}) ⊂ (𝔐/u𝔐)^{⊗} ⊂ D_{cris}(V)^{⊗}.

(2) If k^{sep} denotes a separable closure of k then there exists a W (k^{sep})-linear isomorphism

L ⊗_{ℤ_{p}} W (k^{sep}) ⟶^{∼} 𝔐/u𝔐 ⊗_{W (k)} W (k^{sep})

taking s_{α} to s_{α}. In particular, (s_{α}) ⊂ (𝔐/u𝔐)^{⊗} defines a reductive subgroup G′ of GL(𝔐/u𝔐), which is a pure inner form of G.

(3) If k is finite and G is connected then there exists a W -linear isomorphism

L ⊗_{ℤ_{p}} W ⟶^{∼} 𝔐/u𝔐

taking s_{α} to s_{α}. In particular (s_{α}) ⊂ (𝔐/u𝔐)^{⊗} defines a reductive subgroup

G′ of GL(𝔐/u𝔐), which is isomorphic to G ×_{Spec ℤ_{p}} Spec W.

Proof. (1) and (2) follow from (1.3.4); in fact (1) holds for any G_{K}-invariant tensors, without assuming that G is reductive. To see that G′ is a pure inner form of G in (2), note that specializing the torsor P which appears in the proof of (1.3.4) at u = 0 gives a class in H^{1}(Spec W, G), and G′ can be obtained from G by twisting by this class.

Finally, (3) follows from (1.3.5) once we remark that s_{α} ∈ D_{cris}(V)^{⊗} is equal to

s̃_{α}|_{u=0} : 1 → (𝔐/u𝔐)^{⊗} ↪ D_{cris}(V)^{⊗},

the final inclusion being given by the first isomorphism of (1.2.1)(1). The equality is a formal consequence of the functoriality of this isomorphism. □

(1.4) Application to p-divisible groups: We now apply the previous results to p-divisible groups over O_{K}.

If R is a ring which is p-adically separated and complete, and 𝒢 is a p-divisible group over R, we will denote by 𝔻(𝒢) the corresponding (contravariant) Frobenius crystal on the crystalline site of R.

If R′ → R is a surjection whose kernel is equipped with divided powers and is topologically nilpotent for the p-adic topology we denote by 𝔻(𝒢)(R′) the value of 𝔻(𝒢) on R′. In particular, there is a short exact sequence of finite, locally free R-modules

0 → (Lie 𝒢)^{∗} → 𝔻(𝒢)(R) → Lie 𝒢^{∗} → 0

where the term on the left is the R-dual of Lie 𝒢 , and 𝒢^{∗} is the Cartier dual of 𝒢 . If R′ → R is as above, we denote by Fil^{1}𝔻(𝒢)(R′) the preimage of (Lie 𝒢)^{∗} in 𝔻(𝒢)(R′).

Let S be the p-adic completion of W [u, E(u)^{n}/n!]_{n≥1}. We equip it with a Frobenius φ which sends u to u^{p}. The ideal Fil^{1}S = ker (S →^{u↦π} O_{K}) is equipped with divided powers. In particular, if 𝒢 is a p-divisible group over O_{K}, then 𝔻(𝒢)(S) is a finite free module, equipped with a semi-linear Frobenius φ and a submodule Fil^{1}𝔻(𝒢)(S). In fact φ(Fil^{1}𝔻(𝒢)(S)) ⊂ p𝔻(𝒢)(S), but we will not need this explicitly.

(1.4.1) Let BT_{/𝔖}^{φ} denote the full subcategory of Mod_{/𝔖}^{φ} consisting of objects 𝔐 such that 𝔐 is φ-stable and the cokernel of φ^{∗}(𝔐) → 𝔐 is killed by E(u).

We view S as a 𝔖-algebra by sending u to u. If 𝔐 is in BT_{/𝔖}^{φ} then following Breuil [Br], we set M(𝔐) = S ⊗_{𝔖} φ^{∗}(𝔐), and equip M(𝔐) with the induced Frobenius φ, and with an S-submodule

Fil^{1}M(𝔐) = {x ∈ S ⊗_{φ,𝔖} 𝔐 : 1 ⊗ φ(x) ∈ Fil^{1}S ⊗_{𝔖} 𝔐 ⊂ S ⊗_{𝔖} 𝔐}.

One checks easily that

Fil^{1}M(𝔐) = S · Fil^{1}(φ^{∗}(𝔐)) + Fil^{1}S · M.

We denote by (p-div/O_{K}) the category of p-divisible groups over O_{K}.

Theorem (1.4.2). The functor 𝔐 of (1.2.1) induces a fully faithful contravariant functor

𝔐 : (p-div/O_{K}) → BT_{/𝔖}^{φ}; 𝒢 ↦ 𝔐(𝒢) := 𝔐(T_{p}𝒢^{∗}(−1))

which is an equivalence when p > 2. If 𝒢 is a p-divisible group over O_{K} and either p > 2 or 𝒢^{∗} is connected then there is a canonical isomorphism

𝔻(𝒢)(S) ⟶^{∼} M(𝔐(𝒢))

compatible with φ and filtrations.

Proof. That 𝔐 induces a functor as claimed follows from [Ki 1, 2.2.7] and its proof. If p > 2, 𝔐 is in BT_{/𝔖}^{φ} and 𝒢 = 𝒢 (𝔐) is the corresponding p-divisible group, then the construction of the quasi-inverse to 𝔐 in loc. cit shows that 𝔻(𝒢)(S) ⟶^{∼} M(𝔐(𝒢)). If p = 2 then this formula holds if 𝒢^{∗} is connected [Ki 3, 1.1.6(2), 1.2.8].^{10} □ Corollary (1.4.3). Let 𝒢 be a p-divisible group over O_{K}, and if p = 2 assume that 𝒢^{∗} is connected. Let L = T_{p}𝒢^{∗}(−1), 𝔐 = 𝔐(L) = 𝔐(𝒢) and (s_{α}) ⊂ L^{⊗} a collection of G_{K}-invariant tensors defining a reductive subgroup G ⊂ GL(L). Then

10Note that the correspondence of [Ki 3] is covariant.

(1) There is a canonical φ-equivariant isomorphism φ^{∗}(𝔐/u𝔐) ⟶^{∼} 𝔻(𝒢_{0})(W), where 𝒢_{0} = 𝒢 ⊗_{O_{K}} k.

(2) There exists a W (k^{sep})-linear isomorphism

L ⊗_{ℤ_{p}} W (k^{sep}) ⟶^{∼} 𝔻(𝒢_{0})(W) ⊗_{W} W (k^{sep})

taking s_{α} to φ^{∗}(s_{α}) ∈ 𝔻(𝒢_{0})(W)^{⊗}. In particular, (φ^{∗}(s_{α})) ⊂ 𝔻(𝒢_{0})(W)^{⊗} defines a reductive subgroup G_{W} ⊂ GL(𝔻(𝒢_{0})(W)) which is an inner form of G.

(3) If G is connected and k is finite then there exists a W -linear isomorphism

L ⊗_{ℤ_{p}} W ⟶^{∼} 𝔻(𝒢_{0})(W)

taking s_{α} to φ^{∗}(s_{α}) ∈ 𝔻(𝒢_{0})(W)^{⊗}. In particular, (φ^{∗}(s_{α})) ⊂ 𝔻(𝒢_{0})(W)^{⊗} defines a reductive subgroup G_{W} ⊂ GL(𝔻(𝒢_{0})(W)) which is isomorphic to G ×_{Spec ℤ_{p}} W.

(4) The filtration Fil^{1}𝔻(𝒢_{0})(k) ⊂ 𝔻(𝒢_{0})(k) is given by a cocharacter

μ_{0} : 𝔾_{m} → G_{W} ⊗_{W} k.

Proof. For (1) note that if we view W as an S-algebra by sending u to 0, then using (1.4.2) we have

𝔻(𝒢_{0})(W) ⟶^{∼} 𝔻(𝒢)(S) ⊗_{S} W ⟶^{∼} M(𝔐(𝒢)) ⊗_{S} W

= φ^{∗}(𝔐(T_{p}𝒢^{∗}(−1))) ⊗_{𝔖} W = φ^{∗}(𝔐)/uφ^{∗}(𝔐).

In particular, we may view the tensors φ^{∗}(s_{α}) in 𝔻(𝒢_{0})(W)^{⊗} by (1.3.6)(1), and (2) follows from (1.3.6)(2). Similarly, (3) follows from (1.3.6)(3).

By (1.3.4), (s_{α}) ⊂ L^{⊗} gives rise to a collection of φ-invariant tensors (s̃_{α}) ⊂ 𝔐^{⊗}, which define a reductive subgroup G_{𝔖} ⊂ GL(𝔐). Pulling these back by φ and specializing at u = π we obtain tensors

s_{α,K} = φ^{∗}(s̃_{α})|_{u=π} ∈ (φ^{∗}(𝔐) ⊗_{𝔖} O_{K})^{⊗}

which define a reductive subgroup G_{O_{K}} ⊂ GL(φ^{∗}(𝔐) ⊗_{𝔖} O_{K}).

By (1.4.2), we have φ^{∗}(𝔐) ⊗_{𝔖} O_{K} ⟶^{∼} 𝔻(𝒢)(O_{K}) compatible with φ and filtrations. Hence it suffices to show that the filtration on 𝔻(𝒢)(O_{K}) is given by a cocharacter 𝔾_{m} → G_{O_{K}} , that is, in the terminology of §1, that it is G_{O_{K}} -split. By (1.1.4) it suffices to show that the induced filtration on 𝔻(𝒢)(O_{K}) ⊗_{O_{K}} K is G_{O_{K}} |_{K}-split. Now using (1.2.1) and (1.4.2) we have

(1.4.4)  D_{dR}(V) ⟶^{∼} φ^{∗}(𝔐) ⊗_{𝔖} K ⟶^{∼} 𝔻(𝒢)(O_{K}) ⊗_{O_{K}} K

where V = L[1/p]. The compatibility with filtrations in (1.2.1)(1) and (1.4.2) shows that this is an isomorphism of filtered K-vector spaces, so we have to show that the filtration on D_{dR}(V) is G_{O_{K}} |_{K}-split. This follows from (1.4.5) below. □

Lemma (1.4.5). Let D be a weakly admissible filtered K_{0}-module, and let (s_{α}) ⊂ Fil^{0}D^{⊗} be a collection of φ-invariant tensors defining a reductive subgroup G ⊂ GL(D). Then the filtration on D_{K} = D ⊗_{K_{0}} K is G ⊗_{K_{0}} K-split.

Proof. Let ⟨D⟩^{⊗} be the Tannakian category generated by the weakly admissible module D, and ⟨D⟩^{⊗,G} the Tannakian category generated by the G-representation D. By Tannakian duality, ⟨D⟩^{⊗,G} is a subcategory of ⟨D⟩^{⊗}, and the filtration on ⟨D⟩^{⊗} ⊗_{K_{0}} K induces a filtration on ⟨D⟩^{⊗,G} ⊗_{K_{0}} K. This restricts to the given filtration on D ⊗_{K_{0}} K, which is therefore G ⊗_{K_{0}} K-split, by (1.1.3). □

(1.4.6) We remark that the condition in the corollary when p = 2 is one of the two points which force us to assume that p > 2.

The statement of (1.3.6)(2) was conjectured by Milne [Mi 3], without the restriction at p = 2 which should be unnecessary. See also [Va 3].

(1.5) Deformation theory: We keep the notation of the previous section. Let 𝒢_{0} be a p-divisible group over k. We recall the explicit description of the

universal deformation ring of 𝒢_{0} given by Faltings [Fa, §7] (see also [Mo, §4]).

Fix a cocharacter μ : 𝔾_{m} → GL(𝔻(𝒢_{0})(W)) whose reduction mod p, μ_{0}, gives rise to the filtration on 𝔻(𝒢_{0})(k). The filtration on 𝔻(𝒢)(W) defined by μ corresponds to a p-divisible group 𝒢 over W lifting 𝒢_{0} [Me].

Let U^{∘} ⊂ GL(𝔻(𝒢_{0})(W)) be the opposite unipotent defined by μ, and R the complete local ring at the identity section of U^{∘}. Then R ⟶^{∼} W [[t_{1}, …, t_{n}]] is a power series ring over W where

n = dim U^{∘} = dim_{k}gr^{−1}End_{k}𝔻(𝒢_{0})(k).

We equip R with a Frobenius φ = φ_{R} sending t_{i} to t_{i}^{p} and acting as the usual Frobenius on W.

Write M_{0} = 𝔻(𝒢_{0})(W) equipped with the filtration Fil^{1}M_{0}, induced by the chosen character μ and Frobenius φ = φ_{M_{0}}. Let M = M_{0} ⊗_{W} R with the filtration induced from M_{0}. We equip M with a semi-linear Frobenius φ given by the composite

M = M_{0} ⊗_{W} R →^{φ⊗φ} M →^{u} M

where u ∈ U^{∘}(R) is the tautological R-point of U^{∘}.

Then there is a p-divisible group 𝒢_{R} over R such that 𝒢_{R}⊗_{R}R/(t_{1}, …, t_{n}) ⟶^{∼} 𝒢 and 𝒢_{R} is a universal deformation of 𝒢_{0}, and an isomorphism 𝔻(𝒢_{R})(R) ⟶^{∼} M compatible with Frobenius and filtration [Mo, 4.5]. Since R is formally smooth, the structure of F -crystal on 𝔻(𝒢_{R}) is given by a connection ∇ : M → M ⊗ Ω_{R}^{1}, such that φ^{∗}(M) → M is parallel. That is, it is compatible with the connections φ^{∗}(∇) and ∇ on φ^{∗}(M) and M, respectively.

If R′ is any p-adically complete, p-torsion free W -algebra, equipped with a lift of Frobenius φ_{R′} and ι : R → R′ is a map of W -algebras, then the Frobenius on

M_{R′} := M ⊗_{R} R′ ⟶^{∼} 𝔻(𝒢_{R} ⊗_{R} R′)(R′)

is given as follows: Since φ_{R′} ∘ι and ι∘φ_{R} have the same reduction modulo p (we do not assume they are equal), the structure of crystal on 𝔻(𝒢_{R}) induces a canonical isomorphism ε : φ_{R′}^{∗}ι^{∗}M ⟶^{∼} ι^{∗}φ_{R}^{∗}M, and the Frobenius φ_{R′} on M_{R′} is given by

(1.5.1)  φ_{R′}^{∗}(M_{R′}) = φ_{R′}^{∗}ι^{∗}M ⟶^{∼}_{ε} ι^{∗}φ_{R}^{∗}M → ι^{∗}M = M_{R′}.

The map ε is compatible with the connections φ_{R′}^{∗}ι^{∗}(∇) and ι^{∗}φ_{R}^{∗}(∇) on φ_{R′}^{∗}ι^{∗}M and ι^{∗}φ_{R}^{∗}M respectively. This compatibility follows from the cocycle condition in the definition of a crystal, or equivalently the integrability of ∇. In particular, one sees that (1.5.1) is parallel for ∇_{R′} = ι^{∗}(∇).

Explicitly, ε has the following description [Fa, p134], [Mo, 4.3]: If \underline{i} = (i_{1}, …, i_{n}) is a multi-index, write ∇(∂)\underline{i} = ∇(∂t_{1})^{i_{1}} … ∇(∂t_{n})^{i_{n}} and z\underline{i} = z_{1}^{i_{1}} … z_{n}^{i_{n}} where z_{i} = φ_{R′} ∘ ι(t_{i}) − ι ∘ φ_{R}(t_{i}). Then

ε(m ⊗ 1) = ∑_{\underline{i}} ∇(∂)\underline{i}(m ⊗ 1) ⊗ \frac{z\underline{i}}{\underline{i}!} .

We will need the following lemma relating the action of φ and ∇_{R′}.^{13}

Lemma (1.5.2). Let ι : R → R′ be as above. Suppose that ι ∘ φ_{R} = φ_{R′} ∘ ι and that there is a smooth, closed, subgroup H ⊂ GL(M_{0}), which is φ-stable, and such that the composite

θ : M_{0} ⟶^{m↦φ^{−1}(m)⊗1} φ^{∗}(M_{R′}) → M_{R′} = M_{0} ⊗_{W} R′

is given by θ(m) = A · m for some A ∈ H(R′). Then β : M_{0} →^{∇_{R′}} M_{R′} ⊗_{R′} Ω_{R′}^{1}, the linear map induced by ∇_{R′}, is given by an element of Lie H ⊗_{W} Ω_{R′}^{1}.

Proof. Here the action of φ on GL(M_{0}) is via h ↦ φ ∘ h ∘ φ^{−1}. Note that θ is well defined, without inverting p, as ι ∘ φ_{R} = φ_{R′} ∘ ι. It suffices to prove the lemma with ι(R) in place of R′, so we may assume that ι is surjective.

Since the map φ^{∗}(M_{R′}) → M_{R′} is parallel for ∇_{R′} one computes that dA+βA = Aφ(β) as elements of End M_{R′} ⊗_{R′} Ω_{R′}^{1}[1/p], where φ(β) is the composite

M_{0} →^{φ^{−1}} M_{0}[1/p] →^{∇_{R′}} M_{R′} ⊗_{R′} Ω_{R′}^{1}[1/p]

= M_{0} ⊗_{W} Ω_{R′}[1/p] →^{φ⊗φ} M_{0} ⊗_{W} Ω_{R′}[1/p] = M_{R′} ⊗_{R′} Ω_{R′}^{1}[1/p].

Hence φ(β) = A^{−1}dA + A^{−1}βA ∈ End_{R′}M_{R′} ⊗_{R′} Ω_{R′}^{1} and

β = −dAA^{−1} + Aφ(β)A^{−1}.

Repeatedly substituting the right hand side of the above formula for β one obtains a series whose m^{th} term is contained in Lie H ⊗_{W} R′ · φ^{m−1}(Ω_{R′}^{1}). Here we are using that Lie H is stable by φ, as H is. Since ι is φ-compatible and for any ω ∈ Ω_{R}^{1}, φ^{m}(ω) ∈ (t_{1}^{p^{m}−1}, …, t_{r}^{p^{m}−1})Ω_{R}^{1}, the series converges to an element of Lie H ⊗_{W} Ω_{R′}^{1}. □

(1.5.3) Suppose G ⊂ GL(M_{0}) is a connected reductive subgroup defined by a family of φ-invariant tensors (s_{α}) ⊂ M_{0}^{⊗}, such that the filtration on 𝔻(𝒢_{0})(k) is G ⊗_{W} k-split.^{14} Choose a cocharacter μ_{0} : 𝔾_{m} → G ⊗_{W} k inducing this filtration.

13cf. the second remark in [Fa, p136].

14Note that, in general, there is a well defined Frobenius only on M_{0}[1/p]^{⊗} and not M_{0}^{⊗}, as the dual module M_{0}^{∗} need not be Frobenius stable. A tensor in M_{0}^{⊗} is said to be φ-invariant if it is φ-invariant in M_{0}[1/p]^{⊗}.

We may take μ to be a character μ : 𝔾_{m} → G lifting μ_{0}. Such a lifting always exists [DG XI, 4.2]. Let 𝔤 denote the Lie algebra of G. Then μ induces a grading of 𝔤.

Let U_{G}^{∘} ⊂ G be the opposite unipotent defined by μ, and R_{G} the complete local ring at the identity section of U_{G}^{∘}. Then R_{G} is a formally smooth quotient of R. We choose the co-ordinates t_{i} so that R_{G} = R/(t_{r+1}, …, t_{n}) where r = rk_{W} 𝔤/Fil^{0}𝔤.

Let M_{R_{G}} = M ⊗_{R} R_{G}, and let ∇_{R_{G}} be the induced connection on M_{R_{G}}. Since the s_{α} are φ-invariant, G ⊂ GL(M_{0}) is φ-stable. Hence, by (1.5.2), ∇_{R_{G}} induces a map M_{0} → M_{R_{G}} ⊗_{R_{G}} Ω_{R_{G}}^{1} which is in Lie G ⊗_{W} Ω_{R_{G}}^{1}. In particular we have ∇_{R_{G}}(s_{α} ⊗ 1) = 0. Hence s_{α} ⊗ 1 may be viewed as a morphism of crystals 1 → 𝔻(𝒢_{R} ⊗_{R} R_{G})^{⊗}. In particular, for any map ι : R → R′ as in (1.5), if ι factors through R_{G}, then ε(s_{α} ⊗ 1) = s_{α} ⊗ 1. (This may also be seen directly from the explicit description of ε in terms of ∇.).

(1.5.4) We want to describe which points of R come from R_{G}-points. In [Mo, 4.9] this is done for points with values in a power series ring over W, following a remark of Faltings [Fa, p136]. We will need to consider the case of the ring of integers in a finite extension of K_{0}. For this we need some preparation.

Consider the ring K_{0}[[u]] equipped with the Frobenius given by the usual Frobenius on K_{0} and sending u to u^{p}. Let D_{0} be a finite dimensional K_{0}-vector space, and set D = D_{0} ⊗_{K_{0}} K_{0}[[u]]. Given an isomorphism φ^{∗}(D) ⟶^{∼} D, there is a unique φ-equivariant section ξ : D_{0} = D/uD → D. To construct it choose any K_{0}-linear section ξ_{0} : D/uD → D, and set (cf. [Ki 1, 1.2.6])

ξ = \underrightarrow{lim}_{n}φ^{n} ∘ ξ_{0} ∘ φ^{−n}.

The connection ∇_{D} on D, such that ∇_{D}(ξ(D_{0})) = 0 is the unique connection for which φ^{∗}(D) ⟶^{∼} D is parallel.

For n ≥ 1, let D_{n} = D/u^{n+1}D. We denote by ξ_{n} : D_{0} → D_{n} the map induced by ξ. Suppose that D_{0} is equipped with a decreasing filtration D_{0}^{•}, and equip D and D_{n} with the induced filtration D^{•} and D_{n}^{•}.

Lemma (1.5.5). Suppose that D_{0} with its Frobenius and filtration is a weakly admissible K_{0}-module. Let (s_{α}) ⊂ Fil^{0}(D_{0}^{⊗}) be a collection of φ-invariant tensors defining a reductive subgroup G ⊂ GL(D_{0}), and fix a cocharacter μ : 𝔾_{m} → G inducing the filtrations D_{0}^{•}. Suppose that for some n ≥ 1 we have

(1) For each α, ξ_{n}(s_{α}) ∈ Fil^{0}D_{n}^{⊗}.

(2) If U^{∘} ⊂ GL(D_{0}) denotes the opposite unipotent defined by μ, then the K_{0}[[u]]- linear automorphism of D,

θ : D = D_{0} ⊗_{K_{0}} K_{0}[[u]] → D : m_{0} ⊗ a ↦ φ(φ^{−1}(m_{0}) ⊗ 1) ⊗ a

lies in U^{∘}(K_{0}[[u]]). Then ξ_{n}(s_{α}) = s_{α} ⊗ 1 ∈ D_{n}^{⊗} for each α, and the automorphism θ_{n} : D_{n} → D_{n}, induced by θ, lies in the opposite unipotent subgroup U_{G}^{∘} ⊂ G defined by μ.

Proof. We remark that a cocharacter μ as in the lemma always exists by (1.4.5). Since G is defined by a finite subset of the s_{α}, we may assume that the collection of tensors (s_{α}) is indexed by some finite set I.

Each s_{α} generates a φ-invariant line in a weakly admissible submodule of D_{0}^{⊗}, so this line cannot lie in filtration degree 1, and s_{α} gives rise to a morphism 1 → D_{0}^{⊗} in the category of (Ind-)weakly admissible modules. Now consider the map

End_{K_{0}}D_{0} → (D_{0}^{⊗})^{|I|}; g ↦ (g(s_{α}))_{α∈I}

This is a map of (Ind)-weakly admissible filtered φ-modules, and hence strict for filtrations. In particular, if g ∈ End_{K_{0}}D_{0} maps (s_{α})_{α∈I} into Fil^{0}(D_{0}^{⊗})^{|I|} then g ∈ Lie P + Lie G, where P ⊂ GL(D_{0}) is the parabolic corresponding to D_{0}^{•}.

We will show by induction on n that ξ_{n}(s_{α}) = s_{α} ⊗ 1. Note that this implies

θ_{n}(s_{α} ⊗ 1) = φ(s_{α} ⊗ 1) = φ(ξ_{n}(s_{α})) = ξ_{n}(s_{α}) = s_{α} ⊗ 1

so that θ_{n} ∈ U_{G}^{∘}(K_{0}[[u]]/u^{n}).

Suppose ξ_{n−1}(s_{α}) = s_{α} ⊗ 1, and let θ̃_{n−1} ∈ U_{G}^{∘}(K_{0}[[u]]/u^{n+1}) be any lift of θ_{n−1}. Then

θ_{n} ∘ θ̃_{n−1}^{−1}(s_{α} ⊗ 1) = φ(s_{α} ⊗ 1) = ξ_{n}(s_{α}) = w(s_{α} ⊗ 1)

for some w ∈ ker (U^{∘}(K_{0}[[u]]/u^{n+1}) → U^{∘}(K_{0}[[u]]/u^{n})). Writing w = 1 + u^{n}w_{0} with w_{0} ∈ Lie U^{∘}, we have

u^{n}w_{0}(s_{α} ⊗ 1) = ξ_{n}(s_{α}) − s_{α} ⊗ 1 ∈ Fil^{0}(D_{n}^{⊗}),

so w_{0}s_{α} ∈ Fil^{0}D_{0}^{⊗}. It follows that

w_{0} ∈ Lie U^{∘} ∩ (Lie P + Lie G) = Lie U^{∘} ∩ (Lie P + Lie U_{G}^{∘}) = Lie U_{G}^{∘}.

Hence w ∈ U_{G}^{∘}(K_{0}[[u]]/u^{n+1}) so ξ_{n}(s_{α}) = s_{α} ⊗ 1, and θ_{n} ∈ U_{G}^{∘}(K_{0}[[u]]). □

(1.5.6) We now return to the notation and assumptions of (1.5.3). Let A (resp. A_{G}) be the completion of R[1/p] (resp. R_{G}[1/p]) at (t_{1}, …, t_{n}). We write M_{A} = M ⊗_{R} A, and similarly for M_{A_{G}}. Let σ_{α} ∈ M_{A}^{⊗} denote the parallel transport of s_{α}. That is σ_{α} is the unique section such that ∇(σ_{α}) = 0, and σ_{α} modulo (t_{1}, …, t_{n}) is s_{α}.

Lemma 1.5.7. Let Spf A′_{G} ⊂ Spf A denote the largest closed subspace such that σ_{α} ∈ Fil^{0}M_{A′_{G}}^{⊗}. Then A′_{G} = A_{G}.

Proof. Since ∇_{R_{G}} has coefficients in Lie G, σ_{α}|_{A_{G}} = s_{α} ⊗ 1, so Spf A_{G} ⊂ Spf A′_{G}. Since A_{G} is a regular ring, it suffices to show that A_{G} and A′_{G} have the same tangent space, or that any map of K_{0}-algebras ζ_{1} : A → K_{0}[u]/u^{2} which factors through A′_{G}, actually factors through A_{G}.

Let D_{1} = ζ_{1}^{∗}(M_{A}), and D_{0} = D_{1}/u_{1} ≃ M_{0}[1/p]. By definition of M,

D_{1} = ζ_{1}^{∗}(M_{A}) = ζ_{1}^{∗}(M_{0} ⊗_{K_{0}} A) = D_{0} ⊗ K_{0}[u]/u^{2}.

Note that ζ_{1} is compatible with the Frobenii t_{i} ↦ t_{i}^{p} on A and u ↦ u^{p} = 0 on K_{0}[u]/u^{2}. Hence the Frobenius on M_{A} induces a Frobenius on D_{1}. We also give D_{1} the filtration induced by M_{A} via ζ_{1}. By construction of φ on M the linear map

θ_{1} : D_{1} = D_{0} ⊗_{K_{0}} K_{0}[u]/u^{2} → D_{1}; m_{0} ⊗ a ↦ φ(φ^{−1}(m_{0}) ⊗ a)

lies is in U^{∘}(K_{0}[u]/u^{2}), and ζ_{1} factors through A_{G} if and only if θ_{1} ∈ U_{G}^{∘}(K_{0}[u]/u^{2}).

Lift θ_{1} to an element θ ∈ U^{∘}(K_{0}[[u]]), and consider the Frobenius semi-linear map D → D whose restriction to D_{0} is given by θ ∘ φ. Then θ can be recovered from φ^{∗}(D) → D as in (1.5.5)(2). We now consider the map ξ : D_{0} → D, constructed in (1.5.4).

Lift the filtration on D_{1} to one on D. Since ζ_{1} factors through A′_{G}, ξ_{1}(s_{α} ⊗ 1) ∈ Fil^{0}D_{1}^{⊗}, so the condition (1.5.5)(1) is satisfied. Hence θ_{1} ∈ U_{G}^{∘}(K_{0}[u]/u^{2}) by (1.5.5). □

(1.5.8) We will need a variant of the ring S. Let I^{[n]} ⊂ S denote the divided powers of I = ker (S → O_{K}), and set^{16} Ŝ = \underleftarrow{lim}_{n}S/I^{[n]}. If p > 2, the divided powers of p are topologically nilpotent, and this is the same as the completion of S with respect to the topology defined by the divided powers of the closure of the ideal (u^{ei}/i!)_{i≥1}. In particular, when p > 2, we may regard Ŝ as a subring of K_{0}[[u]], containing S.

Proposition (1.5.9). Suppose p > 2 or that 𝒢_{0}^{∗} is connected. Let ϖ : R → O_{K} be a map of W -algebras and denote by 𝒢_{ϖ} the induced p-divisible group over O_{K}. Then ϖ factors through R_{G} if and only if there exists a collection of φ-invariant tensors (s̃_{α}) ⊂ 𝔻(𝒢_{ϖ})(S)^{⊗} lifting (s_{α}) ⊂ 𝔻(𝒢_{0})(W)^{⊗}, such that

(1) If s_{α,O_{K}} denotes the image of s̃_{α} in 𝔻(𝒢_{ϖ})(O_{K})^{⊗}, then

(s_{α,O_{K}}) ⊂ Fil^{0}(𝔻(𝒢_{ϖ})(O_{K})^{⊗}).

(2) The (s̃_{α}) define a reductive subgroup G_{S} ⊂ GL(𝔻(𝒢_{ϖ})(S)).

Proof. If ϖ factors through R_{G}, we may lift it to a map ϖ̃ : R_{G} → S, and set s̃_{α} = ϖ̃(s_{α} ⊗ 1). Then s̃_{α} clearly satisfy (1) and (2). We have to check that the ϖ̃(s_{α} ⊗ 1) are φ-invariant. The Frobenius on M_{S} := 𝔻(𝒢_{ϖ})(S) = M_{R_{G}} ⊗_{R_{G}} S is given by

φ^{∗}(M_{S}) = φ^{∗}ϖ̃^{∗}(M_{R_{G}}) ⟶^{∼}_{ε} ϖ̃^{∗}φ^{∗}(M_{R_{G}}) ⟶^{ϖ̃^{∗}(φ⊗1)} ϖ̃^{∗}(M_{R_{G}})

where ε is as in (1.5). Now we have

ϖ̃^{∗}(φ ⊗ 1) ∘ ε(ϖ̃(s_{α} ⊗ 1)) = ϖ̃^{∗}(φ ⊗ 1)(ϖ̃(s_{α} ⊗ 1)) = ϖ̃(s_{α} ⊗ 1)

where the first equality was seen in (1.5.3) and the second follows from the φinvariance of s_{α}.

Now suppose that there exist s̃_{α} satisfying (1) and (2). Denote by ϖ_{0} : R → W the map given by ϖ_{0}(t_{i}) = 0 for i = 1, …, n. We denote by 𝒢_{ϖ×ϖ_{0}} the p-divisible group over O_{K} ×_{k} W induced by ϖ × ϖ_{0} : R → O_{K} ×_{k} W.

Assume first that p > 2. Then the surjective map W [u] → O_{K} ×_{k} W sending u to (π, 0) induces a map Ŝ → O_{K}×_{k}W. Let GŜ = G_{S}⊗_{S} Ŝ. By (1.1.5) there is a GŜ-split filtration on 𝔻(𝒢_{ϖ}(Ŝ)) which simultaneously lifts the filtration on 𝔻(𝒢_{ϖ}(O_{K})) and the chosen filtration on 𝔻(𝒢)(W). Since the kernel of Ŝ → O_{K} ×_{k} W is equipped with topologically nilpotent divided powers such a filtration corresponds to a pdivisible group^{17} 𝒢ϖ̃ over Ŝ, deforming 𝒢_{ϖ×ϖ_{0}}. Since R is a universal deformation ring for 𝒢_{0}, 𝒢ϖ̃ is induced by a map ϖ̃ : R → Ŝ lifting ϖ × ϖ_{0}.

16This is the ring considered in [Fa]

17By the main result of [Me]. Note that the divided powers on Fil^{1}S are not topologically nilpotent, so we cannot make the following argument with S in place of Ŝ, as the theorem in [Me] does not apply. When 𝒢_{0}^{∗} is connected the main result of loc. cit should hold without assuming the divided powers are nilpotent, but as remarked in [Zi 1, p214] there seems to be no reference for this.

To show that ϖ̃ factors through R_{G} it suffices to show that the induced map ϖ̃ : A → K_{0}[[u]] factors through A_{G}. We may identify

𝔻(𝒢ϖ̃)(Ŝ) = 𝔻(𝒢_{ϖ})(Ŝ) = 𝔻(𝒢_{ϖ}(S)) ⊗_{S} Ŝ

with MŜ := M_{R} ⊗_{R} Ŝ = M_{0} ⊗_{W} Ŝ, and we view s̃_{α} as elements of M ⊗ ; Ŝ. Since the s̃_{α} ∈ M ⊗ ; Ŝ ⊂ M_{K_{0}[[u]]}^{⊗} are Frobenius invariant, they are parallel for the connection on M_{K_{0}[[u]]}^{⊗}. Hence ϖ̃^{∗}(σ_{α}) = s̃_{α} so, by the construction of 𝒢ϖ̃, s̃_{α} ∈ Fil^{0}M_{K_{0}[[u]]}^{⊗}. It follows by (1.5.7) that ϖ̃ factors through A_{G}.

Finally suppose that 𝒢_{0}^{∗} is connected. Then using results of Zink, we can repeat the above argument with S in place of Ŝ, even when p = 2 : Consider the map S → O_{K} ×_{k} W sending u to (π, 0), and choose a G_{S}-split filtration on 𝔻(𝒢_{ϖ})(S) which lifts the filtrations on 𝔻(𝒢)(W) and 𝔻(𝒢_{ϖ})(O_{K}). In the terminology of [Zi 2] this filtration gives 𝔻(𝒢_{ϖ})(S) the structure of an S-window over S, and hence gives rise to a p-divisible group 𝒢ϖ̃ over S which deforms 𝒢_{ϖ×ϖ_{0}}. By [Zi 1, Cor. 97] the canonical isomorphism 𝔻(𝒢ϖ̃)(S) ⟶^{∼} 𝔻(𝒢_{ϖ})(S) respect filtrations. The rest of the argument is as in the case p > 2. □

Corollary (1.5.11). Suppose p > 2 or 𝒢_{0}^{∗} is connected. Let K′/K be a finite extension and ϖ : R → O_{K′} a map of W -algebras inducing a p-divisible group 𝒢_{ϖ} over O_{K′}. Let L = T_{p}𝒢^{∗}(−1) and suppose there exists a family of G_{K′}-invariant tensors (s_{α,ét}) ⊂ L^{⊗} defining a reductive subgroup of GL(L), such that under the p-adic comparison isomorphism

L ⊗_{ℤ_{p}} B_{cris} ⟶^{∼} M_{0} ⊗_{ℤ_{p}} B_{cris}

s_{α,ét} maps to s_{α} ∈ M_{0}^{⊗}.

Then ϖ factors through R_{G}.

Proof. Let k′ be the residue field of K′. After replacing R and R_{G} by R ⊗_{W} W (k′) and R ⊗_{W} W (k′) respectively, we may assume that K′ = K. We use the notation of §1.2. Let 𝔐 = 𝔐(L). The tensors s_{α,ét} give rise to φ-invariant tensors (s̃_{α,ét}) ⊂ 𝔐^{⊗}, which define a reductive subgroup of GL(𝔐). Then we have

(1 ⊗ s̃_{α,ét}) ⊂ M(𝔐) ⟶^{∼} 𝔻(𝒢)(S),

by (1.4.2). We saw in the proof of (1.3.6) that the image of s_{α,ét} under the p-adic comparison isomorphism is s̃_{α,ét}|_{u=0}. Hence 1 ⊗ s̃_{α,ét}|_{u=0} = s_{α} and (1 ⊗ s̃_{α,ét}) ⊂ 𝔻(𝒢)(S) is a collection of tensors satisfying the hypotheses of (1.5.9). □

§2 Integral canonical models of Hodge type

(2.1) Shimura varieties: We recall the definition of a Shimura datum and the associated Shimura variety [De 3, §2.1]. Let G be a connected reductive group over ℚ and X a conjugacy class of maps of algebraic groups over ℝ

h : 𝕊 = Res_{ℂ/ℝ}𝔾_{m} → G_{ℝ}.

On ℝ-points such a map induces a map of real groups ℂ^{×} → G(ℝ).

We require that (G, X) satisfy the following conditions:

(1) Let 𝔤 denote the Lie algebra of G_{ℝ}. We require that the composite

𝕊 → G_{ℝ} → G_{ℝ}^{ad} → GL(𝔤)

defines a Hodge structure of type (−1, 1), (0, 0), (1, −1). This means that under the action of ℂ^{×} on 𝔤_{ℂ} = 𝔤 ⊗_{ℝ} ℂ we have a decomposition

𝔤_{ℂ} = V^{−1,1} ⊕ V^{0,0} ⊕ V^{1,−1}

where z ∈ ℂ^{×} acts on V^{p,q} via z^{−p}z̄^{−q}.

(2) h(i) is a Cartan involution of G_{ℝ}^{ad} (note that adh(−1) = 1 on 𝔤 so h induces an involution of G_{ℝ}^{ad}). This means that we require the real form of G^{ad} defined by the involution g ↦ h(i)ḡh(i)^{−1} to be compact.

(3) G^{ad} has no factor defined over ℚ whose real points form a compact group.

The second condition implies that for any h_{0} ∈ X the stabilizer K_{∞} ⊂ G(ℝ) (acting by conjugation) of h_{0} is compact modulo its center, and G(ℝ)/K_{∞} ⟶^{∼} X has a complex structure. A pair (G, X) satisfying the above conditions is called a Shimura datum.

Let 𝔸_{f} denote the finite adeles over ℚ, and 𝔸_{f}^{p} ⊂ 𝔸_{f} the subgroup of adeles with trivial component at a prime p. Let K = K_{p}K^{p} ⊂ G(𝔸_{f}) be a compact open subgroup, where K_{p} ⊂ G(ℚ_{p}), and K^{p} ⊂ G(𝔸_{f}^{p}) are compact open.

A theorem of Baily-Borel asserts that

Sh_{K}(G, X)_{ℂ} = G(ℚ)\X × G(𝔸_{f})/K

has a natural structure of an algebraic variety over ℂ. Results of Shimura, Deligne, Milne and others imply that Sh_{K}(G, X)_{ℂ} has a model Sh_{K}(G, X) over a number field E = E(G, X) - the reflex field - which does not depend on K [Mi 2, §4,5].

We will sometimes consider the pro-variety

Sh_{K_{p}}(G, X) = \underleftarrow{lim}Sh_{K}(G, X),

where K runs through compact open subgroups as above with a fixed factor K_{p} at p. Similarly we denote by Sh(G, X) the projective limit taken over all compact open subgroups K ⊂ G(𝔸_{f}).

A morphism i : (G_{1}, X_{1}) → (G_{2}, X_{2}) of Shimura data is a map of groups G_{1} → G_{2}, which induces a map X_{1} → X_{2}. A morphism i induces a map

(2.1.1)  Sh_{K_{1}}(G_{1}, X_{1}) → Sh_{K_{2}}(G_{2}, X_{2}),

provided the compact open subgroups are chosen so that K_{1} maps into K_{2}. This map is defined over the composite of the reflex fields E(G_{1}, X_{1}) · E(G_{2}, X_{2}) [De 1, 5.4]. If i is an embedding, then for any K_{1}, the subgroup K_{2} can always be chosen so that (2.1.1) is a closed embedding [De 1, 1.15]. We will need the following refinement of this result.

Lemma (2.1.2). Let i : (G_{1}, X_{1}) ↪ (G_{2}, X_{2}) be an embedding of Shimura data, and K_{2,p} ⊂ G_{2}(ℚ_{p}) a compact open subgroup. Let K_{1} = K_{1,p}K_{1}^{p} be a compact open subgroup of G_{1}(𝔸_{f}) such that K_{1,p} = K_{2,p} ∩ G(ℚ_{p}). Then there exists a compact open subgroup K_{2} = K_{2,p}K_{2}^{p} of G_{2}(𝔸_{f}) with K_{1} ⊂ K_{2}, and such that i induces an embedding

Sh_{K_{1}}(G_{1}, X_{1}) ↪ Sh_{K_{2}}(G_{2}, X_{2}).

Proof. We follow the argument of [De 1.15] which shows that it suffices to check that the map

Sh_{K_{1,p}}(G_{1}, X_{1}) → Sh_{K_{2,p}}(G_{2}, X_{2})

is an embedding. We now compute the complex points of these Shimura varieties.

Let Z_{1} ⊂ G_{1} and Z_{2} ⊂ G_{2} denote the centers of G_{1} and G_{2}. Write Z_{1}(ℚ)^{−} and Z_{2}(ℚ)^{−} for the closures of Z_{1}(ℚ) and Z_{2}(ℚ) in G_{1}(𝔸_{f}^{p}) and G_{2}(𝔸_{f}^{p}) respectively. Let U_{1} ⊂ Z_{1}(ℚ) be the group of units.^{20} By a theorem of Chevalley [Ch] any finite index subgroup of U_{1} is open in the topology induced by the inclusion U_{1} ⊂ Z_{1}(𝔸_{f}^{p}). Hence

Z_{1}(ℚ)^{−} = \underleftarrow{lim}_{n}Z_{1}(ℚ)/U_{1}^{n}

and Z_{1}(ℚ)^{−} is also equal to the closure of Z_{1}(ℚ) in G_{1}(𝔸_{f}). We set

Ĝ_{1}(ℚ) = \underleftarrow{lim}_{n}G_{1}(ℚ)/U_{1}^{n},

and similarly for Ĝ_{2}(ℚ). Then Ĝ_{1}(ℚ) is an extension of G_{1}(ℚ)/Z_{1}(ℚ) by Z_{1}(ℚ)^{−}. One checks easily that Ĝ_{1}(ℚ) injects into G_{1}(𝔸_{f}^{p}). Now the complex points of Sh_{K_{1}}(G_{1}, X_{1}) are given by

G_{1}(ℚ)\X_{1} × G(𝔸_{f})/K_{1} = (G_{1}(ℚ)/Z_{1}(ℚ))\X_{1} × (G_{1}(𝔸_{f})/Z_{1}(ℚ) · K_{1}).

Passing to the inverse limit with K_{1}^{p} we see that Sh_{K_{1,p}}(G_{1}, X_{1})(ℂ) is given by (cf. [De 3, 2.1.10])

(G_{1}(ℚ)/Z_{1}(ℚ))\X_{1} × (G_{1}(𝔸_{f})/Z_{1}(ℚ)^{−} · K_{1,p}) = Ĝ_{1}(ℚ)\X_{1} × G_{1}(𝔸_{f})/K_{1,p}.

To show that

(2.1.3)  Ĝ_{1}(ℚ)\X_{1} × G_{1}(𝔸_{f})/K_{1,p} → Ĝ_{2}(ℚ)\X_{2} × G_{1}(𝔸_{f})/K_{2,p}

is an injection, consider the map

(2.1.4)  Ĝ_{1}(ℚ)\G_{1}(𝔸_{f}^{p}) → Ĝ_{2}(ℚ)\G_{2}(𝔸_{f}^{p}).

Note that for any x ∈ X_{1}, the image of Z_{1}(ℚ) in G_{2}(ℚ) is contained in the centralizer of 𝕊 →^{h_{x}} G_{1}(ℝ) → G_{2}(ℝ), which is compact mod center. Hence U_{2} contains a subgroup which has finite index in the image of U_{1} and (2.1.4) is well defined. An argument as in [De 1, 1.15.3] shows that (2.1.4) is injective. Fix a set of coset representatives of Ĝ_{1}(ℚ) in G_{1}(𝔸_{f}^{p}). Since Ĝ_{1}(ℚ) is a subgroup of G_{1}(𝔸_{f}^{p}), the fibres of the projection

20An element u ∈ Z_{1}(ℚ) is a unit if the sequence {u^{n}}_{n≥1} is contained in a compact subset of Z_{1}(ℚ_{p}) for any prime p. If we embed Z_{1} into a torus of the form ∏_{i} Res_{F_{i}/ℚ}𝔾_{m} with F_{i} a number field, then this condition holds if and only if the image of u in each factor F_{i}^{×} is a unit in the number field F_{i}.

Ĝ_{1}(ℚ)\X_{1} × G_{1}(𝔸_{f})/K_{1,p} → Ĝ_{1}(ℚ)\G_{1}(𝔸_{f}^{p})

may be identified with X_{1} × G_{1}(ℚ_{p})/K_{1,p}. Since K_{1,p} = K_{2,p} ∩ G_{1}(ℚ_{p}), the map

X_{1} × G_{1}(ℚ_{p})/K_{1,p} → X_{2} × G_{2}(ℚ_{p})/K_{2,p}

is injective, and the lemma follows. □

(2.1.5) Fix a ℚ-vector space V with a perfect alternating pairing ψ. For any ℚ-algebra R, we write V_{R} = V ⊗_{ℚ} R. Take G = GSp(V, ψ) the corresponding group of symplectic similitudes, and let X = S^{±} be the Siegel double space, defined as the set of maps h : 𝕊 → G_{ℝ} such that

(1) The ℂ^{×} action on V_{ℝ} gives rise to a Hodge structure of type (−1, 0), (0, 1) :

V_{ℂ} ⟶^{∼} V^{−1,0} ⊕ V^{0,−1}.

(2) (x, y) ↦ ψ(x, h(i)y) is (positive or negative) definite on V_{ℝ}.

The reflex field of (GSp, S^{±}) is ℚ.

If V_{ℤ} ⊂ V is a ℤ-lattice, and h ∈ S^{±}, then V^{−1,0}/V_{ℤ} is an abelian variety. For a ring R we write V_{R} = V_{ℤ} ⊗_{ℤ} R. If Vℤ̂ is stable by K, then for K^{p} sufficiently small^{21} this gives rise to an interpretation of Sh_{K}(G, X) = Sh_{K}(GSp, S^{±}) as a moduli space for abelian varieties [De 1, §4], [RZ, §6]. Here ℤ̂ denotes the profinite completion of ℤ. In particular, for sufficiently small K^{p} there exists an abelian scheme A over Sh_{K}(GSp, S^{±}), defined over ℚ.

(2.2) Absolute Hodge cycles: We recall some of Deligne’s results on absolute Hodge cycles. Suppose we have an embedding of Shimura data i : (G, X) ↪ (GSp, S^{±}). Then (G, X) is said to be of Hodge type. Fix compact open subgroups K = K_{p}K^{p} ⊂ G(𝔸_{f}) and K′ = K′_{p}K^{′p} ⊂ GSp(𝔸_{f}) such that K ⊂ K′. Since (GSp, S^{±}) has reflex field ℚ, the map Sh_{K}(G, X) → Sh_{K′}(GSp, S^{±}) induced by i is defined over E = E(G, X).

Let (s_{α,B}) ⊂ V^{⊗} be a finite collection of tensors defining the subgroup G ⊂ GSp(V) ⊂ GL(V) (see (1.3.2) above). Fix a ℤ-lattice V_{ℤ} ⊂ V such that V_{ℤ} ⊗_{ℤ} ℤ̂ is stable by K′, as in (2.1.5), and let h : A → Sh_{K}(G, X) denote the universal abelian scheme, which exists assuming that K^{p} is sufficiently small. Write V = R^{1}h_{∗}Ω^{•} for the first relative de Rham cohomology of A. This is a vector bundle on Sh_{K}(G, X) equipped with an integrable connection ∇. Let Ṽ_{ℂ} and V_{ℂ} denote the pullbacks of V to X × G(𝔸_{f})/K and Sh_{K}(G, X)_{ℂ} respectively, so that Ṽ_{ℂ} is an analytic vector bundle over a complex analytic space.

Using the de Rham isomorphism, we may view s_{α,B} as a section of Ṽ_{ℂ}^{⊗}. Since s_{α,B} is G(ℚ) equivariant, this section descends to a section s_{α,dR} of V_{ℂ}^{⊗}. More precisely, s_{α,B} descends to a ∇-parallel section of V_{ℂ}^{an⊗} where V_{ℂ}^{an} is the analytic vector bundle attached to V_{ℂ}. Any ∇-parallel section of V_{ℂ}^{an⊗} arises from V_{ℂ}^{⊗}. This follows from the equivalence between algebraic vector bundles equipped with a flat connection with regular singular points, and the category of analytic vector bundles equipped with a flat connection [De 4] (cf. [De 2. p31]).

21which we sometimes tacitly assume in the following

Now let κ ⊃ E be a field of characteristic 0, and κ̄ an algebraic closure of κ. Fix an embedding ℚ_{p} ↪ ℂ and an embedding of E-algebras σ : κ̄ ↪ ℂ. Let x ∈ Sh_{K}(G, X)(κ) and denote by A_{x} the corresponding abelian variety over κ. Denote by H_{B}^{1}(A_{x}(ℂ), ℚ) the Betti cohomology of A_{x}(ℂ). Write H_{dR}^{1}(A_{x}) for its de Rham cohomology and H_{ét}^{1}(A_{x,κ̄}) = H_{ét}^{1}(A_{x,κ̄}, ℚ_{p}) for the p-adic étale cohomology of A_{x,κ̄} = A_{x} ⊗_{κ} κ̄. The embedding σ induces isomorphisms

H_{dR}^{1}(A_{x}) ⊗_{κ,σ} ℂ ⟶^{∼} H_{B}^{1}(A_{x}(ℂ), ℚ) ⊗_{ℚ} ℂ ⟶^{∼} H^{1}(A_{x,κ̄}, ℚ_{p}) ⊗_{ℚ_{p}} ℂ.

Let s_{α,B,x} be the fibre of s_{α,B} at x (regarded as a ℂ-valued point via σ), and denote by s_{α,dR,x} ∈ H_{dR}^{1}(A_{x})^{⊗} ⊗_{κ,σ} ℂ and s_{α,ét,x} ∈ H_{ét}^{1}(A_{x,κ̄})^{⊗} the images of s_{α,B,x} under these two isomorphisms.

Lemma (2.2.1). The action of Gal(κ̄/κ) on H_{ét}^{1}(A_{x,κ̄}, ℚ_{p}) fixes each s_{α,ét,x} and factors through G(ℚ_{p}). Moreover we have s_{α,dR,x} ∈ H_{dR}^{1}(A_{x})^{⊗}.

Proof. Let Sh_{K^{p}}(G, X) = \underleftarrow{lim}_{H_{p}}Sh_{H_{p}K^{p}}(G, X) where H_{p} runs over compact open subgroups of K_{p}, and similarly for Sh_{K^{′p}}(GSp, S^{±}).

The action of Gal(κ̄/κ) on H_{ét}^{1}(A_{x,κ̄}, ℚ_{p}) is induced by the map Gal(κ̄/κ) → K′_{p}, obtained by pulling back to κ̄ the K′_{p}-torsor Sh_{K^{′p}}(GSp, S^{±}) → Sh_{K′}(GSp, S^{±}). On the other hand, we have a commutative, K_{p}-equivariant diagram

\xymatrix{Sh_{K^{p}}(G, X) \ar[r] \ar[d] & Sh_{K^{′p}}(GSp, S^{±}) \ar[d] \\ Sh_{K}(G, X) \ar[r] & Sh_{K′}(GSp, S^{±})}

which shows that the restriction of Sh_{K^{′p}}(GSp, S^{±}) to Sh_{K}(G, X) descends to a K_{p}-torsor. This shows that the action of Gal(κ̄/κ) on H_{ét}^{1}(A_{x,κ̄}, ℚ_{p}) is induced by a map Gal(κ̄/κ) → K_{p} ⊂ G(ℚ_{p}). In particular this action fixes each s_{α,ét,x}.

To see the final statement note that, by a result of Deligne [De 2, 2.11], the Hodge cycle (s_{α,dR,x}, s_{α,ét,x}) is an absolute Hodge cycle, for each α. In particular, this implies [De 2, 2.7] that s_{α,dR,x} ∈ H_{dR}^{1}(A_{x})^{⊗} ⊗_{κ} κ̄. Moreover, since an absolute Hodge cycle is determined by either its de Rham or étale component Gal(κ̄/κ) fixes s_{α,dR,x} as it fixes s_{α,ét,x}. Hence s_{α,dR,x} ∈ H_{dR}^{1}(A_{x})^{⊗}. □

Corollary (2.2.2). The section s_{α.dR} of V^{⊗} is defined over E, and not just over ℂ.

Proof. This follows by applying (2.2.1) with κ the generic point of each component of Sh_{K}(G, X) (equipped with any complex embedding of κ.). □

(2.2.3) Our main results will concern the situation where K_{p} is a hyperspecial subgroup of G(ℚ_{p}). Recall that this means that there is a reductive group scheme G_{ℤ_{p}} over ℤ_{p} with generic fibre G_{ℚ_{p}} = G ⊗_{ℚ} ℚ_{p} such that K_{p} = G_{ℤ_{p}}(ℤ_{p}) ⊂ G(ℚ_{p}). Any such subgroup is maximal compact in G(ℚ_{p}) [Ti, 1.10, 3.8].

Hyperspecial subgroups exist if and only G_{ℚ_{p}} is quasi-split and split over an unramified extension. Under these conditions the reflex field E = E(G, X) is unramified at p [Mi 4, 4.7]. For the rest of this section we assume that K_{p} is hyperspecial.

Fix an algebraic closure Ē of E, and let π_{0}(Sh_{K}(G, X)) denote the set of connected components of Sh_{K}(G, X) ⊗_{E} Ē. We refer to these as the geometrically connected components of Sh_{K}(G, X).

Proposition (2.2.4). The geometrically connected components of Sh_{K}(G, X) are defined over an extension of E which is unramified at primes dividing p.

Proof. One can extract this from Deligne’s description of the action of Gal(Ē/E) on the geometrically connected components [De 3, 2.6.3]. We give a more direct argument, though still based on the reciprocity law for special points.

Let

π_{0}(Sh_{K_{p}}(G, X)) = \underleftarrow{lim}_{K^{p}}π_{0}(Sh_{K^{p}K_{p}}(G, X)).

By (2.2.5) below, G(𝔸_{f}^{p}) acts transitively on π_{0}(Sh_{K_{p}}(G, X)). As this action commutes with that of Gal(Ē/E), it suffices to exhibit an element of π_{0}(Sh_{K_{p}}(G, X)) whose stabilizer in Gal(Ē/E) corresponds to an (infinite) extension of E which is unramified at each v|p. We will do this by showing the stronger statement that there is a point of Sh_{K_{p}}(G, X) defined over such an extension. The argument is a variant of that of [De 2, p75].

Let x_{∞} ∈ Sh_{K}(G, X)(ℂ) be any point, and T_{∞} ⊂ G_{ℝ} a maximal torus containing the image of the cocharacter h_{x_{∞}} : 𝕊 → G_{ℝ} corresponding to x_{∞}. Let λ_{∞} ∈ Lie T_{∞} be a regular element such that T_{∞} is the centralizer of λ_{∞} in G_{ℝ}. Let T_{p} ⊂ G_{ℚ_{p}} be a maximal torus which is the generic fibre of a maximal torus T_{p,ℤ_{p}} ⊂ G_{ℤ_{p}}. Then T_{p} splits over an unramified extension of ℚ_{p}. Let λ_{p} ∈ Lie T_{p} be a regular element so that T_{p} is the centralizer of λ_{p} in G_{ℚ_{p}}.

Since the map h ↦ [h, λ_{p}] induces an automorphism of Lie G_{ℚ_{p}}/Lie T_{ℚ_{p}}, the image of the map

G(ℚ_{p}) → Lie G_{ℚ_{p}}/Lie T_{p}; g ↦ ad(g)(λ_{p})

contains an open neighbourhood of 0. Hence, if λ ∈ Lie G is a regular element which is sufficiently close to λ_{∞} in the real vector space Lie G_{ℝ} and to λ_{p} in the ℚ_{p}-vector space Lie G_{ℚ_{p}}, then there exists g_{p} ∈ G_{ℤ_{p}}(ℤ_{p}) such that ad(g_{p})(λ) ∈ Lie T_{p} and similarly, there exists g_{∞} ∈ G(ℝ)^{+}, the connected component of the identity in G(ℝ), such that ad(g_{∞})(λ) ∈ Lie T_{∞}. Thus, if T ⊂ G is the centralizer of λ, then

T = g_{p}^{−1}T_{p}g_{p} = g_{∞}^{−1}T_{∞}g_{∞}.

The cocharacter h = ad(g_{∞}^{−1})(h_{x_{∞}}) factors through T_{ℝ}. Let J_{p} = T (ℚ_{p}) ∩ K_{p} = g_{p}^{−1}T_{p,ℤ_{p}}g_{p}(ℤ_{p}), J^{p} = T (𝔸_{f}) ∩ K^{p} and set J = J_{p}J^{p}. The reflex field E(T, h) is contained in the splitting field of T, which is unramified over p. The points of Sh_{J}(T, h) are defined over the abelian extension of E(T, h) corresponding to the quotient T (𝔸_{f})/T (ℚ)T (ℝ)^{+}J, where T (ℝ)^{+} ⊂ T (ℝ) denotes the connected component of the identity [De 3, 2.6]. Since J_{p} ⊂ T (ℚ_{p}) is maximal compact this is an extension of E(T, h) which is unramified at any prime over p. It follows that any point in the image of

Sh_{J_{p}}(T, h) → Sh_{K_{p}}(G, X)

is defined over an extension of E in which p is unramified. □

Lemma (2.2.5). G(𝔸_{f}^{p}) acts transitively on π_{0}(Sh_{K_{p}}(G, X)).

Proof. Fix a cocharacter h_{0} ∈ X, so that X ⟶^{∼} G(ℝ)/K_{∞} as in (2.1). Let x ∈ Sh_{K_{p}}(G, X) and [h, g_{p}, g^{p}] be a representative of x where h ∈ G(ℝ), g_{p} ∈ G(ℚ_{p}) and g^{p} ∈ G(𝔸_{f}^{p}). We have to show that we can choose this representative so that h ∈ G(ℝ)^{+} and g_{p} ∈ K_{p}.

By the real approximation theorem G(ℚ) is dense in G(ℝ), so we may assume h ∈ G(ℝ)^{+}. By (2.2.6) below, we may write g_{p} = g_{0}g′_{p} with g_{0} ∈ G(ℚ)^{+} = G(ℚ)∩G(ℝ)^{+} and g_{p} ∈ K_{p}. Then the triple [g_{0}^{−1}h, g′_{p}, g_{0}^{−1}g^{p}] represents x and has the required property. □

Lemma (2.2.6). Let H be a connected reductive group over ℤ_{(p)}. Then we have H(ℤ_{p}) · H(ℚ)^{+} = H(ℚ_{p}), where H(ℚ)^{+} = H(ℚ) ∩ H(ℝ)^{+}.

Proof. Suppose first that H is a torus over ℤ_{(p)}. As H splits over cyclic extensions of ℚ_{p} and ℝ, a result of Colliot-Thélène and Suresh^{24} [CSu, 2.1,2.2] (see especially the argument just before Proposition 2.2) shows that

H(ℚ) · [H(ℝ)^{+} × H(ℤ_{p})] = H(ℝ) × H(ℚ_{p}).

Hence

H(ℚ)^{+} · [H(ℝ)^{+} × H(ℤ_{p})] = H(ℝ)^{+} × H(ℚ_{p})

and the lemma follows for H a torus.

In general, let T ⊂ H be a maximal torus such that T_{ℚ_{p}} contains a maximal split torus in H_{ℚ_{p}}. Such a torus can be constructed using the same argument as in the proof of (2.2.4): Take a maximal torus T_{p,ℤ_{p}} ⊂ H_{ℤ_{p}} which is the centralizer of a maximal split torus in H_{ℤ_{p}} = H ⊗_{ℤ_{(p)}} ℤ_{p}. Let λ_{p} ∈ Lie H_{ℤ_{p}} be a regular element whose centralizer in H_{ℤ_{p}} is T_{p,ℤ_{p}} and define T to be the centralizer of an element λ ∈ Lie H which is sufficiently close to λ_{p}.

Let H̃ be the universal cover of H^{der}. Then by the strong approximation theorem H̃(ℤ_{p}) · H̃(ℚ)^{+} = H̃(ℚ_{p}), so H(ℤ_{p}) · H(ℚ)^{+} contains the image of H̃(ℚ_{p}). Since H(ℚ_{p})/H̃(ℚ_{p}) is abelian,

H(ℤ_{p}) · H(ℚ)^{+} = H(ℤ_{p})H(ℚ)^{+}H(ℤ_{p}) ⊃ H(ℤ_{p})T (ℚ_{p})H(ℤ_{p}) = H(ℚ_{p}).

where the inclusion follows from the case of a torus considered above, and the final equality follows from the Cartan decomposition [Ti, 3.3.3]. □

(2.3) Integral models: We retain the notation introduced above, so that i : (G, X) ↪ (GSp, S^{±}), and K = K_{p}K^{p} with K_{p} ⊂ G(ℚ_{p}) hyperspecial and equal to G_{ℤ_{p}}(ℤ_{p}) for a reductive group G_{ℤ_{p}} over ℤ_{p}. We will need the following

Lemma (2.3.1). Let W be a ℚ_{p}-vector space and i : G_{ℚ_{p}} ↪ GL(W) a closed embedding of algebraic groups. If p = 2 assume that G_{ℚ_{p}}^{ad} has no factors of type B.^{25} Suppose that G_{ℤ_{p}} is a reductive group over ℤ_{p} with generic fibre G_{ℚ_{p}}.

24We warn the reader that in loc. cit for v real T (O_{v}) should be defined as the connected component of the identity in T (O_{v}) and not the maximal compact subgroup.

25This restriction which arises from the necessary restriction in the result of Prasad-Yu [PY, 1.3] used in the proof, is one of reasons for the restrictions in our results when p = 2.

Then there exists a ℤ_{p}-lattice W_{ℤ_{p}} ⊂ W such that i is induced by a closed embedding i_{ℤ_{p}} : G_{ℤ_{p}} ↪ GL(W_{ℤ_{p}}).

Proof. Let ℤ_{p}^{ur} denote a strict henselisation of ℤ_{p}, and write ℚ_{p}^{ur} = ℤ_{p}^{ur}[1/p]. Write W^{ur} = W ⊗_{ℤ_{p}} ℤ_{p}^{ur} and G_{ℤ_{p}^{ur}} = G_{ℤ_{p}} ⊗_{ℤ_{p}} ℤ_{p}^{ur}. Then G_{ℤ_{p}}(ℤ_{p}^{ur}) is a bounded subgroup of G_{ℚ_{p}}(ℚ_{p}^{ur}) in the sense that any regular function on G_{ℤ_{p}^{ur}} is bounded on G_{ℤ_{p}^{ur}}(ℤ_{p}^{ur}) [Ti, 3.2, 3.4, 3.8].

Let L ⊂ W^{ur} be any ℤ_{p}^{ur}-lattice. The translates g · L for g ∈ G_{ℤ_{p}^{ur}}(ℤ_{p}^{ur}) are contained in a common ℤ_{p}^{ur}-lattice in W^{ur}. Indeed, otherwise there exists v ∈ L and a linear form λ : W^{ur} → ℚ_{p}^{ur} such that the function g ↦ λ(g · v) is unbounded, which contradicts the boundedness of G_{ℤ_{p}^{ur}}(ℤ_{p}^{ur}).

Let Γ = Gal(ℚ_{p}^{ur}/ℚ_{p}). Since Γ is a compact group, the above shows that the translates γ · L for γ ∈ G_{ℤ_{p}^{ur}}(ℤ_{p}^{ur}) ⋊ Γ are contained in a common ℤ_{p}^{ur}-lattice. Hence

W_{ℤ_{p}^{ur}} = ∑_{γ} γ · L

is a lattice in W^{ur}, where γ runs over elements of G_{ℤ_{p}^{ur}}(ℤ_{p}^{ur}) ⋊ Γ. Since W_{ℤ_{p}^{ur}} is G_{ℤ_{p}^{ur}}-stable, i induces a map of algebraic groups over ℤ_{p}^{ur}

i_{ℤ_{p}^{ur}} : G_{ℤ_{p}^{ur}} → GL(W_{ℤ_{p}^{ur}})

by [BT, 1.7.6]. Since W_{ℤ_{p}^{ur}} is Γ-stable it arises from a ℤ_{p}-lattice W_{ℤ_{p}} ⊂ W by étale descent. The map i_{ℤ_{p}^{ur}} is compatible with the descent data on the source and target, as this can be checked on generic fibres, so it descends to a map i_{ℤ_{p}} : G_{ℤ_{p}} → GL(W_{ℤ_{p}}). Finally our hypotheses when p = 2 insure that i_{ℤ_{p}} is a closed embedding as i is [PY, 1.3] (cf. [Va 2, 3.1.2.1]). □

(2.3.2) Now assume that if p = 2 then G has no factors of type B. By (2.3.1) there is a lattice V_{ℤ} ⊂ V such that i is induced by an embedding G_{ℤ_{p}} ↪ GL(V_{ℤ_{p}}). Fix such a V_{ℤ}. Since G_{ℤ_{p}} has generic fibre G⊗_{ℚ}ℚ_{p}, flat base change implies that the closure of G in GL(V_{ℤ_{(p)}}) is a reductive subgroup G_{ℤ_{(p)}} with G_{ℤ_{(p)}} ⊗_{ℤ_{(p)}} ℤ_{p} = G_{ℤ_{p}}. By (1.3.2) G_{ℤ_{(p)}} is defined by a finite collection of tensors (s_{α}) ⊂ V_{ℤ_{(p)}}^{⊗}, which we now fix.

Since K_{p} leaves V_{ℤ_{(p)}} stable by construction, K leaves Vℤ̂ stable provided K^{p} is small enough. Fix such a K^{p}. Let K′_{p} ⊂ GSp(ℚ_{p}) denote the stabilizer of V_{ℤ_{p}}. This is a maximal compact subgroup of GSp(ℚ_{p}) but is not, in general, hyperspecial. By (2.1.2) we may choose K′ = K′_{p}K^{′p} so that i induces an embedding

Sh_{K}(G, X) ↪ Sh_{K′}(GSp, S^{±}).

After replacing K^{′p} be an open subgroup containing K^{p}, we may also assume that K′ leaves Vℤ̂ stable.

(2.3.3) Recall that the interpretation of Sh_{K′}(GSp, S^{±}) as a moduli space of polarized abelian varieties gives rise to a natural model 𝒮_{K′}(GSp, S^{±}) of this scheme over ℤ. For simplicity, we describe this only over ℤ_{(p)}, as this is all we will need.

After scaling the lattice V_{ℤ} ⊂ V_{ℚ}, we may assume that ψ induces an inclusion

V_{ℤ} →^{ψ} V_{ℤ}^{∗} into the dual lattice V_{ℤ}^{∗} ⊂ V_{ℚ}. Let d = |V_{ℤ}^{∗}/V_{ℤ}| and write 2g = dim_{ℚ}V. For a ℤ_{(p)}-scheme T and an abelian scheme B over T we set

V̂^{p}(B) = \underleftarrow{lim}_{p∤n}B[n]

viewed as an étale local system on T. Denote by \underline{Isom}(Vℤ̂^{p}, V̂^{p}(A)) the étale sheaf on T consisting of isomorphisms Vℤ̂^{p} ⟶^{∼} V̂^{p}(A) which are compatible with the pairings induced by ψ and λ up to a ℤ̂^{p×}-scalar. Here ℤ̂^{p} = ∏_{ℓ≠p} ℤ_{ℓ}.

We denote by A_{g,d,K′}(T) the set of isomorphisms classes of triples (A, λ, ε^{p}) consisting of an abelian scheme equipped with a polarization λ : A → A^{∗} of degree d, and a section

ε^{p} ∈ Γ(T, \underline{Isom}(Vℤ̂^{p}, V̂^{p}(A))/K^{′p}).

If K^{′p} is sufficiently small then A_{g,d,K′} is representable by a quasi-projective ℤ_{(p)}-scheme which we denote by the same symbol [MFK, Thm. 7.9]. There is a natural embedding of ℤ_{(p)}-schemes Sh_{K′}(GSp, S^{±}) ↪ A_{g,d,K′}, and we define^{27} 𝒮_{K′}(GSp, S^{±}) as the closure of Sh_{K′}(GSp, S^{±}) in A_{g,d,K′}. For any ring R we will denote by 𝒮_{K′}(GSp, S^{±})_{R} the base change of 𝒮_{K′}(GSp, S^{±}) to R.

Let O denote the ring of integers of E = E(G, X). Let v|p be a prime of E, and denote by O_{(v)} the localization of O at v, and by O_{v} the v-adic completion of O_{(v)}. Denote by 𝒮_{K}^{−}(G, X) the closure of Sh_{K}(G, X) in 𝒮_{K′}(GSp, S^{±})_{O_{(v)}}.

From now on we make the following assumption when p = 2.

(2.3.4) If p = 2 then G has no factors of type B, and the abelian variety over

any characteristic p point of 𝒮_{K}^{−}(G, X) has connected p-divisible group.

These restrictions in the results below arise from those in (1.4.3) and (2.3.1).

Proposition (2.3.5). Let x ∈ 𝒮_{K}^{−}(G, X) be a closed point with residue field of characteristic p, and write Û_{x} for the completion of 𝒮_{K}^{−}(G, X) at x. Then the irreducible components of Û_{x} are formally smooth over O_{(v)}.

Proof. By (2.2.2) the tensors s_{α,dR} ∈ V^{⊗} are defined over E, and for F/E finite and y ∈ Sh_{K}(G, X)(F), the tensors s_{α,ét,y} are Gal(Ē/F)-invariant by (2.2.2).

Let k be the residue field of x, write W = W (k) and denote by 𝒢_{0} the p-divisible group of the abelian variety corresponding to x. Let F/E be a finite extension and x̃ ∈ 𝒮_{K}^{−}(G, X)(F) a point specializing to x. By (1.4.3) the Gal(Ē/F)-invariant tensors s_{α,ét,x̃} give rise, via the p-adic comparison isomorphism, to φ-invariant tensors (s_{α,0}) ⊂ 𝔻(𝒢_{0})(W)^{⊗}, which define a reductive subgroup G_{W} ⊂ GL(𝔻(𝒢_{0})(W)) and such that the filtration on 𝔻(𝒢_{0})(W) ⊗_{W} k is induced by a cocharacter μ_{0} : 𝔾_{m} → G_{W} ⊗_{W} k. Applying the theory of (1.5.3) we obtain a quotient R_{G_{W}} of the universal deformation ring R of 𝒢_{0}.

Extend the valuation v to Ē. Let Û′_{x} be the completion of 𝒮_{K′}(GSp, S^{±}) at x, and let j : Û′_{x} → Spf R be the map defining the p-divisible group over Û_{x} which arises from the universal family of polarized abelian varieties over 𝒮_{K′}(GSp, S^{±}). Then j is an embedding, since a polarization on a deformation of 𝒢_{0} is determined by its restriction to 𝒢_{0}. Let Z ⊂ Û_{x} be the irreducible component containing (the image of) x̃. We claim that the composite

Z ↪ Û_{x} ↪ Û′_{x} ↪ Spf R.

27One can give an explicit description of the image of this embedding and its closure as in [RZ, §6], by considering triples (A, λ, ε^{p}) with λ satisfying a more precise condition than merely being of degree d.

factors through Spf R_{G_{W}} . By (1.5.11) it suffices to check that for any finite extension F′/F in Ē and x̃′ ∈ Sh_{K}(G, X)(F′) lying in Z(F′_{v}), the tensor s_{α,ét,x̃′} maps to s_{α,0} under the p-adic comparison isomorphism for the abelian variety A_{x̃′,Ē}.

A result of Blasius and Wintenberger [Bl] asserts that the p-adic comparison isomorphism takes s_{α,ét,x̃′} to s_{α,dR,x̃′}, so it suffices to check that the isomorphism

H_{cris}^{1}(A_{x}/W) ⊗_{F′} F′_{v} ⟶^{∼} H_{dR}^{1}(A_{}_{x̃′}) ⊗_{F′} F′_{v}

takes s_{α,0} to s_{α,dR,x̃′}, or equivalently that the composite

(2.3.6)  H_{dR}^{1}(Ax̃) ⊗_{F} F′_{v} ⟶^{∼} H_{cris}^{1}(A_{x}/W) ⊗_{F′} F′_{v} ⟶^{∼} H_{dR}^{1}(A_{}_{x̃′}) ⊗_{F′} F′_{v}

takes s_{α,dR,x̃} to s_{α,dR,x̃′}. Now the relative crystalline cohomology of the universal abelian variety over 𝒮_{K′}(GSp, S^{±})_{O_{v}} is an isocrystal on 𝒮_{K′}(GSp, S^{±})_{O_{v}} [BO §2] and the composite in (2.3.6) is given by parallel transport with respect to the Gauss-Manin connection [BO, 2.9]. Since the generic fibre Z_{η} of Z is connected and s_{α,dR}|_{Z_{η}} is parallel, (2.3.6) sends s_{α,dR,x̃} to s_{α,dR,x̃′} as required, and this proves the claim.

Since Z and R_{G_{W}} both have relative dimension gr^{−1}Lie G_{W} ovr W, this shows that Z ⟶^{∼} Spf R_{G_{W}} . As x̃ was an arbitrary point of 𝒮_{K}^{−}(G, X) lifting x, this proves the proposition. □

(2.3.7) We are now ready to prove the first main theorem of this paper. To explain it we recall the extension property of integral canonical models. This was formulated by Milne [Mi 1, §2], in order to give a characterization of the integral models for non-proper Shimura varieties whose existence had been conjectured by Langlands. We follow, in part, the treatment in [Mo, §3].

A scheme or pro-scheme over O_{(v)} is said to have the extension property if for any regular, formally smooth O_{(v)}-scheme S, a map S ⊗ E → X extends to S.

For each K^{p} ⊂ G(𝔸_{f}^{p}) sufficiently small, we now fix K^{′p} ⊂ GSp(𝔸_{f}^{p}) so that there is an embedding

Sh_{K}(G, X) ↪ Sh_{K′}(GSp, S^{±}).

If p = 2 then we continue to assume that the condition (2.3.4) holds.

Theorem (2.3.8). For K = K_{p}K^{p}, let 𝒮_{K}(G, X) denote the normalization of 𝒮_{K}^{−}(G, X), and set

𝒮_{K_{p}}(G, X) = \underleftarrow{lim}_{K^{p}}𝒮_{K_{p}K^{p}}(G, X),

where K^{p} ⊂ G(𝔸_{f}^{p}) runs over sufficiently small compact open subgroups of G(𝔸_{f}^{p}). Then

(1) 𝒮_{K_{p}}(G, X) is an inverse limit of smooth O_{(v)}-schemes with finite étale transition maps, whose restriction to E may be G(𝔸_{f}^{p})-equivariantly identified with Sh_{K_{p}}(G, X).

(2) 𝒮_{K_{p}}(G, X) has the extension property, and in particular depends only on (G, X) and K_{p}, and not on the symplectic embedding i.

Proof. The first claim follows directly from (2.3.5). For the second, suppose that S is regular and formally smooth over O_{(v)}. A morphism S ⊗E → 𝒮_{K′_{p}}(GSp, S^{±}) can be extended to the height 1 primes by the argument of [Mi 1, Prop. 2.13] and then to all of S by a result of Faltings [Mo, 3.6].^{28} Hence a morphism S ⊗E → Sh_{K_{p}}(G, X) extends to a map S → 𝒮_{K_{p}}^{−}(G, X) and this map lifts to 𝒮_{K_{p}}(G, X) since S is formally smooth. This proves (2). □

Corollary (2.3.9). Let V^{∘} be the vector bundle on 𝒮_{K_{p}}(G, X) obtained by pulling back the de Rham cohomology of the universal abelian variety over 𝒮_{K′_{p}}(GSp, S^{±}). Then the sections s_{α,dR} ∈ V^{⊗} extend to G(𝔸_{f}^{p})-invariant sections of V^{∘⊗} over O_{(v)}.

Proof. It suffices to show that for K^{p} sufficiently small, and any closed point x ∈ 𝒮_{K}(G, X) of characteristic p, the s_{α,dR} extend to sections of V^{∘,⊗} over the formal neighbourhood of x, N̂_{x} ⊂ 𝒮_{K}(G, X).

To see this we use the notation of the proof of (2.3.5), which shows that N̂_{x} may be identified with Spf R_{G_{W}} . As in §1.5, let 𝒢_{R_{G_{W}}} denote the tautological deformation of 𝒢 to R_{G_{W}} . By the construction in (1.5.3), the tensors s_{α,0} ∈ 𝔻(𝒢_{0})(W)^{⊗} lift to parallel tensors s′_{α,dR} in 𝔻(𝒢_{R_{G_{W}}})(R_{G_{W}})^{⊗}. The proof of (2.3.5) shows that s_{α,dR} and s′_{α,dR} have the same specialization at a Zariski dense set of closed points of Spec R_{G_{W}} [1/p]. Hence s_{α,dR} = s′_{α,dR}. □

.

§3 Integral canonical models of abelian type

(3.1) Twisting abelian varieties: In this subsection we deduce the existence of integral canonical models for Shimura varieties of abelian type from the results of §2.

We begin with a construction which twists an abelian variety by a torsor. Let Z be an affine group scheme of finite type over ℚ. We will write O_{Z} for the Hopf algebra of Z. Let P be a Z-torsor, and O_{P} the ring of functions on P.

Lemma (3.1.1). Let V be a ℚ-vector space equipped with the structure of an O_{Z} comodule. Then

(1) V is a union of finite dimensional O_{Z}-comodules.

(2) The natural map

(V ⊗_{ℚ} O_{P})^{Z} ⊗_{ℚ} O_{P} ⟶^{∼} V ⊗_{ℚ} O_{P}.

is an isomorphism.

Proof. The first claim is in [Wa, 3.3]. This shows that it is enough to check (2) for finite dimensional comodules, which is well known. □

(3.1.2) Let S be a scheme and A/S an abelian S-scheme. We will consider A in the category of abelian schemes up to isogeny, which is obtained from the category of abelian S-schemes by tensoring the Hom groups by ⊗ℚ. An object in this category will be called an abelian scheme up to isogeny. An isomorphism in this category will be called a quasi-isogeny. For T an S-scheme we set A(T) = Hom_{S}(T, A)⊗_{ℤ} ℚ.

28This is stated in [Mo] for p > 2, but when the second condition in (2.3.4) holds the same proof works for p = 2.

Denote by \underline{Aut}_{ℚ}(A) the ℚ-group whose points in a ℚ-algebra R are given by

\underline{Aut}_{ℚ}(A)(R) = ((End_{S}A) ⊗_{ℤ} R)^{×}.

Let Z and P be as above, and suppose that we are given a map of ℚ-groups Z → \underline{Aut}_{ℚ}(A). We define a pre-sheaf A^{P} in the fppf topology of S by setting

A^{P}(T) = (A(T) ⊗_{ℚ} O_{P})^{Z}.

Lemma (3.1.3). A^{P} is a sheaf, represented by an abelian scheme up to isogeny, which is unique up to canonical isomorphism.

Proof. Using (3.1.1) one sees that V ↦ (V ⊗_{ℚ} O_{P})^{Z} is an exact functor, which implies that A^{P} is a sheaf. A standard argument using the Yoneda lemma shows that the functor taking an abelian scheme up to isogeny to its fppf sheaf is fully faithful. Thus the abelian scheme representing A^{P}, if it exists, is unique up to canonical isomorphism.

Let E/ℚ be a finite Galois extension such that P admits an E-valued point γ̃. Specializing the map in (3.1.1) by γ̃ yields an isomorphism of abelian sheaves

(3.1.3)  A^{P} ⊗_{ℚ} E ⟶^{∼} A ⊗_{ℚ} E.

Now Gal(E/ℚ) acts on the left hand side via its action on E. Transferring this action to the right hand side of (3.1.3), we see that A^{P} = (A ⊗_{ℚ} E)^{Gal(E/ℚ)} so that A^{P} is the kernel of a map α of abelian schemes up to isogeny. Such a kernel is represented by an abelian scheme up to isogeny, which can be constructed by realizing α as a map of abelian schemes, and taking the connected component of the identity of the kernel. □

(3.1.4) Keeping the above assumptions, denote by A^{∗} the dual abelian scheme. Recall that for any ℚ-algebra R, an R-isogeny ϕ : A → A^{∗} is an element of Hom(A, A^{∗}) ⊗ R which has an inverse in Hom(A^{∗}, A) ⊗ R. If R is a subring of ℝ, we say that ϕ is an R-polarization if it is an R-linear combination of polarizations, with positive coefficients. We say ϕ is a weak R-polarization if ϕ or −ϕ is a polarization [De 1, 4.4]. Thus a weak ℚ-polarization is the same thing as a weak polarization, and the set of weak ℝ-polarizations is stable under multiplication by ℝ^{×}. As the set of ℚ-polarizations is a convex cone in Hom(A, A^{∗}) ⊗ ℚ, a ℚ-isogeny ϕ_{0} : A → A^{∗} is a a weak polarization if and only if it is a weak ℝ-polarization when viewed as an ℝ-isogeny.

There is a natural isomorphism α : End_{S}A ⟶^{∼} (End_{S}A^{∗})^{op} and hence an isomorphism

\underline{Aut}_{ℚ}(A) ⟶^{∼} \underline{Aut}_{ℚ}(A^{∗}); g ↦ α(g)^{−1}.

In particular we have a map Z → \underline{Aut}_{ℚ}(A^{∗}). Let c : Z → 𝔾_{m} be a character. We will denote by A(c) the abelian scheme A equipped with the map Z → \underline{Aut}_{ℚ}A obtained by multiplying the map in (3.1.2) by c.

Let λ : A → A^{∗} be a weak polarization . We say that λ is a c-polarization if the induced map A → A^{∗}(c) is compatible with Z-actions.

Lemma (3.1.5). Suppose that P_{ℝ} is a trivial torsor. Then there is a natural isomorphism (A^{∗})^{P} ⟶^{∼} A^{P∗}. If λ : A → A^{∗} is a c-polarization, then there is a unique weak polarization λ^{P} : A^{P} → A^{P∗} such that the diagram

\xymatrix{A^{P} ⊗_{ℚ} O_{P} \ar[r]^{λ^{P} ⊗1} \ar[d]^{∼} & A^{P∗} ⊗_{ℚ} O_{P} \ar[d]^{∼} \\ A ⊗_{ℚ} O_{P} \ar[r]^{λ⊗1} & A^{∗} ⊗_{ℚ} O_{P}}

commutes up to an element of O_{P}^{×}. Here the map on the right is obtained by composing A^{P∗} ⟶^{∼} (A^{∗})^{P} with the isomorphism of (3.1.1)(2).

Proof. Let \underline{Ext}^{1}(A, 𝔾_{m}) denote the sheaf in the fppf topology of S whose value on an S-scheme T is the group of extensions of A by 𝔾_{m}, viewed as fppf sheaves on Spec T. Then there is a canonical isomorphism A^{∗} ⟶^{∼} \underline{Ext}^{1}(A, 𝔾_{m}) [GRR, VIII §3].

Since the functor V ↦ (V ⊗ O_{P})^{Z} of (3.1.1) is exact we have,

\underline{Ext}^{1}(A, 𝔾_{m}) ⊗ O_{P} ⟶^{∼} \underline{Ext}_{O_{P}}^{1} (A ⊗ O_{P}, 𝔾_{m} ⊗ O_{P})

⟶^{∼} \underline{Ext}_{O_{P}}^{1} (A^{P} ⊗ O_{P}, 𝔾_{m} ⊗ O_{P}) ⟶^{∼} \underline{Ext}^{1}(A^{P}, 𝔾_{m}) ⊗ O_{P}.

Taking Z-invariants on both sides yields an isomorphism

(A^{∗})^{P} ⟶^{∼} (\underline{Ext}^{1}(A, 𝔾_{m}) ⊗ O_{P})^{Z} ⟶^{∼} \underline{Ext}^{1}(A^{P}, 𝔾_{m}) ⊗ O_{P}^{Z} ⟶^{∼} A^{P∗}.

Now let λ : A → A^{∗} be a c-polarization, and let f_{c} ∈ O_{P}^{×} be a function such that Z acts on f_{c} via c. Let λ̃^{P} : A^{P} ⊗_{ℚ} O_{P} → A^{P∗}(c) ⊗_{ℚ} O_{P} be the isomorphism making the diagram

(3.1.6)  \xymatrix{A^{P} ⊗_{ℚ} O_{P} \ar[r]^{f_{c}^{−1}·λ̃^{P}} \ar[d]^{∼} & A^{P∗}(c) ⊗_{ℚ} O_{P} \ar[d]^{∼} \\ A ⊗_{ℚ} O_{P} \ar[r]^{λ⊗1} & A^{∗}(c) ⊗_{ℚ} O_{P}}

commute. Here the map on the right is obtained by twisting the composite

A^{P∗} ⊗_{ℚ} O_{P} ⟶^{∼} A^{∗P} ⊗_{ℚ} O_{P} ⟶^{∼} A^{∗} ⊗_{ℚ} O_{P}

by c. By definition, the bottom map is Z-equivariant, as are the restrictions of the vertical maps to A^{P} and A^{P∗} respectively. (Here we regard A^{P} and A^{P∗} as equipped with the trivial Z-action, so Z acts on A^{P∗}(c) via c.) Hence λ̃^{P} sends A^{P} to A^{P∗}(c) and induces an isomorphism λ^{P} : A^{P} ⟶^{∼} A^{P∗}(c).

Finally to see that λ^{P} : A^{P} ⟶^{∼} A^{P∗} is a weak polarization, specialize (3.1.6) by some point of x ∈ P(ℝ), which is non-empty, as we are assuming that P_{ℝ} is trivial. Then one sees that x^{∗}(f_{c}^{−1})λ^{P} is a weak ℝ-polarization. As x^{∗}(f_{c}^{−1}) ∈ ℝ^{×} this implies that λ^{P} is a weak ℝ-polarization, and hence a weak polarization. □

(3.2) The action of G^{ad}(ℚ) : Let (G, X) be a Shimura datum. Denote by G^{ad}(ℝ)_{1} the image of G(ℝ) in G^{ad}(ℝ), and set G^{ad}(ℚ)_{1} = G^{ad}(ℚ) ∩ G^{ad}(ℝ)_{1}. Recall [De 3, §2] that there is an action of G^{ad}(ℚ)_{1} on

Sh(G, X) = \underleftarrow{lim}_{K}Sh_{K}(G, X)

defined as follows. On complex points γ ∈ G^{ad}(ℚ)_{1} takes

(h, g) ∈ G(ℚ)\X × G(𝔸_{f})

to (γ(h), γgγ^{−1}), and this action is defined over E = E(G, X) [De 3, 2.7.12].

Our first task is to give another description of this action in the case when (G, X) is of Hodge type. Suppose that there is an embedding (G, X) ↪ (GSp, S^{±}), where GSp is defined by a symplectic ℚ-vector space V = V_{ℚ}, as in §2. We fix compact open subgroups K ⊂ G(𝔸_{f}) and K′ ⊂ GSp(𝔸_{f}) such that there is an induced embedding of E-schemes.

Sh_{K}(G, X) ↪ Sh_{K′}(GSp, S^{±}).

Let A denote the abelian scheme over the E-scheme Sh_{K}(G, X), obtained by pulling back the universal abelian scheme on Sh_{K′}(GSp, S^{±}). If x ∈ Sh_{K}(G, X)(ℂ), then a lifting of x to X × G(𝔸_{f}) determines an isomorphism V_{ℚ} ⟶^{∼} H_{1}(A_{x}(ℂ), ℚ). For any two liftings these isomorphisms differ by an element of G(ℚ) acting on V_{ℚ}. Hence the composite

(3.2.1)  Z = Z(G) ↪ G ↪ GL(H_{1}(A_{x}(ℂ), ℚ))

depends only on x.

The following lemma is presumably well known.

Lemma (3.2.2). There exists an embedding Z ↪ \underline{Aut}_{ℚ}A which induces the embedding (3.2.1) for any x ∈ Sh_{K}(G, X)(ℂ).

Proof. We will write End A(ℂ) for the endomorphisms of A(ℂ) viewed as a family of complex tori over Sh_{K}(G, X)(ℂ), and we set End_{ℚ}A(ℂ) = End A(ℂ) ⊗_{ℤ} ℚ.

Let \underline{Aut}_{ℚ}A(ℂ) denote the ℚ-group such that for a ℚ-algebra R we have

(\underline{Aut}_{ℚ}A(ℂ))(R) = (End_{ℚ}A(ℂ) ⊗_{ℚ} R)^{×}.

Then \underline{Aut}_{ℚ}A is a closed subgroup of \underline{Aut}_{ℚ}A(ℂ), and a point of \underline{Aut}_{ℚ}A(ℂ) lies on \underline{Aut}_{ℚ}A if and only if the corresponding (R-valued) endomorphism of A(ℂ) is induced by an endomorphism of the abelian scheme A over the E-scheme Sh_{K}(G, X).

For x ∈ Sh_{K}(G, X)(ℂ), the image of the composite

(3.2.3)  G ↪^{(3.2.1)} GL(H^{1}(A_{x}(ℂ), ℚ)) → End_{ℚ}(H^{1}(A_{x}(ℂ), ℚ))

depends only on x. Its commutant is a subalgebra of End_{ℚ}A(ℂ). In particular we see that there is an inclusion Z ⊂ \underline{Aut}_{ℚ}A(ℂ) which induces the embedding (3.2.1) for any complex point x.

To see that this makes Z a subgroup of \underline{Aut}_{ℚ}A note that under the action of G(𝔸_{f}) on the tower \underleftarrow{lim}_{K}Sh_{K}(G, X), Z(ℚ) acts trivially. Hence we see that

Z(ℚ) ⊂ (\underline{Aut}_{ℚ}A)(ℚ).

Let Z^{0} ⊂ Z denote the connected component of the identity. As Z^{0}(ℚ) is Zariski dense in Z^{0} (for example using the real approximation theorem), we see that Z^{0} ⊂ \underline{Aut}_{ℚ}A.

Now suppose that z ∈ Z(R) for some finite ℚ-algebra R. Fix an embedding Ē ⊂ ℂ. Since an R-valued endomorphism of A(ℂ) is determined by its action on the l-adic Tate module \underleftarrow{lim}_{n}A[l^{n}], for any prime l, z is induced by an automorphism of A over Ē. We have to check that σ^{∗}(z) = z for every σ ∈ Gal(Ē/E). Let y ∈ Sh_{K}(G, X)(E_{y}) be a special point defined over its reflex field E_{y}, and associated to a maximal torus T ⊂ G. The observation of the previous paragraph applied to T, implies that for τ ∈ Gal(Ē/E_{y}), τ^{∗}(z) = z over the connected component of Sh_{K}(G, X) containing y. Since G(𝔸_{f}) acts transitively on the connected components of Sh(G, X), this implies that τ^{∗}(z) = z.

Finally, by [De 1, 5.1] and its proof, y may be chosen so that E_{y} is linearly disjoint from any finite extension of E. This implies that the subgroups Gal(Ē/E_{y}) generate Gal(Ē/E) (cf. [De 3, 2.6.3]). □

(3.2.4) We will need a description of the points of Sh(GSp, S^{±}) in terms of abelian varieties up to isogeny. This is slightly different from that sketched in §2 but is more convenient for describing the actions of GSp(𝔸_{f}) and G^{ad}(ℚ) on Sh(G, X).

For an E-scheme T, and an abelian scheme B over T, we set

V̂(B) = \underleftarrow{lim}_{n}B[n]

viewed as a local system on T. Let V̂(B)_{ℚ} = V̂(B) ⊗_{ℤ} ℚ. Note that B ↦ V̂(B)_{ℚ} is functorial for quasi-isogenies.

Recall [De 1, 4.20] that Sh_{K′}(GSp, S^{±})(T) is in canonical bijection with the set of isomorphism class of triples (A_{x}, λ, ε_{K′}), consisting of an abelian scheme up to isogeny A_{x} over T, equipped with a weak polarization λ, and a section

ε_{K′} ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}} , V̂(A_{x})_{ℚ})/K′).

Here, as in (2.3.3), \underline{Isom}(V_{𝔸_{f}} , V̂(A_{x})_{ℚ}) denotes the étale sheaf of isomorphisms V_{𝔸_{f}} = V ⊗_{ℚ} 𝔸_{f} ⟶^{∼} V̂(A_{x})_{ℚ} which is compatible with the pairings induced by ψ and λ, up to a 𝔸_{f}^{×}-scalar.

Now suppose x ∈ Sh_{K}(G, X)(T). Then for any finite index subgroup K′_{1} ⊂ K′ containing K, there is an embedding

Sh_{K}(G, X) ↪ Sh_{K′_{1}}(GSp, S^{±}),

so x gives rise to a point of Sh_{K′_{1}}(GSp, S^{±})(T), and hence to a section

ε_{K′_{1}} ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}} , V̂(A_{x})_{ℚ})/K′_{1}).

Passing to the limit over all such K′_{1} we obtain a section

ε ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}} , V̂(A_{x})_{ℚ})/K).

Now let γ ∈ G^{ad}(ℚ)_{1}, and let P ⊂ G be the fibre of γ. Note that, as γ comes from a point of G(ℝ), P_{ℝ} is trivial. By (3.2.2) we may apply the construction of (3.1) to A_{x}, and obtain an abelian scheme (up to isogeny) A_{x}^{P} over T. Moreover by (3.1.5) λ induces a weak polarization λ^{P} : A^{P} ⟶^{∼} A^{P∗}.

Lemma (3.2.5). Let F be a finite Galois extension of ℚ such that P(F) is nonempty, and let γ̃ ∈ P(F). For σ ∈ Gal(F/ℚ) let cγ̃(σ) = σ(γ̃)γ̃^{−1} ∈ Z(F), and let

ιγ̃ : A_{x}^{P} ⊗_{ℚ} F ⟶^{∼} A_{x} ⊗_{ℚ} F

be the isomorphism obtained by specializing the map of (3.1.1)(2) at γ̃. Then (1) σ ∘ ιγ̃ ∘ σ^{−1} = cγ̃(σ)^{−1} · ιγ̃.

(2) The composite (defined étale locally on T)

V ⊗_{ℚ} 𝔸_{f} ⊗_{ℚ} F →^{γ̃^{−1}·} V ⊗_{ℚ} 𝔸_{f} ⊗_{ℚ} F →^{ε} V̂(A_{x})_{ℚ} ⊗_{ℚ} F →^{ι−1 ; γ̃} V̂(A_{x}^{P})_{ℚ} ⊗_{ℚ} F

is Gal(F/ℚ)-invariant and induces a section

ε^{P} ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}} , V̂(A_{x}^{P})_{ℚ})/γKγ^{−1}).

Proof. Consider the commutative diagram

\xymatrix{A_{x}^{P} \ar[r] \ar[dr]^{ιγ̃} & A_{x} ⊗ F ⊗ O_{P} \ar[r]^{1⊗cγ̃(σ)} \ar[d]^{1⊗γ̃} & A_{x} ⊗ F ⊗ O_{P} \ar[d]^{1⊗γ̃} \\  & A_{x} ⊗ F \ar[r]^{1⊗σ} & A_{x} ⊗ F}

Since A_{x}^{P} = (A_{x} ⊗ O_{P})^{Z}, we see that the composite of the maps in the top line is the composite

A_{x}^{P} →^{(3.1.1)(2)} A_{x} ⊗ F ⊗ O_{P} →^{cγ̃(σ)^{−1}⊗1} A_{x} ⊗ F ⊗ O_{P}.

The formula in (1) now follows from the commutativity of the diagram. The claim in (2) follows from (1) by a simple calculation. □

Lemma (3.2.6). Suppose that γ ∈ G^{ad}(ℚ)_{1}, and let K′_{γ} ⊂ GSp(𝔸_{f}) be a compact open subgroup containing γKγ^{−1} such that the induced map

Sh_{γKγ^{−1}}(G, X) → Sh_{K′_{γ}} (GSp, S^{±})

is a closed embedding. Then the map

Sh_{K}(G, X) → Sh_{K′_{γ}} (GSp, S^{±})

given on T -valued points by the assignment

(A_{x}, λ, ε) ↦ (A_{x}^{P}, λ^{P}, ε^{P})

factors through Sh_{γKγ^{−1}}(G, X). The induced map

Sh_{K}(G, X) → Sh_{γKγ^{−1}}(G, X)

is the one attached to γ in (3.2).

Proof. It suffices to check both statements on ℂ-valued points. Thus we may assume that T = Spec ℂ. Recall that the point (h, g) ∈ GSp(ℚ)\S^{±} ×GSp(𝔸_{f})/K attached to a triple (A_{x}, λ, ε) as above may be computed as follows: Choose an isomorphism α : H_{1}(A_{x}(ℂ), ℚ) ⟶^{∼} V_{ℚ} respecting the polarizations induces by λ and ψ up to a ℚ^{×}-scalar. The Hodge decomposition then gives a grading of V_{ℂ} which determines the cocharacter h, while the element g is given by the composite

V_{𝔸_{f}} →^{ε} V̂(A_{x})_{ℚ} ⟶^{∼} H_{1}(A_{x}(ℂ), ℚ) ⊗_{ℚ} 𝔸_{f} →^{α⊗1} V_{𝔸_{f}} .

The resulting element of GSp(ℚ)\S^{±} × GSp(𝔸_{f})/K does not depend on the choice of α.

Now let F/ℚ be a finite Galois extension and γ̃ ∈ P(F). The same argument as in (3.2.5) shows that there exists an isomorphism α^{P} : H_{1}(A_{x}^{P}(ℂ), ℚ) ⟶^{∼} V_{ℚ} such that the diagram

\xymatrix{V ⊗ 𝔸_{f} ⊗ F \ar[d]^{γ̃^{−1}} \ar[r]^{ε^{P} ⊗1} & V̂(A_{x}^{P})_{ℚ} ⊗ F \ar[r]^{∼} \ar[d]^{ιγ̃} & H_{1}(A_{x}^{P}(ℂ), 𝔸_{f} ⊗ F) \ar[r]^{α^{P} ⊗1} \ar[d]^{ιγ̃} & V ⊗ 𝔸_{f} ⊗ F \ar[d]^{γ̃^{−1}} \\ V ⊗ 𝔸_{f} ⊗ F \ar[r]^{ε⊗1} & V̂(A_{x})_{ℚ} ⊗ F \ar[r]^{∼} & H_{1}(A_{x}(ℂ), 𝔸_{f} ⊗ F) \ar[r]^{α⊗1} & V ⊗ 𝔸_{f} ⊗ F}

commutes. By (3.1.5) γ̃^{−1} ∘ (α^{P} ⊗ 1) is compatible with polarizations up to a F^{×}-scalar, so α^{P} is compatible with polarizations up to a ℚ^{×}-scalar. Hence the maps obtained by taking Gal(F/ℚ)-invariants in the top line may be used to compute the point of GSp(ℚ)\S^{±} × GSp(𝔸_{f})/K′_{γ} corresponding to (A_{x}^{P}, λ^{P}, ε^{P}). The commutativity of the diagram shows that this point is γ(x). □

(3.3) Connected Shimura varieties: We now recall the relationship between Shimura varieties and connected Shimura varieties. This is a variant of [De 3, §2], which was already sketched in [Mo, §3].

(3.3.1) We recall the notation of [De 3, 2.0]. Let H be a group equipped with an action of a group Δ, and Γ ⊂ H a Δ-stable subgroup. Suppose given a Δequivariant map φ : Γ → Δ where Δ acts on itself by inner automorphisms, and suppose that for γ ∈ Γ, φ(γ) acts on H as inner conjugation by γ. Then the elements of the form (γ, φ(γ)^{−1}) form a normal subgroup of the semi-direct product H ⋊ Δ. We denote by H ∗_{Γ} Δ the quotient of H ⋊ Δ by this subgroup.

Converting the left action of G^{ad}(ℚ)_{1} on Sh(G, X) into a right action one obtains a right action of G(𝔸_{f})/Z(ℚ)^{−} ∗_{G(ℚ)/Z(ℚ)} G^{ad}(ℚ)_{1} on Sh(G, X), where Z(ℚ)^{−} denotes the closure of Z(ℚ) in G(𝔸_{f}).

For a subgroup H ⊂ G(ℝ) denote by H_{+} the preimage in H of G^{ad}(ℝ)^{+}, the connected component of the identity in G^{ad}(ℝ). As usual, we write G^{ad}(ℚ)^{+} = G^{ad}(ℚ) ∩ G^{ad}(ℝ)^{+}. Since G(ℚ) is dense in G(ℝ) we have

G(𝔸_{f})/Z(ℚ)^{−} ∗_{G(ℚ)/Z(ℚ)} G^{ad}(ℚ)_{1} ⟶^{∼} G(𝔸_{f})/Z(ℚ)^{−} ∗_{G(ℚ)_{+}/Z(ℚ)} G^{ad}(ℚ)^{+}.

We denote this group by 𝒜 (G).

Let G(ℚ)_{+}^{−} denote the closure of G(ℚ)_{+} in G(𝔸_{f}) and set

𝒜 (G)^{∘} = G(ℚ)_{+}^{−}/Z(ℚ)^{−} ∗_{G(ℚ)_{+}/Z(ℚ)} G^{ad}(ℚ)^{+}.

This group depends only on G^{der} and not on G; it is equal to the completion of G^{ad}(ℚ)^{+} with respect to the topology whose open sets are images of congruence subgroups in G^{der}(ℚ) [De 3, 2.1.15].

(3.3.2) Now let G_{ℤ_{(p)}} be a reductive group over ℤ_{(p)} with generic fibre G. To avoid overloading notation we will usually write G(ℤ_{(p)}) and G(ℤ_{p}) for G_{ℤ_{(p)}}(ℤ_{(p)}) and G_{ℤ_{(p)}}(ℤ_{p}).

Let Z_{ℤ_{(p)}} be the center of G_{ℤ_{(p)}} and G_{ℤ_{(p)}}^{ad} = G_{ℤ_{(p)}}/Z_{ℤ_{(p)}}. Write G(ℤ_{(p)})_{+} = G(ℤ_{(p)}) ∩ G(ℚ)_{+} and G^{ad}(ℤ_{(p)})^{+} = G^{ad}(ℤ_{(p)}) ∩ G^{ad}(ℚ)^{+}. We denote by Z(ℤ_{(p)})^{−} the closure of Z(ℤ_{(p)}) in Z(𝔸_{f}^{p}), and by G(ℤ_{(p)})_{+}^{−} the closure of G(ℤ_{(p)})_{+} in G(𝔸_{f}^{p}). Set

𝒜 (G_{ℤ_{(p)}}) = G(𝔸_{f}^{p})/Z(ℤ_{(p)})^{−} ∗_{G(ℤ_{(p)})_{+}/Z(ℤ_{(p)})} G^{ad}(ℤ_{(p)})^{+}

and

𝒜 (G_{ℤ_{(p)}})^{∘} = G(ℤ_{(p)})_{+}^{−}/Z(ℤ_{(p)})^{−} ∗_{G(ℤ_{(p)})_{+}/Z(ℤ_{(p)})} G^{ad}(ℤ_{(p)})^{+}.

As in (3.3.1), using [De 3, 2.0.12] one sees that the latter group depends only on G_{ℤ_{(p)}}^{der} and not on G_{ℤ_{(p)}}.

Now let G_{2,ℤ_{(p)}} be a reductive group over ℤ_{(p)}, equipped with a central isogeny ψ : G_{ℤ_{(p)}}^{der} → G_{2,ℤ_{(p)}}^{der}. We denote by Z_{2,ℤ_{(p)}} the center of G_{2,ℤ_{(p)}} and by G_{2} and Z_{2} the generic fibres of G_{2,ℤ_{(p)}}, and Z_{2,ℤ_{(p)}} respectively.

Lemma (3.3.3). The natural map

(3.3.4)  𝒜 (G_{ℤ_{(p)}})^{∘}\𝒜 (G_{2,ℤ_{(p)}}) → 𝒜 (G)^{∘}\𝒜 (G_{2})/G_{2}(ℤ_{p}),

where G_{2}(ℤ_{p}) acts on 𝒜 (G_{2}) via right multiplication on G_{2}(ℚ_{p}), is an isomorphism of abelian groups.

Proof. We remind the reader that 𝒜 (G)^{∘} acts on 𝒜 (G_{2}) via

𝒜 (G)^{∘} ⟶^{∼} 𝒜 (G^{der})^{∘} → 𝒜 (G_{2}^{der})^{∘} → 𝒜 (G_{2})

and similarly for 𝒜 (G_{ℤ_{(p)}})^{∘}. Recall that 𝒜 (G_{ℤ_{(p)}})^{∘}\𝒜 (G_{2,ℤ_{(p)}}) and 𝒜 (G)^{∘}\𝒜 (G_{2}) are abelian groups since, as in [De 3, 2.5.1], the strong approximation theorem, implies that G(ℚ)_{+}^{−} (resp. G(ℤ_{(p)})_{+}^{−}) contains the image of G̃(𝔸_{f}) (resp. G̃(𝔸_{f}^{p})) where G̃ is the universal cover of G^{der}. This shows that the map in the lemma is a morphism of abelian groups.

To check that (3.3.4) is surjective let (g, γ) ∈ 𝒜 (G_{2}) with g ∈ G_{2}(𝔸_{f}) and γ ∈ G_{2}^{ad}(ℚ)^{+}. By (2.2.6), after multiplying (g, γ) on the right by an element of G_{2}(ℤ_{p}), we may assume that g_{p}, the p-component of g, is trivial. Then multiplying on the left by an element of 𝒜 (G)^{∘}, we may assume in addition that γ = 1. Then (g, 1) is clearly in the image of (3.3.4)

Next we check injectivity. Let (g, γ) ∈ 𝒜 (G_{2,ℤ_{(p)}}), where we view g ∈ G_{2}(𝔸_{f}) having trivial p-component. Multiplying by an element of 𝒜 (G_{ℤ_{(p)}})^{∘}, we may assume γ = 1. Suppose that (g, 1) is in the kernel of (3.3.4). Then there exists (g_{1}, γ_{1}) ∈ 𝒜 (G)^{∘} and h ∈ G_{2}(ℤ_{p}) such that

(g, 1) = (g_{1}, γ_{1}) · h = (g_{1}, γ_{1})(h, 1) = (g_{1}γ_{1}hγ_{1}^{−1}, γ_{1})

in 𝒜 (G_{2}). Hence g = g_{1}γ_{1}h in G_{2}(𝔸_{f})/Z_{2}(ℚ)^{−}.

Since g_{1} may be approximated by elements of G(ℚ)_{+}, we may replace (g_{1}, γ_{1}) by a pair which represents the same element of 𝒜 (G)^{∘} and such that g_{1p} ∈ G_{2}(ℤ_{p}). Since γ_{1} = g_{1}^{−1}gh^{−1} comparing p-components then shows that γ_{1} ∈ G^{ad}(ℤ_{(p)})^{+}.

Hence we have (g, 1) = (g_{1}γ_{1}hγ_{1}^{−1}, γ_{1}) in

[G_{2}(𝔸_{f}^{p}) × G_{2}(ℤ_{p})]/Z_{2}(ℤ_{(p)})^{−} ∗_{G_{2}(ℤ_{(p)})_{+}/Z_{2}(ℤ_{(p)})} G_{2}^{ad}(ℤ_{(p)})^{+} ⊂ 𝒜 (G_{2}).

Dividing both sides by G_{2}(ℤ_{(p)}) shows that (g^{p}, 1) = (g_{1}^{p}, γ_{1}) in 𝒜 (G_{2,ℤ_{(p)}}). □

(3.3.5) As in [De 3, 2.7], for a locally compact, totally disconnected group Γ, a scheme with continuous Γ-action is an inverse system of quasi-projective^{34} schemes S = \underleftarrow{lim}_{K}S_{K} indexed by the compact open subgroups K ⊂ Γ, and equipped with a right action of Γ, induced by isomorphisms γ : S_{K} ⟶^{∼} S_{γ^{−1}Kγ}, which are the identity if γ ∈ K, and such that S_{K}/(K′/K) ⟶^{∼} S_{K′} for K ⊂ K′ ⊂ Γ compact open. In particular the maps in the inverse limit are finite, so S may also be viewed as a scheme.

If Γ → Γ′ is a map of locally compact, totally disconnected groups, with compact kernel, then Γ acts on the scheme Γ′ × S by (γ′, s)γ = (γ^{−1}γ′, sγ). The quotient scheme S′ = Γ′ × S/Γ is associated to a scheme with continuous Γ′-action, \underleftarrow{lim}S′_{K′} where S′_{K′} = S′/K′. The quotient S′_{K′} also has an explicit description in terms of the S_{K}.

(3.3.6) As in §2, we let E = E(G, X), and we fix an algebraic closure Ē of E. Let E^{p} ⊂ Ē be the maximal extension of E which is unramified at primes dividing p. We set K_{p} = G(ℤ_{p}) and K_{2,p} = G_{2}(ℤ_{p}), and we write K = K_{p}K^{p} for a compact open subgroup of G(𝔸_{f}).

We now assume given a Shimura datum (G_{2}, X_{2}) such that the isomorphism G^{ad} ⟶^{∼} G_{2}^{ad} induces (G^{ad}, X^{ad}) ⟶^{∼} (G_{2}^{ad}, X_{2}^{ad}). Here, as usual, X^{ad} denotes the G^{ad}(ℝ)-conjugacy class which contains the image of X.

Fix a connected component X^{+} ⊂ X. For any compact open subgroup K′ ⊂ G(𝔸_{f}) we denote by Sh_{K′}(G, X)^{+} ⊂ Sh_{K′}(G, X) the geometrically connected component which is the image of X^{+} × 1. By (2.2.4), the geometrically connected components of Sh_{K}(G, X) are defined over E^{p}. In particular Sh_{K}(G, X)^{+} is defined over E^{p}. We write

Sh_{K_{p}}(G, X)^{+} := \underleftarrow{lim}_{K^{p}}Sh_{K}(G, X)^{+},

and Sh(G, X)^{+} for the limit over all compact open subgroups of G(𝔸_{f}).

34over some base, which will be clear in applications

Lemma (3.3.7). Let ℰ (G_{ℤ_{(p)}}^{der}) ⊂ 𝒜 (G_{ℤ_{(p)}}) × Gal(E^{p}/E) denote the stabilizer of Sh_{K_{p}}(G, X)^{+} ⊂ Sh_{K_{p}}(G, X) (viewed as E^{p}-schemes). Then

(1) ℰ (G_{ℤ_{(p)}}^{der}) is an extension of Gal(E^{p}/E) by 𝒜 (G_{ℤ_{(p)}})^{∘} which depends (as an extension) only on G_{ℤ_{(p)}}^{der} and X^{ad}, and not on G_{ℤ_{(p)}}.

(2) There is a canonical isomorphism

𝒜 (G_{ℤ_{(p)}}) ∗_{𝒜 (G_{ℤ_{(p)}} )^{∘}} ℰ (G_{ℤ_{(p)}}^{der}) ⟶^{∼} 𝒜 (G_{ℤ_{(p)}}) × Gal(E^{p}/E).

where an element of ℰ (G_{ℤ_{(p)}}^{der}) acts on 𝒜 (G_{ℤ_{(p)}}) via conjugation by its image

in 𝒜 (G_{ℤ_{(p)}}).

Proof. Consider the exact sequence

(3.3.8)  0 → 𝒜 (G)^{∘} → 𝒜 (G) → 𝒜 (G)^{∘}\𝒜 (G) → 0.

The conjugacy class X determines a map r : Gal(Ē/E) → 𝒜 (G)/𝒜 (G)^{∘} [De 3, 2.5.9], and pulling (3.3.8) back by this map produces an extension ℰ (G^{der}) of Gal(Ē/E) by 𝒜 (G)^{∘}. Clearly 𝒜 (G) ∗_{𝒜 (G)^{∘}} ℰ (G^{der}) ⟶^{∼} 𝒜 (G) × Gal(Ē/E), and this isomorphism identifies ℰ (G^{der}) with the stabilizer of Sh(G, X)^{+} in 𝒜 (G) × Gal(Ē/E) [De 3, 2.6.3].

Now let

𝒜̃(G_{ℤ_{(p)}}) = [G(𝔸_{f}^{p}) × G(ℤ_{p})]/Z(ℤ_{(p)})^{−} ∗_{G(ℤ_{(p)})_{+}/Z(ℤ_{(p)})} G^{ad}(ℤ_{(p)})^{+} ⊂ 𝒜 (G)

and

𝒜̃(G_{ℤ_{(p)}})^{∘} = G(ℤ_{(p)})_{+}^{∼}/Z(ℤ_{(p)})^{−} ∗_{G(ℤ_{(p)})_{+}/Z(ℤ_{(p)})} G^{ad}(ℤ_{(p)})^{+} ⊂ 𝒜 (G)^{∘}

where G(ℤ_{(p)})_{+}^{∼} denotes the closure of G(ℤ_{(p)})_{+} in G(𝔸_{f}^{p}) × G(ℤ_{p}).

By (3.3.3) we have a bijection

𝒜̃(G_{ℤ_{(p)}})^{∘}\𝒜̃(G_{ℤ_{(p)}}) ⟶^{∼} 𝒜 (G)^{∘}\𝒜 (G).

Hence (3.3.8) is obtained by pushout from the sequence

0 → 𝒜̃(G_{ℤ_{(p)}})^{∘} → 𝒜̃(G_{ℤ_{(p)}}) → 𝒜̃(G_{ℤ_{(p)}})^{∘}\𝒜̃(G_{ℤ_{(p)}}) → 0,

and pulling the latter sequence back by r gives rise to an extension ℰ̃(G_{ℤ_{(p)}}^{der}) of Gal(Ē/E) by 𝒜̃(G_{ℤ_{(p)}})^{∘}, which may be identified with the stabilizer of Sh(G, X)^{+} ⊂ Sh(G, X) in 𝒜̃(G_{ℤ_{(p)}}) × Gal(Ē/E). The same argument as in [De 3, 2.5.6] shows that this extension depends only on G_{ℤ_{(p)}}^{der}.

Now the isomorphism 𝒜̃(G_{ℤ_{(p)}})/G(ℤ_{p}) ⟶^{∼} 𝒜 (G_{ℤ_{(p)}}) induces an isomorphism

𝒜̃(G_{ℤ_{(p)}})^{∘}/(𝒜̃(G_{ℤ_{(p)}})^{∘} ∩ G(ℤ_{p})) ⟶^{∼} 𝒜 (G_{ℤ_{(p)}})^{∘},

so the quotient ℰ̃(G_{ℤ_{(p)}}^{der})/(𝒜̃(G_{ℤ_{(p)}})^{∘} ∩ G(ℤ_{p})) is an extension of Gal(Ē/E) by 𝒜 (G_{ℤ_{(p)}})^{∘} which may be identified with the stabilizer of

Sh(G, X)^{+}/[𝒜 (G)^{∘} ∩ G(ℤ_{p})] = Sh_{K_{p}}(G, X)^{+} ⊂ Sh_{K_{p}}(G, X) = Sh(G, X)/G(ℤ_{p}) in 𝒜 (G_{ℤ_{(p)}}) × Gal(Ē/E).

Finally, since Sh_{K_{p}}(G, X)^{+} is defined over E^{p}, Gal(Ē/E^{p}) ⊂ ℰ̃(G_{ℤ_{(p)}}^{der}), and the

quotient Gal(Ē/E^{p})\ℰ̃(G_{ℤ_{(p)}}^{der})/(𝒜̃(G_{ℤ_{(p)}})^{∘} ∩ G(ℤ_{p})) is the required extension of Gal(E^{p}/E) by 𝒜 (G_{ℤ_{(p)}})^{∘}. □

(3.3.9) Consider the right action of ℰ (G_{ℤ_{(p)}}^{der}) on 𝒜 (G_{2,ℤ_{(p)}}) × Sh_{K_{p}}(G, X)^{+} given by (a, s)e = (ē^{−1}aē, se), where ē denotes the image of e in 𝒜 (G_{2,ℤ_{(p)}}). This extends uniquely to an action of the semi-direct product 𝒜 (G_{2,ℤ_{(p)}}) ⋊ ℰ (G_{ℤ_{(p)}}^{der}) in which 𝒜 (G_{2,ℤ_{(p)}}) acts by (a, s) · a′ = (aa′, s). This action descends to an action of 𝒜 (G_{2,ℤ_{(p)}}) ∗_{𝒜 (G_{ℤ_{(p)}} )^{∘}} ℰ (G_{ℤ_{(p)}}^{der}) on [𝒜 (G_{2,ℤ_{(p)}}) × Sh_{K_{p}}(G, X)^{+}]/𝒜 (G_{ℤ_{(p)}})^{∘} where we use the notation of (3.3.5).

Proposition (3.3.10). There is an isomorphism of E^{p}-schemes with continuous G_{2}(𝔸_{f}^{p})-action

Sh_{K_{2,p}}(G_{2}, X_{2}) ⟶^{∼} [𝒜 (G_{2,ℤ_{(p)}}) × Sh_{K_{p}}(G, X)^{+}]/𝒜 (G_{ℤ_{(p)}})^{∘}

which is compatible with the action of Gal(E^{p}/E) when the right hand side is equipped with the action of

𝒜 (G_{2,ℤ_{(p)}}) ∗_{𝒜 (G_{ℤ_{(p)}} )^{∘}} ℰ (G_{ℤ_{(p)}}^{der}) ⟶^{∼} 𝒜 (G_{2,ℤ_{(p)}}) × Gal(E^{p}/E).

Proof. We will use the notation introduced in the proof of (3.3.7). By [De 3, 2.7.11, 2.7.13] there is an isomorphism of Ē-schemes with continuous G_{2}(𝔸_{f})-action

Sh(G_{2}, X_{2}) ⟶^{∼} [𝒜 (G_{2}) × Sh(G, X)^{+}]/𝒜 (G)^{∘}

⟶^{∼} [𝒜̃(G_{2,ℤ_{(p)}}) × Sh(G, X)^{+}]/𝒜̃(G_{ℤ_{(p)}})^{∘}

compatible with the action of Gal(Ē/E) on both sides. Here the second isomorphism follows from (3.3.3). The Gal(Ē/E) actions on the second and third terms are induced by the action of ℰ (G^{der}) and ℰ̃(G_{ℤ_{(p)}}^{der}) respectively, which are defined as in (3.3.9).

Dividing by G_{2}(ℤ_{p}) induces Gal(Ē/E)-compatible isomorphisms of Ē-schemes with continuous G_{2}(𝔸_{f}^{p})-action.

Sh_{K_{2,p}}(G_{2}, X_{2}) ⟶^{∼} [𝒜 (G_{2,ℤ_{(p)}}) × Sh(G, X)^{+}/(𝒜 (G)^{∘} ∩ G(ℤ_{p}))]/𝒜 (G_{ℤ_{(p)}})^{∘}

⟶^{∼} [𝒜 (G_{2,ℤ_{(p)}}) × Sh_{K_{p}}(G, X)^{+}]/𝒜 (G_{ℤ_{(p)}})^{∘}

Finally, descending the first and last term to E^{p}, gives the isomorphism of the proposition. □

(3.4) Integral models II: Now assume that, if p = 2, then (G, X) satisfies the condition (2.3.4). In particular, we have the integral canonical model 𝒮_{K}(G, X) over O_{(v)}, constructed in (2.3.8) as the normalization of the closure of

Sh_{K}(G, X) ↪ 𝒮_{K′}(GSp, S^{±}).

We again denote by (A, λ) the pullback to 𝒮_{K}(G, X) of the universal polarized abelian scheme over 𝒮_{K′}(GSp, S^{±}).

Lemma (3.4.1). The embedding of (3.2.2) induces an embedding

Z ↪ \underline{Aut}_{ℚ}A.

Proof. This follows from the fact that any endomorphism of A|_{Sh_{K}(G,X)} extends to an endomorphism over 𝒮_{K}(G, X) [FC,I 2.7]. Note that this shows that the group \underline{Aut}_{ℚ}A in the statement of the lemma agrees with the one denoted by the same symbol in (3.1.2). □

(3.4.2) Let B be an abelian scheme up to isogeny over an O_{(v)} scheme T. Using the notation of (2.3.3), we write

V̂^{p}(B)_{ℤ_{(p)}} = V̂^{p}(B) ⊗_{ℤ} ℤ_{(p)} = V̂^{p}(B) ⊗_{ℤ} ℚ.

Note that this is functorial for quasi-isogenies.

Now consider the category obtained from the category of abelian schemes over T, by tensoring the Hom groups by ⊗ℤ_{(p)}. An object in this category will be called an abelian scheme up to prime to p isogeny. An isomorphism in this category will be called a p′-quasi-isogeny.

Now fix a ℤ-lattice V_{ℤ} ⊂ V as in (2.3.2). By (2.3.3) any x ∈ Sh_{K′}(GSp, S^{±})(T) gives rise to a triple (A_{x}, λ, ε_{K′}^{p}), consisting of an abelian scheme, a polarization and a section of Γ(T, \underline{Isom}(Vℤ̂^{p}, V̂^{p}(A_{x}))/K^{′p}). We will view A_{x} as an abelian scheme up to prime to p-isogeny, λ as a weak polarization and ε_{K′}^{p} as a section of Γ(T, \underline{Isom}(V_{𝔸_{f}^{p}} , V̂^{p}(A_{x})_{ℤ_{(p)}}/K^{′p})). (When viewed in this way, the triple depends, up to isomorphism, only on V_{ℤ_{(p)}} and not on V_{ℤ}.)

If x ∈ Sh_{K}(G, X)(T) then, as in (3.2.4), we may promote ε_{K′}^{p} to a section

ε^{p} ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}^{p}} , V̂^{p}(A_{x})_{ℤ_{(p)}})/K^{p}).

Fix a point x as above. Let γ ∈ G^{ad}(ℚ) and let P be the corresponding Z-torsor, as in (3.2.4).

Lemma (3.4.3). Fix a finite Galois extension F/ℚ with P(F) non-empty and let γ̃ ∈ P(F). For σ ∈ Gal(F/ℚ) let cγ̃(σ) = σ(γ̃)γ̃^{−1} ∈ Z(F), and let

ιγ̃ : A_{x}^{P} ⊗_{ℚ} F ⟶^{∼} A_{x} ⊗_{ℚ} F

be the isomorphism obtained by specializing the map of (3.1.1)(2) at γ̃. Then (1) σ ∘ ιγ̃ ∘ σ^{−1} = cγ̃(σ)^{−1} · ιγ̃.

(2) The composite (defined étale locally on T)

V ⊗_{ℚ} 𝔸_{f}^{p} ⊗_{ℚ} F →^{γ̃^{−1}·} V ⊗_{ℚ} 𝔸_{f}^{p} ⊗_{ℚ} F →^{ε^{p}} V̂^{p}(A_{x}) ⊗_{ℚ} F →^{ι−1 ; γ̃} V̂^{p}(A_{x}^{P}) ⊗_{ℚ} F

is Gal(F/ℚ)-invariant and induces a section

ε^{p,P} ∈ Γ(T, \underline{Isom}(V_{𝔸_{f}^{p}} , V̂^{p}(A_{x}^{P})_{ℤ_{(p)}})/γK^{p}γ^{−1})

Proof. It suffices to prove the lemma when A_{x} = A is the universal abelian scheme over 𝒮_{K}(G, X). Since 𝒮_{K}(G, X) is flat over O_{(v)} it suffices to check (1) and (2) over Sh_{K}(G, X). In this case the lemma is a formal consequence of (3.2.5). □

(3.4.4) In the following we will consider the triple (A_{x}^{P}, λ^{P}, ε^{p,P}). Note that A_{x}^{P} is defined only in the isogeny category of abelian T -schemes. If (A′, λ′, ε^{p′}) is a triple of the form discussed in (3.4.2), so that A′ is an abelian scheme up to prime to p isogeny, we will say that (A′, λ′, ε^{p′}) is isogenous to (A_{x}^{P}, λ^{P}, ε^{p,P}) if these triples become isomorphic when one considers A′ as an abelian scheme up to isogeny.

Lemma (3.4.5). The action of 𝒜 (G) on Sh(G, X) induces an action of 𝒜 (G_{ℤ_{(p)}}) on 𝒮_{K_{p}}(G, X). More precisely, if (h, γ^{−1}) ∈ 𝒜 (G_{ℤ_{(p)}}) then the above action induces a map

(h, γ^{−1}) : 𝒮_{K}(G, X) → 𝒮_{γKγ^{−1}}(G, X)

such that the triple attached to (h, γ^{−1})(x) is isogenous to (A_{x}^{P}, λ^{P}, ε^{p,P} ∘ γ̃hγ̃^{−1}), where we use the notation of (3.4.3).

Proof. Since γ ∈ G(ℤ_{(p)}), we have γK_{p}γ^{−1} ⊂ K_{p}, so the action of 𝒜 (G) on Sh(G, X) induces an action of 𝒜 (G_{ℤ_{(p)}}) on Sh_{K_{p}}(G, X). More precisely, (h, γ^{−1}) induces a map

Sh_{K}(G, X) → Sh_{γKγ^{−1}}(G, X).

The extension property implies that this map extends to a map

𝒮_{K}(G, X) → 𝒮_{γKγ^{−1}}(G, X).

By (3.2.6) and [FC, I 2.7], the triple attached to (h, γ^{−1})(x) is (A_{x}^{P}, λ^{P}, (ε^{p}∘h)^{P}). As (ε^{p} ∘ h)^{P} = ε^{p,P} ∘ γ̃hγ̃^{−1} the lemma follows □

Proposition (3.4.6). The group^{37}

Δ(G, G^{ad}) = ker (𝒜 (G_{ℤ_{(p)}}) → 𝒜 (G_{ℤ_{(p)}}^{ad}))

acts freely on 𝒮_{K_{p}}(G, X).

Proof. Let (h, γ^{−1}) ∈ Δ(G, G^{ad}) with h ∈ G(𝔸_{f}^{p}) and γ ∈ G^{ad}(ℤ_{(p)})^{+}. As usual, we denote by P the Z-torsor associated to γ, and we fix a Galois extension F/ℚ and a point γ̃ ∈ P(F).

Let x ∈ 𝒮_{K_{p}}(G, X)(T), where T is the spectrum of an algebraically closed field, and suppose that (h, γ^{−1}) fixes x. Write (A_{x}, λ, ε^{p}) for the corresponding triple introduced in (3.4.2). Then by (3.4.5), for every compact open subgroup K^{p} ⊂ G(𝔸_{f}^{p}) there exists a quasi-isogeny α = α(K^{p}) : A_{x} ⟶^{∼} A_{x}^{P} respecting polarizations, and such that the left hand square of the following diagram (defined étale locally on T) commutes modulo K^{p} (That is up to multiplication by an element of K^{p} on the bottom left hand corner.)

(3.4.7)  \xymatrix{V̂^{p}(A_{x}) ⊗ F \ar[r]^{∼}_{α⊗1} & V̂^{p}(A_{x}^{P}) ⊗ F \ar[r]^{ιγ̃} & V̂^{p}(A_{x}) ⊗ F \\ V ⊗ 𝔸_{f}^{p} ⊗ F \ar[r]^{γ̃hγ̃^{−1}} \ar[u]^{ε^{p}} & V ⊗ 𝔸_{f}^{p} ⊗ F \ar[u]^{ε^{p,P}} \ar[r]^{γ̃^{−1}·} & V ⊗ 𝔸_{f}^{p} ⊗ F \ar[u]^{ε^{p}}}

37Contrary to the assertion in [Mo, 3.21.1], this group is not in general finite.

while the right square commutes by the definition of ε^{p,P}.

For K^{p} sufficiently small, the map α(K^{p}) is unique. Hence if K^{p} is sufficiently small then α does not depend on K^{p}, and we may assume that (3.4.7) commutes.

Note that the composite of the maps in the lower row of (3.4.7) is hγ̃^{−1}. Since (h, γ^{−1}) ∈ Δ(G, G^{ad}), we have hγ̃^{−1} ∈ Z(𝔸_{f}^{p} ⊗ F). Hence

ιγ̃ ∘ α ∈ Z(𝔸_{f}^{p} ⊗ F) ∩ (\underline{Aut}_{ℚ}A_{x})(F) = Z(F) ⊂ (\underline{Aut}_{ℚ}A_{x})(𝔸_{f}^{p} ⊗ F).

Hence hγ̃^{−1} ∈ Z(F). Writing

γ̃ = (h^{−1}γ̃) · (γ̃^{−1}hγ̃)

and noting that γ̃^{−1}hγ̃ is Gal(F/ℚ)-invariant, we see that the cocycle cγ̃(σ) is trivial.

It follows that P is trivial. Let P_{ℤ_{(p)}} denote the fibre of G_{ℤ_{(p)}} → G_{ℤ_{(p)}}^{ad} over γ. Then P_{ℤ_{(p)}} is a Z_{ℤ_{(p)}}-torsor with generic fibre P. It follows from (3.4.8) below that P_{ℤ_{(p)}} is the trivial torsor, so that we may assume that γ̃ ∈ G(ℤ_{(p)})_{+}, and after replacing h by hγ̃^{−1}, that γ̃ = 1.

In particular, we may identify A_{x} and A_{x}^{P} and regard the latter as an abelian scheme up to prime to p isogeny. Since (h, 1) fixes x, there exists a p′-quasi-isogeny α : A_{x} ⟶^{∼} A_{x} making the left square in (3.4.7) commute.^{38} Moreover, we have already observed that h ∈ Z(ℚ) (as we can now take F = ℚ above).

It remains to show that, via (3.4.1), an element z ∈ Z(ℚ) induces a p′-isogeny A_{x} → A_{x} if and only if z ∈ Z(ℤ_{(p)}). Since the degree of an isogeny of abelian schemes is a locally constant function, it suffices to check this over the generic fibre of 𝒮_{K}(G, X), and then on complex points. Then z induces a p′-isogeny if and only if

z ∈ Z(ℚ) ∩ K′_{p} = Z(ℚ) ∩ K_{p} = Z(ℚ) ∩ G(ℤ_{(p)}) = Z(ℤ_{(p)}).

□

Lemma (3.4.8). Let P_{ℤ_{(p)}} be a Z_{ℤ_{(p)}}-torsor. If P = P_{ℤ_{(p)}}|_{Spec ℚ} is a trivial Ztorsor, then P_{ℤ_{(p)}} is a trivial Z_{ℤ_{(p)}}-torsor.

Proof. In order not to overload notation, we write Z be Z_{ℤ_{(p)}}. By [DG, XII, Thm. 4.7] Z is of multiplicative type, and hence is an extension of a finite flat multiplicative group scheme by a torus Z^{0} ⊂ Z. Now consider the commutative diagram

\xymatrix{H^{0}(ℤ_{(p)}, Z/Z^{0}) \ar[r] \ar[d] & H^{1}(ℤ_{(p)}, Z^{0}) \ar[r] \ar[d] & H^{1}(ℤ_{(p)}, Z) \ar[r] \ar[d] & H^{1}(ℤ_{(p)}, Z/Z^{0}) \ar[d] \\ H^{0}(ℚ, Z/Z^{0}) \ar[r] & H^{1}(ℚ, Z^{0}) \ar[r] & H^{1}(ℚ, Z) \ar[r] & H^{1}(ℚ, Z/Z^{0})}

where the H^{∗} means cohomology in the fppf topology of the spectrum of the indicated ring.

38Note that when γ̃ ≠ 1, it does not make sense to require that α : A_{x} ⟶^{∼} A_{x}^{P} be a p′-quasi-isogeny, since A_{x}^{P} is defined only up to isogeny.

For a finite flat ℤ_{(p)}-scheme the ℤ_{(p)}-points are in bijection with the ℚ-points. Hence the map on the left is a bijection and the map on the right is an injection. To prove the lemma, we have to show that the third map is an injection. By the five lemma it suffices to prove that the second map is an injection.

To see this note that the kernel of this map is the cokernel of

Z^{0}(ℚ)/Z^{0}(ℤ_{(p)}) → H_{(p)}^{1}(ℤ_{(p)}, Z^{0}),

where H_{(p)}^{1} denotes cohomology with support in (p). By Lang’s lemma we have H^{1}(ℤ_{p}, Z^{0}) = 0, so that

H_{(p)}^{1}(ℤ_{(p)}, Z^{0}) = H_{(p)}^{1}(ℤ_{p}, Z^{0}) ⟶^{∼} Z^{0}(ℚ_{p})/Z^{0}(ℤ_{p}).

Finally, by (2.2.6), Z^{0}(ℚ)/Z^{0}(ℤ_{(p)}) → Z^{0}(ℚ_{p})/Z^{0}(ℤ_{p}) is a surjection and the lemma follows. □

(3.4.9) We are now ready to prove the main theorem of this section. Let (G_{2}, X_{2}) be as in (3.3.6). Thus, G_{2,ℤ_{(p)}} is a reductive ℤ_{(p)}-group equipped with a central isogeny G_{ℤ_{(p)}}^{der} → G_{2,ℤ_{(p)}}^{der} which induces an isomorphism (G^{ad}, X^{ad}) ⟶^{∼} (G_{2}^{ad}, X_{2}^{ad}), and K_{2,p} = G_{2,ℤ_{(p)}}(ℤ_{p}) ⊂ G_{2}(ℚ_{p}) is a hyperspecial subgroup. We will denote by E_{2} = E(G_{2}, X_{2}) ⊂ E the reflex field of (G_{2}, X_{2}), and by O_{2} its ring of integers.

Theorem (3.4.10). Let v|p be a prime of O_{2}. Then the tower of E_{2}-schemes Sh_{K_{2,p}}(G_{2}, X_{2}) admits a G_{2}(𝔸_{f}^{p})-equivariant extension to a tower of O_{2,(v)}-schemes

𝒮_{K_{2,p}}(G_{2}, X_{2}) = \underleftarrow{lim}_{K_{2}^{p}}𝒮_{K_{2,p}K_{2}^{p}}(G_{2}, X_{2})

which has the extension property.

Proof. For each K = K_{p}K^{p} with K^{p} ⊂ G(𝔸_{f}^{p}) a compact open subgroup, let K′ = K′_{p}K^{′p} be as in (2.3.7), and denote by 𝒮_{K}(G, X) the normalization of the closure of

Sh_{K}(G, X) ↪ 𝒮_{K′}(GSp, S^{±})_{O_{(p)}}.

Note that the O_{(v)}-scheme denoted by the same symbol in (2.3.8) is the localization of 𝒮_{K}(G, X) at (v). By (2.3.8)

𝒮_{K_{p}}(G, X) = \underleftarrow{lim}_{K^{p}}𝒮_{K}(G, X)

is a G(𝔸_{f}^{p})-equivariant tower of smooth O_{(p)}-schemes having the extension property. In particular, the action of 𝒜 (G_{ℤ_{(p)}}) on Sh_{K_{p}}(G, X) extends to 𝒮_{K_{p}}(G, X), as already noted in (3.4.5).

As in (3.3.6), let E^{p} ⊂ Ē be the maximal subfield in which p is unramified, and write O_{(p)}^{p} for its ring of integers localized at (p). Denote by 𝒮_{K}(G, X)^{+} the closure of Sh_{K}(G, X)^{+} in 𝒮_{K}(G, X) ⊗_{O_{(p)}} O_{(p)}^{p}. Set

𝒮_{K_{p}}(G, X)^{+} = \underleftarrow{lim}_{K^{p}}𝒮_{K}(G, X)^{+}.

The action of 𝒜 (G_{ℤ_{(p)}}) × Gal(E^{p}/E) on 𝒮_{K_{p}}(G, X) ⊗_{E} E^{p}, induces an action of ℰ (G_{ℤ_{(p)}}^{der}) on 𝒮_{K_{p}}(G, X)^{+}. Write

(3.4.11)  𝒮_{K_{2,p}}(G_{2}, X_{2}) = [𝒜 (G_{2,ℤ_{(p)}}) × 𝒮_{K_{p}}(G, X)^{+}]/𝒜 (G_{ℤ_{(p)}})^{∘}.

Since ker (𝒜 (G_{ℤ_{(p)}})^{∘} → 𝒜 (G_{2,ℤ_{(p)}})) is a subgroup of Δ(G, G^{ad}), (3.4.6) and the description of the quotients of the form (3.4.11) given in [De 3, 2.7.3] implies that (3.4.11) is a tower of smooth O_{(p)}^{p}-schemes. More precisely a connected component of an element in this tower is a quotient of 𝒮_{K}(G, X)^{+}, for some K, by the free action of a finite subgroup of Δ(G, G^{ad}). The ℰ (G_{ℤ_{(p)}}^{der})-action on 𝒮_{K_{p}}(G, X)^{+} induces a 𝒜 (G_{2,ℤ_{(p)}}) × Gal(E^{p}/E)-action on (3.4.11). By (3.3.10), the generic fibre of (3.4.11) may be 𝒜 (G_{2,ℤ_{(p)}}) × Gal(E^{p}/E)-equivariantly identified with the E^{p}- scheme Sh_{K_{2,p}}(G_{2}, X_{2}). Hence (3.4.11) descends to a tower of smooth O_{(p)}-schemes, again denoted 𝒮_{K_{2,p}}(G_{2}, X_{2}) whose generic fibre may be G_{2}(𝔸_{f}^{p})-equivariantly identified with Sh_{K_{2,p}}(G, X).

Let 𝒮_{K_{2,p}}(G_{2}, X_{2})^{+} be the image of 𝒮_{K_{p}}(G, X)^{+} in 𝒮_{K_{2,p}}(G_{2}, X_{2}). By what we saw above, the morphism 𝒮_{K_{2,p}}(G_{2}, X_{2})^{+} → 𝒮_{K_{p}}(G, X)^{+} is pro-finite étale. As 𝒮_{K_{2,p}}(G_{2}, X_{2})^{+} satisfies the extension property, the argument in [Mo, Prop. 3.21.4] shows that 𝒮_{K_{p}}(G, X)^{+} does also. It follows that 𝒮_{K_{2,p}}(G_{2}, X_{2})_{O_{(p)}^{p}} satisfies the extension property, and then one sees easily that so does 𝒮_{K_{2,p}}(G_{2}, X_{2}) as an O_{(p)}- scheme.

Finally, since Sh_{K_{2,p}}(G_{2}, X_{2}) has a canonical model defined over E_{2}, the tower of E-schemes Sh_{K_{2,p}}(G_{2}, X_{2}) is equipped with a descent datum relative to E/E_{2}. The extension property implies that this induces a descent datum on 𝒮_{K_{2,p}}(G_{2}, X_{2}) relative to O_{(p)}/O_{2,(p)}. Since descent for a quasi-projective schemes relative to a finite flat morphism is effective, 𝒮_{K_{2,p}}(G_{2}, X_{2}) descends to a G_{2}(𝔸_{f}^{p})-equivariant tower of smooth O_{2,(p)}-schemes having the extension property. □

(3.4.12) Let (G_{2}, X_{2}) be a Shimura datum of abelian type and K_{2} = K_{2,p}K_{2}^{p} ⊂ G_{2}(𝔸_{f}) a compact open with K_{2,p} hyperspecial. By (3.4.10), Sh_{K_{2,p}}(G_{2}, X_{2}) has a canonical integral model at any prime v|p of E(G_{2}, X_{2}), provided K_{2,p} can be constructed starting with a hyperspecial subgroup of G(ℚ_{p}), where (G, X) is a Shimura datum of Hodge type as in (3.3.6). It remains to show that we can always choose (G, X) so that this condition is satisfied.^{40}

To do this we recall Deligne’s classification of Shimura data of abelian type.

Lemma (3.4.13). Let (H, Y) be a Shimura datum of abelian type with H an adjoint group. Then there exists a central isogeny H^{♯} → H such that

(1) If (G, X) is a Shimura datum of Hodge type such that (G^{ad}, X^{ad}) is isomorphic to (H, Y) then G^{der} is a quotient of H^{♯}.

(2) There exists a Shimura datum of Hodge type (G, X) such that (G^{ad}, X^{ad}) is isomorphic to (H, Y) and G^{der} = H^{♯}.

(3) If H is quasi-split and unramified at a prime p, then (G, X) in (2) can be chosen so that G is quasi-split and unramified at p.

Proof. The first two claims are implicit in [De 3], but we explain how to reduce them to the results proved by Deligne in loc. cit.

Let (H, Y) = (H_{1}, Y_{1}) × ⋯ × (H_{r}, Y_{r}) with H_{i} adjoint and ℚ-simple. For each i = 1, …, r let H_{i,v}, for v = 1, …, j_{i}, denote the simple factors of H_{i} over ℝ. Write H̃_{i,v} (resp. H̃_{i}, resp. H̃) for the universal cover of H_{i,v}, (resp. H_{i}, resp. H). Let H_{i}^{♯} be the cover of H_{i} constructed in [De 3, 2.3.7, 2.3.8]. Then H_{i}^{♯} satisfies (1) with (H_{i}, Y_{i}) in place of (H, Y). (See the remark at the end of [De 3, 2.3.7].) We fix h ∈ Y, and a fractional lifting h̃ : 𝕊 → H̃_{ℝ} of h. We denote by h̃_{i,v} : 𝕊 → H̃_{i,v} the components of h̃. Finally we denote by μ̃_{h} and μ̃_{h_{i,v}} the fractional cocharacters attached to h̃ and h̃_{i,v} respectively [De 3, 1.2.4].

40At least when p > 2. When p = 2 whether (G, X) can be made to satisfy (2.3.4) depends on (G_{2}, X_{2}).

Suppose that (G, X) is as in (1) and consider an embedding of Shimura data (G, X) ↪ (GSp(V), S^{±}). We think if V as a H̃-representation via H̃ → G^{der} ⊂ G. Let W ⊂ V_{ℂ} be an irreducible H̃-subrepresentation. Write W = ⊗_{i,v}W_{i,v} where W_{i,v} is an irreducible complex representation of H̃_{i,v}. If W_{i,v} is non-trivial, let w_{i,v} be the highest weight of W_{i,v} as a H̃_{i,v}-representation, and for each i, v, let S_{i,v} be the set of w_{i,v} with W_{i,v} non-trivial as W runs over irreducible factors of V_{ℂ}. For each i the absolute Galois group of ℚ permutes the H_{i,v} transitively. If a Galois element sends H_{i,v} to H_{i,v′} then it maps S_{i,v} bijectively onto S_{i,v′}. Note that the S_{i,v} are non-empty since V is a faithful representation of G.

As in [De 3, 1.3.5], μ̃_{h} acting on W has two weights, a, a + 1, for some a ∈ ℚ. Hence for each W there is at most one pair (i, v) such that the action of μ̃_{h_{i,v}} on W_{i,v} is non-trivial. For such a pair the weights of μ̃_{h_{i,v}} on W_{i,v} have the form a, a+1, and hence w_{i,v} is a fundamental weight which appears in the table of [De 3, 1.3.9]. For each i at least one of the factors H_{i,v} is non-compact. Hence the remark on transitivity of the Galois action shows that all the S_{i,v} consist of fundamental weights which appear in loc. cit.

Now for each i let 𝒮_{i} be the union of the S_{i,v}, v = 1, …, j_{i}. where we now consider the elements of S_{i,v} as fundamental weights of H_{i}. Then the set 𝒮_{i} satisfies^{42} the conditions (b),(c),(d) of [De 3, 2.3.7]: It is a non-empty, Galois stable, set of fundamental weights of H_{i} such that any element which is a fundamental weight for a non-compact H_{i,v} appears in [De 3, 1.3.9]. Hence, by definition, the action of H̃_{i} on V factors through H_{i}^{♯}.

This proves that H^{♯} satisfies (1), and (2) follows from [De 3, 2.3.10]. Finally, to see (3), note that H_{i} has the form Res_{F/ℚ}H_{i}^{s} with F a totally real field, and H_{i}^{s} absolutely simple [De 3, 2.3.4]. If H is quasi-split and unramified at p, then p is unramified in F. Now (3) follows from the proof of [De 3, 2.3.10] since the field K in loc. cit can be chosen to be an extension of F which is unramified at primes above p. □

Corollary (3.4.14). Let (G_{2}, X_{2}) be a Shimura datum of abelian type, and K_{2} = K_{2,p}K_{2}^{p} ⊂ G_{2}(𝔸_{f}) compact open with K_{2,p} hyperspecial. If p > 2,^{43} and v|p is a prime of the ring of integers O_{2} ⊂ E(G_{2}, X_{2}), then the tower of E_{2}-schemes Sh_{K_{2,p}}(G_{2}, X_{2}) admits a G_{2}(𝔸_{f}^{p})-equivariant extension to a tower of O_{2,(v)}-schemes

𝒮_{K_{2,p}}(G_{2}, X_{2}) = \underleftarrow{lim}_{K_{2}^{p}}𝒮_{K_{2,p}K_{2}^{p}}(G_{2}, X_{2})

which has the extension property.

Proof. Let (G, X) be a Shimura datum of Hodge type and G^{der} → G_{2}^{der} a central isogeny inducing an isomorphism (G^{ad}, X^{ad}) ⟶^{∼} (G_{2}^{ad}, X_{2}^{ad}). By (3.4.13)(1), G^{der}, and hence G_{2}^{der}, is a quotient of G^{ad♯}. Hence by (3.4.13)(3), we may assume that G is quasi-split and unramified at p.

42In fact, to be strictly compatible with Deligne’s notation we should take 𝒮_{i} = {{w} : w ∈ S_{i,v}, v = 1, … j_{i}}

43But see (3.4.15) below.

Now let G_{2,ℤ_{(p)}} be a reductive group over ℤ_{(p)} with generic fibre G_{2}, and such that K_{2,p} = G_{2,ℤ_{(p)}}(ℤ_{p}). The central isogeny G^{der} → G_{2}^{der} extends uniquely to a central isogeny G_{ℤ_{(p)}}^{der} → G_{2,ℤ_{(p)}}^{der}, of reductive groups over ℤ_{(p)}.

Let Z ⊂ G denote the center of G. Since G admits an extension to a reductive group over ℤ_{(p)}, Z admits an extension to a flat ℤ_{(p)}-group Z_{ℤ_{(p)}} of multiplicative type. The kernel of G^{der} × Z → G is a finite, multiplicative ℤ_{(p)}-group, and extends to a finite flat, multiplicative subgroup C ⊂ G_{ℤ_{(p)}}^{der} × Z_{ℤ_{(p)}}. The quotient G_{ℤ_{(p)}} = G_{ℤ_{(p)}}^{der} × Z_{ℤ_{(p)}}/C is a reductive group over ℤ_{(p)} with generic fibre G, and derived group G_{ℤ_{(p)}}^{der}.

Thus the assumptions introduced at the end of (3.3.2) and in (3.3.6) are satisfied, and we may apply (3.4.10) to conclude the corollary. □

(3.4.15) If p = 2 then the conclusion of the corollary still holds provided the pair (G, X) with G quasi-split and unramified at p, which was constructed in the proof of (3.4.14), can be chosen so as to satisfy the conditions (2.3.4).

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