We use Lau's classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

When p > 2, we construct a Hodge-type analogue of Rapoport-Zink spaces under the unramifiedness assumption, as formal schemes parametrizing 'deformations' (up to quasiisogeny) of p-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of 'local Shimura varieties' conjectured by Rapoport and Viehmann.

Let (G, X) be a Shimura datum of Hodge type. Let p be an odd prime such that G Qp splits after a tamely ramified extension and p ∤ |π 1 (G der )|. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and G Qp is either unramified or residually split, we prove the generalisation of Mantovan's formula for the l-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo p of the Kisin-Pappas integral model. CONTENTS

Assume that p > 2, and let O K be a p-adic discrete valuation ring with residue field admitting a finite p-basis, and let R be a formally smooth formally finite-type O K -algebra. (Indeed, we allow slightly more general rings R.) We construct an anti-equivalence of categories between the categories of p-divisible groups over R and certain semi-linear algebra objects which generalise (ϕ, S)-modules of height 1 (or Kisin modules). A similar classification result for p-power order finite flat group schemes is deduced from the classification of p-divisible groups. We also show compatibility of various construction of (Zp-lattice or torsion) Galois representations, including the relative version of Faltings' integral comparison theorem for p-divisible groups. We obtain partial results when p = 2.Ki S (ϕ, ∇ 0 ) of certain "torsion Kisin S-modules" (Definition 9.2.1). Furthermore, we have a natural G R∞ -equivariant isomorphism H(R) ∼ = T * (M * (H)).Let us review previous results when p > 2. When R = O K with perfect residue field, all our main results are known already. Theorem 4(1) is proved by Faltings [Fal99, §6], while the first assertion was already proved by Fontaine [Fon82, 1 Indeed, we will work with slightly more general rings than Brinon [Bri06,Bri08]. See §4.4 for more details.2 In this paper, we only consider commutative finite locally free group schemes, so we will often suppress the adjective "commutative". Definition 6.1.6) as follows:Lemma 6.2.2. Assume that p = 2 and R satisfies the p-basis condition ( §2.2.1). Then the functors Mod S (ϕ) ϕ nilp → MF S (ϕ) ϕ nilp and Mod S (ϕ) ψ nilp → MF S (ϕ) ψ nilp , defined by S ⊗ ϕ,S (·), are fully faithful. The same statement holds for the full subcategories of ϕand ψnilpotent objects in Mod S (ϕ, ∇) and Mod KiThe same proof of Lemma 6.2 shows that it suffices to prove the lemma when R is a p-adic discrete valuation ring with perfect residue field, which follows from combining Proposition 1.1.9 and Theorem 1.2.8 in [Kis09]. Proposition 6.3. Assume that R satisfies the p-basis condition ( §2.2.1). Then the functors Mod, are essentially surjective. In particular, they are equivalences of categories if p > 2 or if they are restricted to ϕand ψnilpotent objects, and are equivalence of categories up to isogeny if p = 2.We prove the proposition later in §6.4. Let us record some interesting corollaries. The following is immediate from Corollary 6.3.1. Assume that R satisfies the p-basis condition ( §2.2.1). If p > 2 then there exists an exact contravariant functorsatisfies the normality assumption ( §2.2.4), and an anti-equivalence of categories if R satisfies the formally finite-type assumption ( §2.2.2). If p = 2 then we have an exact contravariant functor M * [ 1 p ] : G → M * (G)[ 1 p ] on the isogeny categories {p-divisible groups over R}[1/p] → Mod S (ϕ, ∇)[1/p], which is fully faithful if R satisfies the formally finite-type assumption ( §2.2.2).When p = 2, we have the following strengthening of Corollary 6.3.1 for ϕand ψnilpotent objects: Corollary 6.3.2. Let ...

Let O K be a 2-adic discrete valuation ring with perfect residue field k. We classify p-divisible groups and p-power order finite flat group schemes over O K in terms of certain Frobenius modules over S := W (k) [[u]]. We also show the compatibility with crystalline Dieudonné theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, who independently obtained our result using display theory. G K∞ := Gal(K/K ∞ ). Set S := W (k) [[u]] and we extend the Witt vector Frobenius map to a ring endomorphism ϕ : S → S by ϕ(u) = u p . Breuil has conjectured that p-divisible groups and finite flat group schemes over O K are classified by a certain Frobeniusmodule over S; namely, Mod S (ϕ) 1 and (Mod /S) 1 , respectively. (The precise definitions will be given later in Definitions 2.1 and 2.5.) Also note that there exist contravariant functors T * S from Mod S (ϕ) 1 and (Mod /S) 1 to the category of finitely generated Z p -modules with continuous G K∞ -action (2.2.1, 2.6.2). The main result of this paper is the proof of the following conjecture of Breuil. Conjecture 1.1 (Breuil). There exists an exact anti-equivalence of categories G M G between the category of p-divisible groups over O K and Mod S (ϕ) 1 such that the contravariant Dieudonné crystal D * (G) can be recovered from M G and there exists a natural G K∞ -equivariant isomorphism T p (G) ∼ = T * S (M G ). There exists an exact anti-equivalence of categories H M H between the category of p-power order finite flat group schemes over O K and (Mod /S) 1 such that 2000 Mathematics Subject Classification. 11S20, 14F30.