Abstract. We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5.

We prove that an $F$-crystal $(M,\vph)$ over an algebraically closed field $k$ of characteristic $p>0$ is determined by $(M,\vph)$ mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of $(M,\vph)$. We also extend this result to triples $(M,\vph,G)$, where $G$ is a flat, closed subgroup scheme of ${\bf GL}_M$ whose generic fibre is connected and has a Lie algebra normalized by $\vph$. We get two purity results. If ${\got C}$ is an $F$-crystal over a reduced ${\bf F}_p$-scheme $S$, then each stratum of the Newton polygon stratification of $S$ defined by ${\got C}$, is an affine $S$-scheme (a weaker result was known before for $S$ noetherian). The locally closed subscheme of the Mumford scheme ${\Ma_{d,1,N}}_k$ defined by the isomorphism class of a principally quasi-polarized $p$-divisible group over $k$ of height 2d, is an affine ${\Ma_{d,1,N}}_k$-scheme.Comment: Final version (63 pages) accepted for publication in Ann. Sci. Ec. Norm. Su

We also provide a criterion of when s D (m) satisfies the purity property (i.e., it is an affine A-scheme). Similar results are proved for quasi Shimura p-varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic (0, p).

Abstract. Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k. Let n D be the smallest non-negative integer for which the following statement holds: if C is a p-divisible group over k of the same codimension and dimension as D and suchTo the Dieudonné module of D we associate a non-negative integer`D which is a computable upper bound ofif the set I has at least two elements we also show that n D Ä maxf1; n D i ; n D i C n D j 1 j i; j 2 I; j ¤ i g. We show that we have n D Ä 1 if and only if`D Ä 1; this recovers the classification of minimal p-divisible groups obtained by Oort. If D is quasi-special, we prove the Traverso truncation conjecture for D. If D is F -cyclic, we explicitly compute n D . Many results are proved in the general context of latticed F -isocrystals with a (certain) group over k.

Let k be an algebraically closed field of characteristic p > 0. Let W (k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, theétale cohomology with Z p coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with arbitrary families of crystalline tensors over a finite, discrete valuation ring extension of W (k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W (k) whose special fibers over k have a Zariski dense set of k-valued points.The first main goal of this paper is to construct global deformations of the pairs (D, (t α ) α∈J ) and (D, (v α ) α∈J ) over certain regular, affine, formally smooth Spec W (k)schemes Spec Q whose special fibres are geometrically connected and have a Zariski dense set ofk-valued points (see Theorem 3.4.1 and Subsubsection 3.4.2). The deformations of (D, (t α ) α∈J ) will be obtained from deformations of the quadruple (M, F 1 , φ, (t α ) α∈J ) chosen in such a way that the induced deformations of the triple (M, F 1 , (t α ) α∈J ) are constant. If p = 2, then for these constructions we will assume that either D or D t is connected. Each such global deformation of (D, (t α ) α∈J ) over Spec Q will consist in a pdivisible group over Spec Q endowed naturally with a family of crystalline tensors indexed by the set J, whose pull-back at a suitable k-valued point of Spec Q is (D, (t α ) α∈J ) (cf. Subsection 3.4). Each Q will be the p-adic completion of a particular type of ind-étale algebra over a polynomial W (k)-algebra, cf. Subsection 3.1 and Theorem 3.2.The second main goal of this paper is to use the mentioned global deformations in order to prove the following Main Theorem.