Crystalline boundedness principle

Abstract

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

Minimal truncations of supersingular $p$-divisible groups

“…This is a special case of [Va1,Lemma 3.2.2] for the group G = GL GL GL M , but for the sake of completeness we include a self-contained proof which works for all p-divisible groups over k. Let us first show that…”

“…Part (b) is a particular case of [Va1,proof of Cor. 3.3.4], but we provide here a simpler argument which works for all isoclinic p-divisible groups.…”

“…Let t λ ∈ N be the i-number of (M, φ, G) defined in [Va1,3.1.4] (i.e., the smallest natural number such that for each element g ∈ G(W (k)) congruent mod p t λ to 1 M , there exists an isomorphism between (M, gφ) and (M, φ) which is an element of G(W (k))). From an argument entirely analogous to the proof of Theorem 2.2 (a) (cf.…”

Reconstructing $p$-divisible groups from their truncations of small level

“…Each estimate of n D represents progress towards the classification of p-divisible groups over k; implicitly, it represents progress towards the understanding of the ultimate stratifications defined in [Va1,Section 5.3] and (thus also) of the special fibres of all integral canonical models of Shimura varieties of Hodge type. The goal of the paper is to put forward basic principles that compute either n D or some very sharp upper bounds of n D .…”

“…[Va1]. Let m G WD T .g; / be the Fontaine-Dieudonné torsion of .g; / introduced in [Va1, Definitions 2.2.2 (a) and (b)].…”