Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups

Abstract

International audienceThe isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D

Minimal F-crystals and isomorphism numbers of isosimple F-crystals

“…Proof This is a generalization of [, Proposition 5.19] and we use the same strategy to prove this proposition. We first prove the proposition in the isoclinic case.…”

“…In this section, we briefly recall this notion and present some of its properties that will be useful to study minimal F ‐crystals. We refer to , , and for detail expositions.…”

“…Optimal upper bounds of general F ‐crystals are not known but some results in certain special cases have been found. For example, Lau, Nicole and Vasiu proved that if is a Dieudonné module with dimension d and codimension c , then and the inequality is optimal in the sense that there exists an isoclinic Dieudonné module such that . In the case of isoclinic F ‐crystals, upper bounds that are optimal in many cases but not in general have also been found in .…”

“…We define the minimal height to be the smallest non‐negative integer so that there exists an isogeny whose cokernel is annihilated by . This is the generalization of minimal heights of p ‐divisible groups defined in . It turns out that just as in the case of p ‐divisible groups; see Corollary .…”

Computing isomorphism numbers of F -crystals using the level torsions

“…Since then, the conjecture has been verified in various cases, for example, in the cases of supersingular p-divisible groups [9, Theorem 1.2] and quasi-special p-divisible groups [14, Theorem 1.5.2]. Only recently, Lau, Nicole and Vasiu [6,Theorem 1.4] found an optimal upper bound n D 2cd/(c + d) which proves a corrected version of Traverso's conjecture. In the search for optimal upper bounds for n D , the following play important roles:…”