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

Abstract

The 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 $D$.