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

TL;DR

This paper investigates invariants of p-divisible groups related to their Newton polygons, proving lower semicontinuity, refining stratifications, and disproving Traverso's conjecture, with applications to endomorphism liftability.

Contribution

It establishes lower semicontinuity of isomorphism numbers and isogeny cutoffs, refines Newton polygon strata, and disproves Traverso's conjecture.

Findings

Invariants are lower semicontinuous in families of p-divisible groups.

Determined maximal isogeny cutoffs within each isogeny class.

Provided an upper bound for isomorphism numbers, optimal in isoclinic cases.

Abstract

The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field 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.