Traverso's Isogeny Conjecture for Some Unitary p-Divisible Groups

TL;DR

This paper investigates invariants of supersingular unitary p-divisible groups, providing a complete classification of minimal heights, establishing bounds on isogeny cutoffs, and relating these to stratifications in unitary Shimura varieties.

Contribution

It offers a complete description of minimal heights for supersingular unitary p-divisible groups and bounds on their isogeny cutoffs, connecting these to Shimura variety stratifications.

Findings

Complete classification of minimal heights for supersingular unitary p-divisible groups.

Bounds established on isogeny cutoffs for these groups.

Reformulation of results in the context of Shimura variety stratifications.

Abstract

The isogeny cutoff of a $p$-divisible group $X$ (defined over an algebraically closed field of characteristic $p$) measures the amount of $p$-torsion necessary to determine its isogeny class. The minimal height of $X$ measures its distance to the closest minimal $p$-divisible group (in the sense of Oort). In this paper, we study these invariants for supersingular unitary $p$-divisible groups of signature $(a,b)$. We provide a complete description of the possible minimal heights. As an application, we establish bounds on the isogeny cutoffs for these $p$-divisible groups. Finally, we rephrase our results in the language of the $\mathrm{BT}_m$ stratifications of unitary Shimura varieties of signature $(a,b)$.