Oort's conjecture on automorphisms of generic supersingular abelian varieties

TL;DR

This paper proves Oort's conjecture that, generically, the automorphism group of the universal principally polarized abelian variety over the supersingular locus is trivial except in specific low-dimensional cases, and provides explicit descriptions of related loci.

Contribution

It confirms Oort's conjecture for generic automorphism groups of supersingular abelian varieties and describes the a=1-locus in Rapoport-Zink spaces explicitly.

Findings

Automorphism group is only ±1 generically, except for g=2,3 and p=2.

Explicit description of the a=1-locus in Rapoport-Zink space.

Results extend to moduli spaces of supersingular p-divisible groups.

Abstract

We prove Oort's conjecture that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of genus g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of $\pm 1$, unless g=2 or 3 and p=2. On the way, we provide an explicit description of the a=1-locus in the Rapoport-Zink space of principally polarized supersingular p-divisible groups of any dimension g. We also prove analogous results for generic automorphism groups on moduli spaces of supersingular p-divisible groups with and without polarization.