Supersingular Ekedahl-Oort strata and Oort's conjecture

TL;DR

This paper proves Oort's conjecture for the automorphism groups of supersingular abelian varieties in certain cases, confirming that generic members have minimal automorphism groups, and extends the result to specific low-dimensional cases.

Contribution

It establishes the automorphism group structure for generic supersingular abelian varieties in the maximal Ekedahl-Oort stratum, confirming Oort's conjecture for even dimensions when p ≥ 5 and for g=4.

Findings

Generic supersingular abelian varieties have automorphism group {±1} for even g and p ≥ 5.

Oort's conjecture holds for g=4 and any prime p.

The result confirms minimal automorphism groups in the specified cases.

Abstract

Let $\mathcal{A}_g$ be the moduli space over $\overline{\mathbb{F}}_p$ of $g$-dimensional principally polarised abelian varieties, where $p$ is a prime. We show that if $g$ is even and $p\geq 5$, then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in $\mathcal{A}_g$ has automorphism group $\{ \pm 1\}$. This confirms Oort's conjecture in the case of $p\geq 5$ and even $g$. We also separately prove Oort's conjecture for $g=4$ and any prime $p$.