Oort's conjecture and automorphisms of supersingular curves of genus four

TL;DR

This paper proves that supersingular genus 4 curves in characteristic p>2 have trivial automorphism groups and confirms Oort's conjecture for supersingular abelian fourfolds, also providing a new proof for genus 3.

Contribution

It establishes the triviality of automorphism groups for supersingular genus 4 curves and confirms Oort's conjecture for these cases, offering a new proof for genus 3.

Findings

All components of the supersingular genus 4 locus have trivial automorphism groups.

Confirmed Oort's conjecture for supersingular abelian fourfolds in characteristic p>2.

Provided a new proof of Oort's conjecture for genus 3.

Abstract

We show that every component of the locus of smooth supersingular curves of genus $4$ in characteristic $p>2$ has a trivial generic automorphism group. As a result, we prove Oort's conjecture about automorphism groups of supersingular abelian fourfolds for $p>2$. Our main idea consists of estimating dimensions of the loci of smooth supersingular curves that admit an automorphism of prime order by considering possible choices of the corresponding quotient curves. This reasoning also results in a new proof of Oort's conjecture for $g = 3$ and $p>2$, previously proved by Karemaker, Yuboko, and Yu.