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

2024/05/02 by Dušan Dragutinović, Dragutinović, Dušan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2405.01282

openalex publication_date 2024/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• An unusual family of supersingular curves of genus five

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