Automorphism groups of superspecial curves of genus $4$ over $\mathbb{F}_{11}$

TL;DR

This paper explicitly determines automorphism groups of all nonhyperelliptic superspecial genus 4 curves over 11, providing a computational method applicable to similar curves and linking enumeration results via Galois cohomology.

Contribution

It introduces an algorithm to compute automorphism groups of nonhyperelliptic genus 4 curves over finite fields, and confirms compatibility between computational and theoretical enumerations.

Findings

Automorphism groups explicitly determined for all such curves.

Algorithm applicable to any nonhyperelliptic genus 4 curve over finite fields.

Confirmed consistency between computational and Galois cohomology enumerations.

Abstract

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus $4$ over $\mathbb{F}_{11}$. Our algorithm determining automorphism groups works for any nonhyperelliptic curves of genus $4$ over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus $4$ over $\mathbb{F}_{11}$ obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.