The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

TL;DR

This paper classifies non-hyperelliptic genus 3 curves with cyclic automorphism group of order 6, analyzing their isomorphism classes and $a$-numbers using hypergeometric series and Hasse-Witt matrices.

Contribution

It provides necessary and sufficient conditions for isomorphism of these curves and determines their possible $a$-numbers, including the count of classes with maximal $a$-number.

Findings

Curves are described by a parameter r with specific restrictions.

Conditions for isomorphism between curves are established.

The $a$-number distribution among these curves is determined.

Abstract

In this paper, we study non-hyperelliptic curves of genus $3$ with cyclic automorphism group of order $6$. Over an algebraically closed field $K$ of characteristic $\neq 2,3$, such curves are written as plane quartics $C_r: x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ with one parameter $r$. As the first main theorem, we show that $r\neq 0,\pm 2$ and give a necessary and sufficient condition with respect to $r$ and $r'$ such that $C_r \cong C_{r'}$. By describing the Hasse-Witt matrix of $C_r$ in terms of a certain Gauss' hypergeometric series, we obtain the second main theorem, where we determine the possible $a$-number of $C_r$, and give the exact number of isomorphism classes over $K$ of such curves attaining the possible maximal $a$-number.