The $a$-numbers of non-hyperelliptic curves of genus three with large cyclic automorphism group

Abstract

In the study of algebraic curves and their moduli spaces, it is important to determine the a-numbers of curves over a field of positive characteristic. It is known that non-hyperelliptic curves of genus 3 are classified by the structures of their automorphism groups as finite groups. In this paper, we determine the a-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6 or 9. Moreover, we also find the exact number of the isomorphism classes of such curves attaining the possible maximal a-number.