The automorphism group of a cyclic $p$-gonal curve

Abstract

Let $M$ be a cyclic $p$-gonal curve with a positive prime number $p$, and let $V$ be the automorphism of order $p$ satisfying $M/ \lt V) \simeq \bf{P}^{1}$. It is well-known that finite subgroups $H$ of $\operatorname{Aut}(\bf{P}^{1})$ are classified into five types. In this paper, we determine the defining equation of $M$ with $H \subset \operatorname{Aut}(M/ \lt V \gt)$ for each type of $H$, and we make a list of hyperelliptic curves of genus 2 and cyclic trigonal curves of genus 5, 7, 9 with $H = Aut(M/ \lt V \gt)$.