On the maximality of hyperelliptic Howe curves of genus 3

Abstract

In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbf{F}_{p^2}$ without taking its $\mathbf{F}_{p^2}$-form.