On the maximality of hyperelliptic Howe curves of genus 3

TL;DR

This paper investigates hyperelliptic Howe curves of genus 3 over finite fields, demonstrating that superspecial cases are either maximal or minimal over the quadratic extension, revealing their extremal properties.

Contribution

It establishes a criterion for hyperelliptic Howe curves of genus 3 to be maximal or minimal over finite fields based on superspeciality.

Findings

Superspecial hyperelliptic Howe curves are maximal or minimal over _{p^2}.

Standard forms of these curves exhibit extremal point counts.

Results apply for characteristic p  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 $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.