On the maximality of genus-3 nonhyperelliptic curves of Ciani type

TL;DR

This paper investigates Ciani curves of genus 3 in positive characteristic, demonstrating that superspecial cases are either maximal or minimal over quadratic extensions of finite fields, revealing their extremal properties.

Contribution

It establishes a criterion linking superspecial Ciani curves to their maximal or minimal status over finite fields, expanding understanding of their arithmetic properties.

Findings

Superspecial Ciani curves are maximal or minimal over _{p^2}

Standard forms of these curves exhibit extremal point counts

Results apply for characteristic p rom 3 onwards

Abstract

In this paper, we study a Ciani curve $C: x^4 + y^4 + z^4 + rx^2y^2 + sy^2z^2 + tz^2x^2 = 0$ in positive characteristic $p \geq 3$. 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.