On the Rosenhain forms of superspecial curves of genus two

TL;DR

This paper investigates superspecial genus-2 curves over finite fields, revealing conditions on their defining parameters and their maximality or minimality over quadratic extensions, with implications for higher genus hyperelliptic curves.

Contribution

It establishes a criterion linking the parameters of superspecial genus-2 curves to their maximality/minimality over finite fields, extending to higher genus cases.

Findings

Difference between parameters is a square in _{p^2}

Curves are maximal or minimal over _{p^2} based on a specific criterion

Applications to superspecial hyperelliptic curves of genus 3 and 4

Abstract

In this paper, we examine superspecial genus-2 curves $C: y^2 = x(x-1)(x-\lambda)(x-\mu)(x-\nu)$ in odd characteristic $p$. As a main result, we show that the difference between any two elements in $\{0,1,\lambda,\mu,\nu\}$ is a square in $\mathbb{F}_{p^2}$. Moreover, we show that $C$ is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form (we also give a criterion in terms of $p$ that tells whether $C$ is maximal or minimal). As these applications, we study the maximality of superspecial hyperelliptic curves of genus $3$ and $4$ whose automorphism groups contain $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.