On the Rosenhain forms of superspecial curves of genus two

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 $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$.

15 pages