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 $F_{p^2}$. Moreover, we show that C is maximal or minimal over $F_{p^2}$ without taking its $F_{p^2}$-form (we give an explicit criterion in terms of p that tells whether C is maximal or minimal). As an application, we also study the maximality of superspecial hyperelliptic curves of genera 3 and 4 whose automorphism groups contain $Z/2Z\times Z/2Z$.