On superspecial hyperelliptic curves of Rosenhain forms

Abstract

Any genus-$g$ hyperelliptic curve $C$ defined over an algebraically closed field of characteristic $p \geq 3$ can be written in a Rosenhain form as $y^2 = x(x-1)\prod_{i=1}^{2g-1}(x-λ_i)$. In this paper, we first show that, if $C$ is superspecial, then each of $λ_i,1-λ_i$, and $λ_i-λ_j$ is a square in $\mathbb{F}_{p^2}$. As an application, we propose a new algorithm for enumerating superspecial hyperelliptic curves in small characteristic. By implementing our algorithm, we successfully computed the number of isomorphism classes of such curves of genera $4$ and $5$ in all characteristics $p \leq 41$, and of genus $6$ in all characteristics $p \leq 31$.