On superspecial hyperelliptic curves of Rosenhain forms

Abstract

Any genus-$g$ hyperelliptic curve $C$ defined over an algebraically closed field of characteristic $p\ge 3$ can be written in a Rosenhain form as $y^2=x(x-1){\prod }_{i=1}^{2g-1}(x-{\lambda }_i)$. In this paper, we first show that, if $C$ is superspecial, then each of ${\lambda }_i,1-{\lambda }_i$, and ${\lambda }_i-{\lambda }_j$ 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 $p\le 41$, and of genus $6$ in all characteristics $p\le 31$.

11 pages, comments welcome!