Listing superspecial curves of genus three using Richelot isogeny graphs

Abstract

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small $p$. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in $\mathbb{F}_{p^2}$. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes $11 \leq p < 100$.