Some explicit arithmetic on curves of genus three and their applications

Abstract

A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetic on curves of genus $3$, including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus $3$ are also given. Using the formulae, we shall construct an algorithm with complexity $\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus $3$.