Some explicit arithmetics on curves of genus three and their applications

09/07/2022

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by   Tomoki Moriya, et al.

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A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the 2-torsion subgroup of the domain. In particular, a Richelot isogeny whose codomain is the product of two or more principally porlalized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetics 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 Õ(p^3) (resp. Õ(p^4)) to enumerate all hyperelliptic (resp.non-hyperelliptic) superspecial Howe curves of genus 3.