Explicit Constructions for Genus 3 Jacobians

Abstract

Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in $JX$. Finally we introduce a related construction by means of an imbedding of the open set previously described in a Grassmanian variety.