Three-Torsion Subgroups and Wild Conductors of Genus 3 Hyperelliptic Curves

Abstract

We give a practical method for computing the 3-torsion subgroup of the Jacobian of a genus 3 hyperelliptic curve. We define a scheme for the 3-torsion points of the Jacobian and use complex approximations, homotopy continuation and lattice reduction to find precise expression for the 3-torsion. In the latter stages of the paper, we explain how the 3-torsion subgroup can be used to compute the wild part of the local exponent of the conductor at 2.