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

TL;DR

This paper presents a practical method for computing the 3-torsion subgroup of Jacobians of genus 3 hyperelliptic curves and explores its application in determining wild conductors at prime 2.

Contribution

It introduces a scheme for identifying 3-torsion points and combines complex approximations, homotopy continuation, and lattice reduction techniques for precise computation.

Findings

Effective computation of 3-torsion subgroups for genus 3 hyperelliptic Jacobians

Application of 3-torsion data to analyze wild conductors at prime 2

Enhanced methods for explicit arithmetic of hyperelliptic Jacobians

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.