Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3

TL;DR

This paper presents a new algorithm for computing the rational torsion subgroup of Jacobians of genus 3 hyperelliptic curves, extending previous methods for genus 2, and applies it to discover new torsion structures.

Contribution

It introduces a generalized algorithm for genus 3 hyperelliptic Jacobians, implementing it in Magma and identifying previously unknown torsion structures.

Findings

Discovered new torsion structures in genus 3 Jacobians

Successfully applied the algorithm to a database of low discriminant curves

Extended existing genus 2 algorithms to genus 3 cases

Abstract

We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points over finite fields using the Kummer variety and to check whether they are rational using heights. Both have been made explicit for Jacobians of hyperelliptic curves of genus 3 by Stoll. This article is partially based on the second-named author's Master thesis.