Computing torsion for plane quartics without using height bounds

TL;DR

This paper presents a provable algorithm to compute the rational torsion subgroup of Jacobians of plane quartic curves without height bounds, using reduction modulo primes and torsion point searches.

Contribution

It introduces a novel method combining reduction and torsion point searches, implemented in Magma, to efficiently compute torsion subgroups for a large dataset of curves.

Findings

Successfully computed torsion subgroups for over 98% of the dataset.

The method avoids reliance on height bounds, improving computational efficiency.

Implemented in Magma and validated on a large set of plane quartic curves.

Abstract

We describe an algorithm that provably computes the rational torsion subgroup of the Jacobian of a curve without relying on height bounds. Instead, the strategy is to find upper bounds for the torsion subgroup using reduction modulo primes, and searching for torsion points, not just over Q but also over small number fields, until the two bounds meet. Both complex analytic and Chinese remainder theorem based methods are used to find such torsion points. The method has been implemented in Magma for plane quartic curves over Q with a rational point and used to provably compute the rational torsion subgroup for more than 98% of Jacobians of curves in a data set due to Sutherland consisting of 82240 plane quartic curves.