Computing torsion for plane quartics without using height bounds

2023/01/19 by Raymond van Bommel, van Bommel, Raymond · 1 citation
Engineering · #11F80 #11G10 #11G30 #11G40 #14H40 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.08312

openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Rational and isolated quadratic points on hyperelliptic curves of rank 0 and small genus

Related

• Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3
• Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics
• Large torsion subgroups of split Jacobians of curves of genus two or three
• Torsion of \mathbb Q-curves over number fields of small odd prime degree
• Torsion of elliptic curves with rational j-invariant over quartic number fields