Volume 4, Issue 8
Special volume AJAX in honor of René Schoof
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 to search for torsion points, not just over 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 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 82 240 plane quartic curves.
Keywords: Abelian varieties, Galois representations, Birch–Swinnerton-Dyer conjecture, Jacobians, curves.
Citation:
Raymond van Bommel. Computing torsion for plane quartics without using height bounds. Polynesian Journal of Mathematics, Volume 4, Issue 8, Pages 1–22. (Sept. 2026) DOI: 10.69763/polyjmath.4.8
Milestones:
Received 31 Jan 2026
Revised 3 May 2026
Accepted 17 May 2026
Published 28 Sep 2026
Communicated by Samuele Anni
The above content is published under the terms of the Creative Commons Attribution 4.0 International license.