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Computing torsion for plane quartics without using height bounds

Raymond van Bommel

TL;DR

The paper introduces a provable method to compute the rational torsion subgroup of the Jacobian of a plane quartic without relying on height bounds. It combines upper-bound techniques from reduction modulo primes with explicit searches for torsion points over small degree number fields, using both CRT-based and complex-analytic reconstruction, and handles fake torsion phenomena. The approach is implemented in Magma and demonstrated on a large dataset of plane quartics, achieving provable results for the vast majority of Jacobians and providing detailed verification data. This yields practical, scalable torsion computations that contribute to understanding torsion structures and their role in BSD-related arithmetic, while offering robust tools for explicit Jacobian arithmetic and torsion analysis.

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.

Computing torsion for plane quartics without using height bounds

TL;DR

The paper introduces a provable method to compute the rational torsion subgroup of the Jacobian of a plane quartic without relying on height bounds. It combines upper-bound techniques from reduction modulo primes with explicit searches for torsion points over small degree number fields, using both CRT-based and complex-analytic reconstruction, and handles fake torsion phenomena. The approach is implemented in Magma and demonstrated on a large dataset of plane quartics, achieving provable results for the vast majority of Jacobians and providing detailed verification data. This yields practical, scalable torsion computations that contribute to understanding torsion structures and their role in BSD-related arithmetic, while offering robust tools for explicit Jacobian arithmetic and torsion analysis.

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.
Paper Structure (15 sections, 9 theorems, 13 equations, 2 tables, 6 algorithms)

This paper contains 15 sections, 9 theorems, 13 equations, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weil pairing
  4. Weil polynomials
  5. Newton-Raphson method and precision
  6. Jacobian arithmetic
  7. Over exact fields
  8. Over the complex numbers
  9. Changing base points
  10. Fake torsion points
  11. Methods
  12. Algebraic reconstruction
  13. Finding torsion points: the CRT method
  14. Finding torsion points: the analytic method
  15. Results

Key Result

Lemma 1

Let $K$ be a number field, let $\mathfrak{p}$ be a prime of $\mathcal{O}_K$ over the prime number $p$, and let $A$ be an abelian variety over $K$. Suppose that $A$ has good reduction at $\mathfrak{p}$ and that $A_{\mathfrak{p}}$ is its reduction. Moreover, suppose that $p > e(\mathfrak{p}/p) + 1$, w is injective for any integer $n$ such that $p \nmid n$.

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Example 2
  • Definition 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • Proposition 7
  • ...and 15 more