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Classification of torsion of elliptic curves over quartic fields

Maarten Derickx, Filip Najman

TL;DR

We determine all possible torsion subgroups $E(K)_{\mathrm{tor}}$ for elliptic curves over quartic fields $K$ and provide the explicit list of groups that occur, together with a proof that no sporadic degree-4 points arise on the relevant modular curves. The authors deploy a three-pronged strategy: a rank-0 analysis via a Hecke sieve on $X_1(m,n)$, a global method for cases with positive rank, and ad hoc arguments for a few remaining stubborn instances. They also establish data and software reproducibility, and briefly discuss extending the approach to higher-degree number fields, notably quintics, with substantial preliminary eliminations. Overall, the work yields a complete, explicit quartic torsion classification and showcases scalable methods for higher-degree analogues.

Abstract

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and $E$ ranges over all elliptic curves over $K$. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves $E$. Proving this requires showing that numerous modular curves $X_1(m,n)$ have no non-cuspidal degree $4$ points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

Classification of torsion of elliptic curves over quartic fields

TL;DR

We determine all possible torsion subgroups for elliptic curves over quartic fields and provide the explicit list of groups that occur, together with a proof that no sporadic degree-4 points arise on the relevant modular curves. The authors deploy a three-pronged strategy: a rank-0 analysis via a Hecke sieve on , a global method for cases with positive rank, and ad hoc arguments for a few remaining stubborn instances. They also establish data and software reproducibility, and briefly discuss extending the approach to higher-degree number fields, notably quintics, with substantial preliminary eliminations. Overall, the work yields a complete, explicit quartic torsion classification and showcases scalable methods for higher-degree analogues.

Abstract

Let be an elliptic curve over a quartic field . By the Mordell-Weil theorem, is a finitely generated group. We determine all the possibilities for the torsion group where ranges over all quartic fields and ranges over all elliptic curves over . We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves . Proving this requires showing that numerous modular curves have no non-cuspidal degree points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

Paper Structure

This paper contains 18 sections, 32 theorems, 50 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Data availability and reproducibility
  4. Conventions and notation
  5. Analytic and algebraic ranks of modular abelian varieties
  6. Hecke operators
  7. Previous results
  8. Auxiliary results
  9. The Hecke sieve for the rank 0 cases
  10. The group $\mathbb Z/3\mathbb Z \times \mathbb Z /15 \mathbb Z$
  11. The positive rank cases
  12. The global method
  13. The groups $\mathbb Z /63 \mathbb Z$ and $\mathbb Z /65 \mathbb Z$
  14. Higher degree number fields
  15. Sporadic points on X1(n) and X0(n)
  16. ...and 3 more sections

Key Result

Theorem 1.1

If $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$, the groups that appear as $E(K)_{\mathrm{tors}}$ are exactly the following

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1: JeonKimPark06
  • ...and 59 more