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Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

Barinder S. Banwait, Maarten Derickx

Abstract

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

Abstract

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields for all squarefree with , improving on the previously known range of . We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.
Paper Structure (12 sections, 13 theorems, 20 equations, 3 figures)

This paper contains 12 sections, 13 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The five genus 1 cases
  3. X1(13) or X1(18
  4. Everywhere Local solubility
  5. Congruence conditions on d
  6. Two-cover descent
  7. Positive rank of J1d(N)
  8. X1(16)
  9. Positive rank of J1d(16)
  10. Elliptic curve Chabauty
  11. Comparison of our results with a conjecture of Granville
  12. Future research

Key Result

Theorem 1.1

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then $E(\mathbb{Q})_{tors}$ is isomorphic to one of the following $15$ groups:

Figures (3)

  • Figure 5.1: Graphs showing how $|T_B(N)|$ grows with $B$ for $N = 13$, $16$ and $18$, together with the conjectural growth of $\kappa_f'B^{1/3}$; the values of $\kappa_f'$ here are, respectively, $1.65$, $12.4$ and $1.5$.
  • Figure 5.2: The same graph as \ref{['fig:graph1']} but with smaller values of $\kappa_f'$, viz. respectively, $0.65$, $2.5$, $0.4$.
  • Figure 5.3: Plot of $B$ against $|T_B(N)|/B^{1/3}$; this should conjecturally converge to $\kappa_f'$.

Theorems & Definitions (27)

  • Theorem 1.1: Mazur (1977)
  • Theorem 1.2: Kamienny-Kenku-Momose (1992)
  • Theorem 1.3
  • Theorem 2.1: Kamienny-Najman, Theorems 15 and 16 in kamienny2012torsion
  • Proposition 3.1: Krumm, Theorems 2.6.5 and 2.6.9 in krummthesis
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • ...and 17 more