Sporadic cubic torsion

Derickx, Maarten; Etropolski, Anastassia; van Hoeij, Mark; Morrow, Jackson S.; Zureick-Brown, David

doi:10.2140/ant.2021.15.1837

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Sporadic cubic torsion

Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow and David Zureick-Brown

Vol. 15 (2021), No. 7, 1837–1864

DOI: 10.2140/ant.2021.15.1837

Abstract

Let $K$ be a number field, and let $E ∕ K$ be an elliptic curve over $K$ . The Mordell–Weil theorem asserts that the $K$ -rational points $E (K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E (K)$ for $K$ a cubic number field.

To do so, we determine the cubic points on the modular curves $X_{1} (N)$ for

N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121 .

As part of our analysis, we determine the complete lists of $N$ for which $J_{0} (N)$ , $J_{1} (N)$ , and $J_{1} (2, 2 N)$ have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_{1} (N) (ℚ)$ is generated by $Gal$ ( $\bar{ℚ} ∕ ℚ$ )-orbits of cusps of $X_{1} {(N)}_{\bar{ℚ}}$ for $N \leq 55$ , $N \neq 54$ .

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Keywords

modular curves, elliptic curves, finitely many cubic points

Mathematical Subject Classification

Primary: 11G18

Secondary: 11G05, 11Y50, 14H45

Supplementary material

Code for sporadic cubic torsion computations

Milestones

Received: 10 August 2020

Revised: 5 November 2020

Accepted: 20 December 2020

Published: 1 November 2021

Authors

Maarten Derickx
Mathematisch Instituut Universiteit Leiden Leiden Netherlands

Anastassia Etropolski
Department of Mathematics Rice University Houston, TX United States

Mark van Hoeij
Department of Mathematics Florida State University Tallahassee, FL United States

Jackson S. Morrow
Department of Mathematics Emory University Atlanta, GA United States

David Zureick-Brown
Department of Mathematics Emory University Atlanta, GA United States

Vol. 15, No. 7, 2021