Vol. 15, No. 7, 2021

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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 EK 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 X1(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 J0(N), J1(N), and J1(2,2N) 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 J1(N)() is generated by Gal( ̄)-orbits of cusps of X1(N) ̄ for N 55, N54.

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