Vol. 15, No. 7, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 8, 1865–2122
Issue 7, 1593–1864
Issue 6, 1343–1592
Issue 5, 1077–1342
Issue 4, 821–1076
Issue 3, 569–820
Issue 2, 309–567
Issue 1, 1–308

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
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

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.

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

Received: 10 August 2020
Revised: 5 November 2020
Accepted: 20 December 2020
Published: 1 November 2021
Maarten Derickx
Mathematisch Instituut
Universiteit Leiden
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