Sporadic cubic torsion

Algebra & Number Theory, vol. 15, pp. 1837–1864

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 list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\mathbb{Q})$ is generated by $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of $X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$.

Authors 5

1. Maarten Derickx Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands

2. Anastassia Etropolski

   Rice University

   Affiliation as printed

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

3. Mark van Hoeij

   Florida State University

   Affiliation as printed

   Department of Mathematics, Florida State University, Tallahassee, FL, United States

4. Jackson S. Morrow

   Emory University

   Affiliation as printed

   Department of Mathematics, Emory University, Atlanta, GA, United States

   Department of Mathematics, Emory University, Atlanta, GA, United States#TAB#

5. David Zureick-Brown

   Emory University

   Affiliation as printed

   Department of Mathematics, Emory University, Atlanta, GA, United States

   Department of Mathematics, Emory University, Atlanta, GA, United States#TAB#