Sporadic cubic torsion

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 $E\left(K\right)$ 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 $E\left(K\right)$ for $K$ a cubic number field.

To do so, we determine the cubic points on the modular curves $X_1\left(N\right)$ 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 $J_0\left(N\right)$, $J_1\left(N\right)$, and $J_1\left(2,2N\right)$ 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 $J_1\left(N\right)\left(\mathbb{Q}\right)$ is generated by $Gal(\bar{\mathbb{Q}}∕\mathbb{Q})$-orbits of cusps of $X_1{\left(N\right)}_{\bar{\mathbb{Q}}}$ for $N\le 55$, $N\ne 54$.