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 $E(K)$ 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 $N\leq 55$, $N \neq 54$.

Theorems & Definitions (45)

• Theorem 1.1: Mordell--Weil
• Theorem 1.2: Mazur Mazur:eisenstein
• Theorem 1.3: Kenku--Momose kenku1988torsion; Kamienny kamienny1992torsion
• Theorem A
• Remark 1.5: Enumeration of remaining cases
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.6: Models of $X_H$
• Lemma 2.8