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 $N\le 55$, $N\ne 54$.

24 pages. v2: we corrected an error in Theorem 3.1

References in corpus (5)

• Constructing elliptic curves over finite fields with prescribed torsion
• Gonality of the modular curve X1(N)
• Torsion of elliptic curves over cyclic cubic fields
• Low Degree Places on the Modular Curve X1(N)
• Structure of the cuspidal rational torsion subgroup of J_1(p^n)

Cited by in corpus (6)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Fermat's Last Theorem and modular curves over real quadratic fields
• Torsion of elliptic curves over cyclic cubic fields
• Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions
• Determination of the modular Jacobian varieties $J_1(M,MN)$ with the Mordell-Weil rank zero
• Fields generated by points on superelliptic curves