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

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

让 $钾$ 是一个数字字段，让 $乙∕钾$ 是椭圆曲线 $钾$. Mordell-Weil 定理断言$钾$- 理性点 $乙\left(钾\right)$ 的 $乙$形成一个有限生成的阿贝尔群。在这项工作中，我们完成了有限群的分类，这些群表现为$乙\left(钾\right)$ 为了 $钾$ 三次数字字段。

为此，我们确定模曲线上的三次点 $X_1\left(N\right)$ 为了

作为我们分析的一部分，我们确定了完整的列表 $N$ 为此 $J_0\left(N\right)$, $J_1\left(N\right)$， 和 $J_1\left(2,2N\right)$ 具有等级 0。我们还通过证明对 Conrad、Edixhoven 和 Stein 的猜想的广义版本提供证据 $J_1\left(N\right)\left(\mathbb{Q}\right)$ 是由 $加尔$($\bar{\mathbb{Q}}∕\mathbb{Q}$)-尖端的轨道 $X_1{\left(N\right)}_{\bar{\mathbb{Q}}}$ 为了 $N\le 55$, $N\ne 54$.