论文标题
零星的立方扭转
Sporadic Cubic Torsion
论文作者
论文摘要
让$ k $为一个数字字段，让$ e/k $为$ k $的椭圆曲线。 Mordell- WEOL定理断言，$ K $ - 合理点$ E（k）$ e $ $ e $形成了有限生成的Abelian集团。在这项工作中，我们完成了有限组的分类，这些分类显示为每立方体数字$ k $ $ k $ of $ e（k）$。 为此，我们确定模块化曲线上的立方点$ x_1（n）$ for \ [n = 21、22、24、24、26、26、28、33、32、33、33、33、33、35、39、39、45、65、121。 $ j_1（2,2n）$）的等级为0。我们还为Conrad，Edixhoven和Stein的广义版本提供了证据，证明了$ j_1（n）上的扭矩（\ Mathbb {q}）$是由$ \ text {gal}（\ bar {\ mathbb {q}}/\ mathbb {q}）$ - cusps的$ x_1（n）_ {\ bar {\ bar {\ bar {\ mathbb {q}}}} $ for $ n \ n \ leq 55 $，$ n \ neq 54 $ neq 54 $。
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 $N\leq 55$, $N \neq 54$.
