论文标题
椭圆形曲线与非亚伯纠缠
Elliptic curves with non-abelian entanglements
论文作者
论文摘要
我们考虑对四倍体$（k，e，m_1，m_2）$进行分类的问题，其中$ k $是一个数字字段，$ e $是在$ k $和$ k $和$ k $和$ k $和$ k $和$（m_1，m_2）$的椭圆曲线上，是一对相对主要的积极整数，其中$ k（e [m_1] $ k（e [m_1]）\ cap k（e [m_1） $ k $。模块化曲线有一个无限集$ \ Mathcal {s} $，其$ K $合理点捕获了$ k $上的所有椭圆曲线，而没有具有此属性的复杂乘法。我们的主要定理明确描述了$ \ Mathcal {s} $的（有限）子集，该子集由具有零属的模块化曲线组成。在$ k = \ mathbb {q} $的情况下，这有针对确定$ e $扭转的Galois代表何时尽可能大的申请。我们用特定示例说明了此应用程序。
We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1]) \cap K(E[m_2])$ is a non-abelian extension of $K$. There is an infinite set $\mathcal{S}$ of modular curves whose $K$-rational points capture all elliptic curves over $K$ without complex multiplication that have this property. Our main theorem explicitly describes the (finite) subset of $\mathcal{S}$ consisting of those modular curves having genus zero. In the case $K = \mathbb{Q}$, this has applications to the problem of determining when the Galois representation on the torsion of $E$ is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example.
