从群论角度看可分域的纠缠
A group theoretic perspective on entanglements of division fields
摘要：
本文从群论的角度出发，系统地研究了分裂场的纠缠。对于正整数$n$和具有满射行列式的子群$G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$，给出了$G$表示$(a,b)$ -纠缠的定义，并给出了$G$表示可解释或不可解释的$(a,b)$ -纠缠的附加判据.使用这些新的定义，我们确定元组$((p,q),T)$，与$p<q\in\mathbb{Z}$不同的素数和$T$一个有限的群体，使有无限多个非$\bar{\mathbb{Q}}$ -同构椭圆曲线在$\mathbb{Q}$与一个无法解释的$(p,q)$ -纠缠类型$T$。此外，对于每一种可能的纠缠度$(p,q)$和类型$T$的组合，我们通过构造相应的模曲线和$j$映射，对定义在$\mathbb{Q}$上的椭圆曲线进行了完全分类.
英文摘要：
In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer $n$ and a subgroup $G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$ with surjective determinant, we provide a definition for $G$ to represent an $(a,b)$-entanglement and give additional criteria for $G$ to represent an explained or unexplained $(a,b)$-entanglement. Using these new definitions, we determine the tuples $((p,q),T)$, with $p<q\in\mathbb{Z}$ distinct primes and $T$ a finite group, such that there are infinitely many non-$\bar{\mathbb{Q}}$-isomorphic elliptic curves over $\mathbb{Q}$ with an unexplained $(p,q)$-entanglement of type $T$. Furthermore, for each possible combination of entanglement level $(p,q)$ and type $T$, we completely classify the elliptic curves defined over $\mathbb{Q}$ with that combination by constructing the corresponding modular curve and $j$-map.
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