论文标题
关于分区字段纠缠的小组理论观点
A group theoretic perspective on entanglements of division fields
论文作者
论文摘要
在本文中，我们从组理论的角度开始对分裂场纠缠的系统研究。对于积极的整数$ n $和一个子组$ g \ subseteq \ text {gl} _2（\ mathbb {z}/{n}/{n} \ mathbb {z}）$具有peRecriptive instacivants，我们为$ g $提供了一个定义，以代表$（a，b）$ - n $ g）或$ g）或$ g）and $ g） $（a，b）$ - 纠缠。 使用这些新定义，我们确定元组$（（p，q），t）$，带有$ p <q \ in \ mathbb {z} $不同的素数和$ t $ a是有限的组，使得有很多非non- $ \ bar {\ bar {\ mathbb {q}} $ - iSomorphic elliptic culb a and c $ a n and c} $（p，q）$ - 类型$ t $的纠缠。此外，对于纠缠级别$（p，q）$和$ t $的每种可能组合，我们通过构造相应的模块化曲线和$ j $ -map的组合将定义在$ \ mathbb {q} $上定义的椭圆曲线与该组合进行分类。
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.
