A group theoretic perspective on entanglements of division fields

TL;DR

This paper introduces a group theoretic framework to analyze entanglements of division fields in elliptic curves, classifies such entanglements, and constructs modular curves and j-maps for these classifications.

Contribution

It provides a systematic group theoretic approach to classify and understand entanglements of division fields in elliptic curves, including explicit modular curve constructions.

Findings

Classified tuples with infinite non-isomorphic elliptic curves having unexplained entanglements.

Constructed modular curves and j-maps for each entanglement type.

Identified conditions for explained and unexplained entanglements.

Abstract

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.