A group theoretic perspective on entanglements of division fields

Abstract

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n n and a subgroup G ⊆ G L 2 ( Z / n Z ) G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G G to represent an ( a , b ) (a,b) -entanglement and give additional criteria for G G to represent an explained or unexplained ( a , b ) (a,b) -entanglement. Using these new definitions, we determine the tuples ( ( p , q ) , T ) ((p,q),T) , with p > q ∈ Z p>q\in \mathbb {Z} distinct primes and T T a finite group, such that there are infinitely many non- Q ¯ \overline {\mathbb {Q}} -isomorphic elliptic curves over Q \mathbb {Q} with an unexplained ( p , q ) (p,q) -entanglement of type T T . Furthermore, for each possible combination of entanglement level ( p , q ) (p,q) and type T T , we completely classify the elliptic curves defined over Q \mathbb {Q} with that combination by constructing the corresponding modular curve and j j -map.