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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 ⊆ GL 2 (Z/nZ) 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 ∈ Z distinct primes and T a finite group, such that there are infinitely many non-Q-isomorphic elliptic curves over 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 Q with that combination by constructing the corresponding modular curve and j-map.