Papers › A group theoretic perspective on entanglements of division fields
A group theoretic perspective on entanglements of division fields
Harris B. Daniels, Jackson S. Morrow
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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⊆GL₂(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-ℚ̅-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.
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