Elliptic curves with non-abelian entanglements

Abstract

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1])\cap K(E[m_2])$ is a non-abelian extension of $K$. There is an infinite set $\mathcal{S}$ of modular curves whose $K$-rational points capture all elliptic curves over $K$ without complex multiplication that have this property. Our main theorem explicitly describes the (finite) subset of $\mathcal{S}$ consisting of those modular curves having genus zero. In the case $K=\mathbb{Q}$, this has applications to the problem of determining when the Galois representation on the torsion of $E$ is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example.

32 pages, supporting Magma code in two separate files

References in corpus (4)

• Modular curves of prime-power level with infinitely many rational points
• On the possible images of the mod ell representations associated to elliptic curves over Q
• Surjectivity of mod 2^n representations of elliptic curves
• Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9

Cited by in corpus (1)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$