Elliptic curves with non-abelian entanglements

2020/08/20 by Nathan Jones, Jones, Nathan, Ken McMurdy +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2008.09087

openalex publication_date 2020/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of classifying quadruples (K,E,m₁,m₂) where K is a number field, E is an elliptic curve defined over K and (m₁,m₂) is a pair of relatively prime positive integers for which the intersection K(E[m₁]) ∩ K(E[m₂]) is a non-abelian extension of K. There is an infinite set 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 S consisting of those modular curves having genus zero. In the case K = 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.

Citations

• Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• On the possible images of the mod ell representations associated to elliptic curves over Q

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