 

Nathan Jones and
Ken McMurdy
Elliptic curves with nonabelian entanglements
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Published: 
January 28, 2022. 
Keywords: 
Elliptic curve, Galois representation, entanglement, division field, Serre curve. 
Subject: 
Primary 11G05, 11F80. 


Abstract
In this paper 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}]) ∩ K(E[m_{2}]) is a nonabelian extension of K. There is an infinite set S of modular curves whose Krational points capture all elliptic curves over K without complex multiplication that have this property. Our main theorem explicitly describes the subset S_{0} ⊆ S consisting of those modular curves having genus zero. The subset S_{0} turns out to consist of four modular curves, each isomorphic to P^{1} over its field of definition. 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.


Acknowledgements
The authors would like to thank David ZureickBrown for insightful conversations and also Jackson Morrow and Harris Daniels, as well as Andrew Sutherland, for helpful comments on an earlier version of the paper. Finally, we thank the anonymous referee for a careful reading of the manuscript and many helpful suggestions for improvement.


Author information
Nathan Jones:
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 S Morgan St, 322 SEO, Chicago, IL 60607, USA
ncjones@uic.edu
Ken McMurdy:
Department of Mathematics
Ramapo College of New Jersey
505 Ramapo Valley Road, Mahwah, NJ 07430, USA
kmcmurdy@ramapo.edu

