Elliptic Curves with abelian division fields
Abstract
Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves E/Q such that Q(E[n]) is as small as possible, that is, when Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5. More generally, we classify all curves such that Q(E[n]) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem), when Q(E[n])/Q is an abelian extension. In particular, we prove that this only happens for n=2,3,4,5,6, or 8, and we classify the possible Galois groups that occur for each value of n.
Cite
@article{arxiv.1511.08578,
title = {Elliptic Curves with abelian division fields},
author = {Enrique Gonzalez-Jimenez and Alvaro Lozano-Robledo},
journal= {arXiv preprint arXiv:1511.08578},
year = {2021}
}
Comments
In this version we fix an error in the proof of Proposition 3.7. We thank Tyler Genao for pointing out this error to us. In Table 4 the elliptic curve 46800cw4 has been replaced by the elliptic curve 486720dr3. The elliptic curve 46800cw4 does not have 2-adic image equal to X58f