Elliptic Curves with abelian division fields

TL;DR

This paper classifies elliptic curves over Q with abelian division fields, showing that such cases occur only for specific small n and detailing the structure of their Galois groups.

Contribution

It provides a complete classification of elliptic curves over Q with abelian division fields, identifying all such n and describing their Galois groups.

Findings

Q(E[n]) is minimal only for n=2,3,4,5.

Q(E[n]) is contained in a cyclotomic extension only for n=2,3,4,5,6,8.

The possible Galois groups for these n are explicitly classified.

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.