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.

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 In Table 4 the elliptic curve 46800cw4 has been replaced by the elliptic curve 486720dr3.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

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• Torsion of rational elliptic curves over different types of cubic fields