Fields Generated by a Single Point on Elliptic Curves

We study the finite extensions of a number field generated by a single point on an elliptic curve. We construct geometrically non-isomorphic elliptic curves with complex multiplication that generate the same collection of extensions, giving a negative answer to a question of Mazur and Rubin. For torsion-generated fields, we prove that inclusion of the collections up to finitely many exceptions implies isogeny over the ground field, and in the non-CM case even isomorphism. When all geometric endomorphisms are defined over the ground field, we characterize equality of the torsion-generated collections by the existence, for every prime p, of an isogeny over the ground field of degree prime to p. This condition also implies equality of the full collections of fields generated by a single point.