Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: Global results

Abstract

Given a correspondence between a modular curve $S$ and an elliptic curve $A$, we prove that the intersection of any finite-rank subgroup of $A$ with the set of points on $A$ corresponding to CM points on $S$ is finite. We prove also a version in which $S$ is replaced by a Shimura curve and $A$ is replaced by a higher-dimensional abelian variety