Abstract: We present global and local results about linear independence of CM points on modular elliptic curves. The main global result (proved using equidistribution) states that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite rank subgroup of A with the set of CM-points of A is finite. The local results (proved via a method involving "arithmetic differential equations") give quantitative versions of similar statements. The latter method applies also to certain infinite rank subgroups, and to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura curve analogues of these results. All of this is joint work with B.Poonen.