Given a correspondence between a modular curve and an elliptic curve A we show that there are not too many relations among the CM-points of A. In particular we show that the intersection of any finite rank subgroup of A with the set of CM-points of A is finite. We will also present a local version of this global result with an effective bound valid also for certain infinite rank subgroups. Furthermore we will give similar global and local results for intersections between subgroups of A and isogeny classes in A. The local results are proved using a technique based on "ordinary arithmetic differential equations". All of this is joint work with B. Poonen.