  Given a correspondence between a modular curve and an elliptic curve A we
study the group of relations among the CM points of A. In particular we prove
that the intersection of any finite rank subgroup of A with the set of CM
points of A is finite. We also prove a local version of this global result with
an effective bound valid also for certain infinite rank subgroups. We deduce
the local result from a ``reciprocity'' theorem for CL (canonical lift) points
on A. Furthermore we prove similar global and local results for intersections
between subgroups of A and isogeny classes in A. Finally we prove Shimura curve
analogues and, in some cases, higher-dimensional versions of these results.
