  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 an isogeny class on $S$ is finite. The
question was proposed by A. Buium and B. Poonen in 2009. We follow the strategy
proposed by the authors, using a result about the equidistribution of Hecke
points on Shimura varieties and Serre's open image theorem. The result is an
instance of the Zilber-Pink conjecture.
