Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results

Abstract

AbstractIn the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curveA, the intersection of any finite-rank subgroup ofAwith the set of CM-points ofAis finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to provequantitativeversions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as 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.