Action of Modular Correspondences around CM Points

We study the action of modular correspondences in the p- adic neighborhood of CM points. We deduce and prove two stable and efficient p-adic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over Fq can be computed in probabilistic time ? exp((log q)1/2+?) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.