p-adic equidistribution of CM points

Let X be a modular curve and consider a sequence of Galois orbits of CM points in X, whose p-conductors tend to infinity. Its equidistribution properties in X.C/ and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification X_p^an of X_Qp . We partition the set of CM points of sufficiently high conductor in X_Qp into finitely many explicit basins B_V , indexed by the irreducible components V of the mod-p reduction of the canonical model of X. We prove that a sequence z_n of local Galois orbits of CM points with p-conductor going to infinity has a limit in X_p^an if and only if it is eventually supported in a single basin B_V . If so, the limit is the unique point of X_p^an whose mod-p reduction is the generic point of V . The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross’s theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.