p-adic dynamics of Hecke operators on modular curves
The action of Hecke operators in the complex topology, ranges from easy density arguments, to deep equidistribution results due to Duke, Clozel-Ullmo and others. Passing from archimedean to non-archimedean primes raises many interesting questions. I will first explain several good motivations to study this problem, coming both from geometry (stratifications of Shimura varieties) and from arithmetic (properties of singular moduli). I will report on joint work with P. Kassaei (King's college) on the dynamics of Hecke operators acting on modular curves, considered in the p-adic topology. I will also mention work of Hererro, Menares and Rivera-Letelier and, to the extent time allows, work in progress by the union of these two teams.