Equidistribution de sous-varietes speciales (Equisistribution of special subvarieties)

Abstract

A strongly special subvariety of a Shimura variety $S$ is (essentially) a subvariety associated to a semi-simple sub-Shimura datum. We prove that the set of probability measures canonically associated to to strongly special subvarieties is compact. More precisely: If $\mu_n$ is a sequence of such probability measures associated to strongly special subvarieties $Z_n$ of $S$ there exists a subsequence $\mu_{n_k}$ which converge to a measure $\mu_Z$ canonically associated to a strongly special subvariety $Z$ and for all $k>>0$ $Z_{n_k}$ is contained in $Z$. We give some application to the Andre-Oort conjecture: If $X$ is a subvariety of $S$ then there exists at most finitely many maximal (amongs subvarieties of $X$) strongly special subvarieties of $X$ as predicted by the Andre-Oort Conjecture. The proof uses Ratner's theory, some results of Mozes-Shah and Dani-Margulis.