p-adic equidistribution of CM points

On the moduli space of complex elliptic curves (the modular curve), CM points are those representing elliptic curves with complex multiplication. By work of Duke and of Clozel and Ullmo, it is known that CM points are equidistributed on the modular curve according to the hyperbolic measure. Duke's fundamental work hinges on his solution of a particular problem of Linnik's type.   
In this talk we will explain, for a fixed prime p, the distribution of CM points on the p-adic modular curve. We will focus on the subset of CM points lying inside the supersingular locus. In this case, we solve a p-adic equidistribution problem of Linnik's type. This is joint work with Sebastián Herrero and Juan Rivera-Letelier.   
(30 minute general talk followed by expert talk)