Equidistribution of CM-points on quaternion Shimura varieties

The aim of this paper is to show some equidistribution statements of Galois orbits of CM-points for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CM-points as predicted by André-Oort conjecture. Our main result says that the Galois orbits of CM-points with the maximal Mumford-Tate groups are equidistributed provided that some subconvexity bounds on Rankin-Selberg L-series and on torsions of the class groups. A proof of the subconvexity bound for L-series has been announced by Michel and Venkatesh.