I shall report a joint paper with Ke Chen and Xin Lu. We prove that the Hecke translates of certain CM points in the moduli space of principle polarized abelian varieties do not lie in the open Torelli locus of smooth projective curves under suitable conditions on the number field of definition and dimension of possible simple CM factors. The proof makes use of properties of the stable Faltings height and known cases of the Sato-Tate equidistribution. We also refine our previous work showing that certain weakly special subvarieties in Ag only meet the open Torelli locus in dimension zero. These results are closely related to our generalization on questions of Ekedahl-Serre and Coleman-Oort types.
