On CM points away from the Torelli locus

Abstract

In this paper we prove that certain points in the Hecke orbit of a CM point in the Siegel modular variety do not lie in the open Torelli locus under suitable conditions on the field of definition and the CM factors that arise in the corresponding CM abelian variety. The proof is a combination of properties of stable Faltings height and known cases of the Sato-Tate equidistribution, and is motivated by an analogue over function fields proved by Kukulies. We also discuss the relation of our result with questions of Ekedahl-Serre type and refine a previous result showing that certain Shimura subvarieties of unitary type in the Siegel modular variety only meet the open Torelli locus in dimension zero. Contents