  In this paper we study various aspects of the Ekedahl-Serre problem. We
formulate questions of Ekedahl-Serre type and Coleman-Oort type for general
weakly special subvarieties in the Siegel moduli space, propose a conjecture
relating these two questions, and provide examples supporting these questions.
The main new result is an upper bound of genera for curves over number fields
whose Jacobians are isogeneous to products of elliptic curves satisfying the
Sato-Tate equidistribution, and we also refine previous results showing that
certain weakly special subvarieties only meet the open Torelli locus in at most
finitely many points.

Algebraic Geometry