A special point problem of Andr\'e-Pink-Zannier in the universal family of abelian varieties

Abstract

The Andr\'e-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal family of principally polarized abelian varieties. We explain the moduli interpretation of the conjecture in this case and prove several different cases for this conjecture: its overlap with the Andr\'e-Oort conjecture; when the subvariety is contained in an abelian scheme over a curve and the point is a torsion point on its fiber; when the subvariety is a curve.