A special point problem of André-Pink-Zannier in the universal family of abelian varieties

Abstract

The Andre-Pink-Zannier conjecture predicts that a subvariety of a mixed Shimura variety is weakly special if its intersection with the generalized Hecke orbit of a given point is Zariski dense. 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 Andr´e-Pink- Zannier 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.