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

The Andr-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-Pink-Zannier conjecture in this case and prove several different cases for this conjecture: its overlap with the Andr-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.