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

Abstract

The André-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 André-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.

arXiv commentRemoved the algebraic point assumption. Comments are welcome!

References in corpus (1)

• The modular height of an abelian variety and its finiteness property

Cited by in corpus (2)

• Pink's conjecture on unlikely intersections and families of semi-abelian varieties
• Unlikely Intersections in families of abelian varieties and the polynomial Pell equation