Unlikely intersections between isogeny orbits and curves

TL;DR

This paper investigates the intersection patterns between special algebraic curves and certain orbits generated by isogenies in families of abelian varieties, extending unlikely intersection principles.

Contribution

It characterizes algebraic curves over number fields that intersect isogeny orbits infinitely often within a family of abelian varieties.

Findings

Identifies conditions for infinite intersections with isogeny orbits.

Extends unlikely intersection results to non-isotrivial abelian schemes.

Uses Pila-Zannier strategy to prove the main results.

Abstract

Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila-Zannier strategy.