Isogeny orbits in a family of abelian varieties

TL;DR

This paper proves that in a non-isotrivial family of abelian varieties, a curve containing infinitely many isogeny orbits of a finitely generated subgroup must be a special subvariety.

Contribution

It establishes a new characterization of special subvarieties based on the distribution of isogeny orbits in families of abelian varieties.

Findings

Infinite isogeny orbits imply the curve is special.

Provides criteria for identifying special subvarieties.

Advances understanding of isogeny orbit distributions.

Abstract

We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special.