Local to global principles for homomorphisms of abelian schemes

TL;DR

This paper develops criteria for the existence of homomorphisms between abelian schemes over function fields, using local-to-global principles and advanced tools like Tate's conjecture and Hilbertianity methods.

Contribution

It introduces new local-to-global criteria for homomorphisms of abelian schemes over function fields, extending understanding of when such maps, including isogenies, exist.

Findings

Established criteria for existence of $k(S)$-homomorphisms based on restrictions to subvarieties.

Applied Tate conjecture and Hilbertianity methods to derive main results.

Extended local-to-global principles to the context of abelian schemes over function fields.

Abstract

Let $A$ and $B$ be abelian varieties defined over the function field $k(S)$ of a smooth algebraic variety $S/k.$ We establish criteria, in terms of restriction maps to subvarieties of $S,$ for existence of various important classes of $k(S)$-homomorphisms from $A$ to $B,$ e.g., for existence of $k(S)$-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case, when the base field is finite.