Abelian varieties with isogenous reductions

TL;DR

This paper proves that if two abelian varieties over a number field have reductions that are isogenous at many primes, then they are related by a homomorphism over an algebraic closure, revealing deep connections between local and global properties.

Contribution

It establishes a new link between local isogenies of reductions and global homomorphisms between abelian varieties over algebraic closures.

Findings

Existence of a global homomorphism from local isogenies at many primes.

Bridging local reduction properties to global structure of abelian varieties.

Advancement in understanding the relationship between reductions and global morphisms.

Abstract

If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism from A to B defined over an algebraic closure of K.