On the $\ell$-adic Galois representations attached to nonsimple abelian varieties

TL;DR

This paper investigates the structure of Galois representations associated with nonsimple abelian varieties over finitely generated fields, providing conditions for their decomposition and extending complex case results to arithmetical settings.

Contribution

It introduces criteria for decomposing Galois representations of nonsimple abelian varieties and applies these to establish new arithmetical analogues of known complex results.

Findings

Provided sufficient conditions for Galois representation decomposition.

Extended Moonen and Zarhin's results to positive characteristic fields.

Achieved decomposition results for abelian varieties of dimension up to 5.

Abstract

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).