The Mumford--Tate conjecture for products of abelian varieties

TL;DR

This paper proves that the Mumford--Tate conjecture, relating Galois representations and Hodge structures, holds for product varieties if it holds for each factor, extending the conjecture's validity to product abelian varieties.

Contribution

It establishes that the Mumford--Tate conjecture for abelian varieties is preserved under taking products, a significant step in understanding the conjecture's behavior.

Findings

Mumford--Tate conjecture holds for products if it holds for individual factors.

Results are independent of the embedding of the field into complex numbers.

Extends the validity of the conjecture to product of abelian varieties.

Abstract

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the $\ell$-adic \'etale cohomology groups of~$X$ (namely, a Galois representation) and certain extra structure on the singular cohomology groups of~$X$ (namely, a Hodge structure) convey the same information. The main result of this paper says that if $A_1$ and~$A_2$ are abelian varieties (or abelian motives) over~$K$, and the Mumford--Tate conjecture holds for both~$A_1$ and~$A_2$, then it holds for $A_1 \times A_2$. These results do not depend on the embedding $K \subset \CC$.