The Mumford–Tate conjecture for products of abelian varieties

Let $X$ be a smooth projective variety over a finitely generated field $K$ and fix an embedding $K \subset \CC$. The Mumford–Tate conjecture is a precise way of saying that the extra structure on the $\ell$-adic étale cohomology groups of $X$ (a Galois representation) and the extra structure on the singular cohomology groups of $X$ (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 $K \subset \CC$.

We use techniques from Hodge theory, representation theory, and number theory. A crucial ingredient is a construction of Deligne [Del79].