| Résume | Let $A$ be an abelian variety defined over a number field. The algebraic monodromy groups $H_\ell(A)$ are an $\ell$-adic analogue of the Hodge group of $A_\mathbb{C}$, and they encode a great deal of information about the Galois representations associated with $A$.A natural question is whether we can describe $H_\ell(A \times B)$ in terms of $H_\ell(A)$ and $H_\ell(B)$. While the answer is negative in general, I will describe sufficient conditions (involving the dimensions and endomorphism algebras of $A$ and $B$) to ensure that $H_\ell(A \times B)$ is isomorphic to $H_\ell(A) \times H_\ell(B)$, and show how this can be used to prove the Mumford-Tate conjecture for \textit{nonsimple} abelian varieties of dimension up to 5. |
