The Mumford-Tate conjecture for the product of an abelian surface and a K3 surface

Abstract: In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying that the Hodge structure on singular cohomology conveys the same information as the Galois representation on $\ell$-adic \'{e}tale cohomology. To make this precise, let $G_{\mathrm{B}}$ be the Mumford-Tate group of the Hodge structure $H^{2}_{\text{sing}}(A(\mathbb{C}) \times X(\mathbb{C}), \mathbb{Q})$. Let $G_{\ell}^{\circ}$ be the connected component of the identity of the Zariski closure of the image of the Galois group $\textrm{Gal}(\bar{K}/K)$ in $\mathrm{GL}(H^{2}{\text{\'{e}t}}(A{\bar{K}} \times X_{\bar{K}}, \mathbb{Q}{\ell}))$. The Mumford-Tate conjecture asserts that $G{\mathrm{B}} \otimes \mathbb{Q}{\ell} \cong G{\ell}^{\circ}$. The proof presented in this paper uses input from number theory (Chebotaryov's density theorem), Lie theory, and some facts about K3 surfaces over finite fields.