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

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 $\mathrm{\ell }$-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}$), $\mathbb{Q}$)$\mathrm{.}Let$G{\ell}^{\circ}$ be the connected component of the identity of the Zariski closure of the image of the Galois group $\text{Gal}(\bar{K}\mathrm{/}K)$ in $$\mathrm{G}\mathrm{L}$(H^{2}{$\acute{\text{e}}\text{t}$}(A{\bar{K}} \times X_{\bar{K}}, $\mathbb{Q}$_{\ell}))$. The Mumford-Tate conjecture asserts that $G_{\mathrm{B}}\otimes {\mathbb{Q}}_{\mathrm{\ell }}\cong G_{\mathrm{\ell }}^{\circ }$. 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.