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 étale cohomology. To make this precise, let $G_{\mathrm{B}}$ be the Mumford-Tate group of the Hodge structure $H_{\text{sing}}^2(A(\mathbb{C})\times X(\mathbb{C}),\mathbb{Q})$. Let $G_{\ell }^{\circ }$ be the connected component of the identity of the Zariski closure of the image of the Galois group $\text{Gal}(\bar{K}/K)$ in $\mathrm{GL}(H_{\text{eˊt}}^2(A_{\bar{K}}\times X_{\bar{K}},{\mathbb{Q}}_{\ell }))$. 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.

arXiv comment19 pages, all comments are welcome