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

Abstract

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, for an algebraic variety over a finitely generated field of characteristic 0. This paper presents a proof of the Mumford–Tate conjecture in degree 2 for the product of an abelian surface and a K3 surface.