LGRT 1681
The transcendental lattice T of a projective K3 surface is an irreducible Hodge structure. The ring of endomorphisms of the rational Hodge structure of T is thus a field K that turns out to be either a totally real or a totally complex number field. When K is totally complex, then it is spanned by rational Hodge isometries, which turn out to be algebraic, by a result of Mukai and Buskin. We will review this relationship between number theory and the Hodge conjecture for products of K3 surfaces.