The Tate conjecture, abelian varieties and K3 surfaces

M. Artin and J. Tate conjectured that the Brauer group of a smooth and projective variety over a finite field is a finite group. In his 1966 Bourbaki talk [Tate66b], Tate explains why this is analogous to the conjectured finiteness of the Tate–Shafarevich group of an abelian variety over a number field.