The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds

We prove the Tate conjecture in every codimension on every power of an abelian variety of dimension at most four over a finite field. For geometrically simple fourfolds, we use Broe's theorem and the classification of Frobenius relations. For dihedral pairs of abelian surfaces, we construct algebrai