 For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture was proved by Nygaard, Ogus, Charles, Madapusi Pera, and Maulik.Let "X" be a smooth projective variety over a finitely generated field "k".The semisimplicity conjecture predicts that the representation of the Galois group "G" = Gal("k"s/"k") on the ℓ-adic cohomology of "X" is semisimple (that is, a direct sum of irreducible representations).