for abelian varieties over number fields, part of Faltings's solution of the Mordell conjecture. Zarhin extended these results to any finitely generated base field. The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves C1 × ... × Cn. The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties. Namely, for any abelian varieties A and B over a finitely generated field k, the natural map is an isomorphism. In particular, an abelian variety A is determined up to isogeny by the Galois representation on its Tate module H1(Aks, Zℓ). The Tate conjecture also holds for K3 surfaces over finitely generated fields of characteristic not 2. (On a surface, the nontrivial part of the conjecture is about divisors.) In characteristic zero, the Tate conjecture for K3 surfaces was proved by André and Tankeev. For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture was proved by Nygaard, Ogus, Charles, Madapusi Pera, and Maulik. surveys known cases of the Tate conjecture. Related conjectures 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(ks/k) on the ℓ-adic cohomology of X is semisimple (that is, a direct sum of irreducible representations). For k of characteristic 0, showed that the Tate conjecture (as stated above) implies the semisimplicity of For k finite of order q, Tate showed that the Tate conjecture plus the semisimplicity conjecture would imply the strong Tate conjecture, namely that the order of the pole of the zeta function Z(X, t) at t = q−j is equal to the rank of the group of algebraic cycles of codimension j modulo numerical equivalence. Like the Hodge conjecture, the Tate conjecture would imply most of Grothendieck's standard conjectures on algebraic cycles. Namely, it would imply the Lefschetz standard conjecture (that the inverse of the Lefschetz isomorphism is defined by an algebraic correspondence); that the Künneth components of the diagonal are algebraic; and that numerical equivalence and homological equivalence of algebraic cycles are the same. Notes References External links James Milne, The Tate conjecture over finite fields (AIM talk). Topological methods of algebraic geometry Diophantine geometry Conjectures Unsolved problems in number theory