In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology.

The Tate conjecture states that the subspace WG of W fixed by the Galois group G is spanned, as a Qℓ-vector space, by the classes of codimension-i subvarieties of V. An algebraic cycle means a finite linear combination of subvarieties; so an equivalent statement is that every element of WG is the class of an algebraic cycle on V with Qℓ coefficients.

The Tate conjecture for divisors (algebraic cycles of codimension 1) is a major open problem.

[3] In particular, an abelian variety A is determined up to isogeny by the Galois representation on its Tate module H1(Aks, Zℓ).

For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture was proved by Nygaard, Ogus, Charles, Madapusi Pera, and Maulik.