Tate conjecture

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 is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.

Let V be a smooth projective variety over a field k which is finitely generated over its prime field. Let k_s be a separable closure of k, and let G be the absolute Galois group Gal(k_s/k) of k. Fix a prime number ℓ which is invertible in k. Consider the ℓ-adic cohomology groups (coefficients in the ℓ-adic integers Z_ℓ, scalars then extended to the ℓ-adic numbers Q_ℓ) of the base extension of V to k_s; these groups are representations of G. For any i ≥ 0, a codimension-i subvariety of V (understood to be defined over k) determines an element of the cohomology group

which is fixed by G. Here Q_ℓ(i ) denotes the ith Tate twist, which means that this representation of the Galois group G is tensored with the ith power of the cyclotomic character.

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