In mathematics, specifically arithmetic 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 conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.
Statement of the conjecture Let V {\displaystyle V} be a smooth projective variety over a field k {\displaystyle k} which is finitely generated over its prime field. Let k s {\displaystyle k_{s}} be a separable closure of k {\displaystyle k} , and let G {\displaystyle G} be the absolute Galois group Gal ( k s / k ) {\displaystyle \operatorname {Gal} (k_{s}/k)} of k {\displaystyle k} . Fix a prime number ℓ {\displaystyle \ell } which is invertible in k {\displaystyle k} . Consider the ℓ-adic cohomology groups (coefficients in the ℓ-adic integers Z ℓ {\displaystyle \mathbb {Z} _{\ell }} , scalars then extended to the ℓ-adic numbers Q ℓ {\displaystyle \mathbb {Q} _{\ell }} ) of the base extension of V {\displaystyle V} to k s {\displaystyle k_{s}} ; these groups are representations of G {\displaystyle G} . For any i ≥ 0 {\displaystyle i\geq 0} , a codimension- i {\displaystyle i} subvariety of V {\displaystyle V} (understood to be defined over k {\displaystyle k} ) determines an element of the cohomology group
H 2 i ( V k s , Q ℓ ( i ) ) = W {\displaystyle H^{2i}(V_{k_{s}},\mathbb {Q} _{\ell }(i))=W}
which is fixed by G {\displaystyle G} . Here Q ℓ ( i ) {\displaystyle \mathbb {Q} _{\ell }(i)} denotes the i {\displaystyle i} th Tate twist, which means that this representation of the Galois group G {\displaystyle G} is tensored with the i {\displaystyle i} th power of the cyclotomic character. The Tate conjecture states that the subspace W G {\displaystyle W^{G}} of W {\displaystyle W} fixed by the Galois group G {\displaystyle G} is spanned, as a Q ℓ {\displaystyle \mathbb {Q} _{\ell }} -vector space, by the classes of codimension- i {\displaystyle i} subvarieties of V {\displaystyle V} . An algebraic cycle means a finite linear combination of subvarieties; so an equivalent statement is that every element of W G {\displaystyle W^{G}} is the class of an algebraic cycle on V {\displaystyle V} with Q ℓ {\displaystyle \mathbb {Q} _{\ell }} coefficients.
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