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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. ==Statement of the 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 ''i''th Tate twist, which means that this representation of the Galois group ''G'' is tensored with the ''i''th power of the cyclotomic character. The Tate conjecture states that the subspace ''W''''G'' of ''W'' fixed by the Galois group ''G'' is spanned, as a Q''l''-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 ''W''''G'' is the class of an algebraic cycle on ''V'' with Q''l'' coefficients. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tate conjecture」の詳細全文を読む スポンサード リンク
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