In 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 etale cohomology. The conjecture is a central problem in the theory of algebraic cycles and can be considered an arithmetic analog of the Hodge conjecture. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe 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. 1 Proposed Year Progress Toward ResolutionThe Tate conjecture for divisors (algebraic cycles of codimension 1) is a major open problem. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Tate Conjecture (Wikipedia)
Wikimedia FoundationLead section
a 1963 conjecture of John Tate
Statement of the conjecture
states that the subspace W^G of W fixed by the Galois group G is spanned
Known cases
The Tate conjecture for divisors (algebraic cycles of codimension 1) is a major open problem.
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