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| In [[mathematics]], the '''Tate conjecture''' is a 1963 [[conjecture]] of [[John Tate]] linking [[algebraic geometry]], and more specifically the identification of [[algebraic cycle]]s, with [[Galois module]]s coming from [[étale cohomology]]. It is unsolved in the general case, {{As of|2010|lc=on}}, and, like the [[Hodge conjecture]] to which it is related at the level of some important analogies, it is generally taken to be one of the major problems in the field.
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| Tate's original statement runs as follows. Let ''V'' be a smooth [[algebraic variety]] over a [[field (mathematics)|field]] ''k'', which is finitely-generated over its [[prime field]]. Let ''G'' be the [[absolute Galois group]] of ''k''. Fix a [[prime number]] ''l''. Write ''H''*(''V'') for the [[l-adic cohomology]] (coefficients in the [[p-adic integer|l-adic integer]]s, scalars then extended to the [[l-adic number]]s) of the base extension of ''V'' to the given [[algebraic closure]] of ''k''; these groups are ''G''-modules. Consider
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| :<math>H^{2i}(V)(i) = W\ </math>
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| for the ''i''-fold [[Tate twist]] of the cohomology group in degree 2''i'', for ''i'' = 1, 2, ..., ''d'' where ''d'' is the [[dimension of an algebraic variety|dimension]] of ''V''. Under the Galois action, the image of ''G'' is a [[compact group|compact subgroup]] of ''GL''(''V''), which is an ''l''-adic [[Lie group]]. It follows by the ''l''-adic version of [[Cartan's theorem]] that as a [[closed subgroup]] it is also a [[Lie subgroup]], with corresponding [[Lie algebra]]. Tate's conjecture concerns the subspace ''W'' ′ of ''W'' invariant under this Lie algebra (that is, on which the [[infinitesimal transformation]]s of the [[Lie algebra representation]] act as 0). There is another characterization used for ''W'' ′, namely that it consists of vectors ''w'' in ''W'' that have an open [[Group action#Orbits and stabilizers|stabilizer]] in ''G'', or again have a finite [[orbit (group theory)|orbit]].
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| Then the '''Tate conjecture''' states that ''W'' ′ is also the subspace of ''W'' generated by the cohomology classes of [[algebraic cycle]]s of [[codimension]] ''i'' on ''V''.
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| An immediate application, also given by Tate, takes ''V'' as the [[cartesian product]] of two [[abelian varieties]], and deduces a conjecture relating the morphisms from one abelian variety to another to [[intertwining map]]s for the [[Tate module]]s. This is also known as the ''Tate conjecture'', and several results have been proved towards it.
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| The same paper also contains related conjectures on [[L-function]]s.
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| ==References==
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| *{{Citation |first=John |last=Tate |chapter=Algebraic Cycles and Poles of Zeta Functions |title=Arithmetical Algebraic Geometry |year=1965 |editor-first=O. F. G. |editor-last=Schilling |location=New York |publisher=Harper and Row }}.
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| == External links ==
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| *[[James Milne]], [http://www.jmilne.org/math/articles/2007e.pdf The Tate conjecture over finite fields (AIM talk)].
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| *Keerthi Madapusi Pera, [http://www.math.harvard.edu/~keerthi/papers/tate.pdf The Tate conjecture for K3 surfaces in odd characteristic]
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Diophantine geometry]]
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| [[Category:Conjectures]]
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