In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of John Torrence Tate Jr born March 13, 1925 in Minneapolis Minnesota, is an American Mathematician, distinguished for many fundamental Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, an algebraic cycle on an Algebraic variety V is roughly speaking a Homology class on V that is represented by a In Mathematics, and in particular in Algebraic number theory, a Galois module is a module for a Galois group G. In Mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological It is unsolved in the general case, as of 2005, 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. Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. The Hodge conjecture is a major unsolved problem in Algebraic geometry which relates the Algebraic topology of a Non-singular complex Algebraic
Tate's original statement runs as follows. Let V be a smooth algebraic variety over a field k, which is finitely-generated over its prime field. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's Let G be the absolute Galois group of k. In Mathematics, the absolute Galois group GK of a field K is the Galois group of K sep over K Fix a prime number l. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Write H*(V) for the l-adic cohomology (coefficients in the l-adic integers, scalars then extended to the l-adic numbers) of the base extension of V to the given algebraic closure of k; these groups are G-modules. In Mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is Consider
- H2i(V)(i) = W
for the i-fold Tate twist of the cohomology group in degree 2i, for i = 1, 2, . In Mathematics, a Tate module is a Galois module constructed from an Abelian variety A over a field K. . . , d where d is the dimension of V. In Mathematics, the dimension of an Algebraic variety V in Algebraic geometry is defined informally speaking as the number of independent Under the Galois action, the image of G is a compact subgroup of GL(V), which is an l-adic Lie group. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the 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. In Mathematics, there are two basic results in Lie group theory that go by the name Cartan's theorem. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a Subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Tate's conjecture concerns the subspace W′ of W invariant under this Lie algebra (that is, on which the infinitesimal transformations of the Lie algebra representation act as 0). In Mathematics, an infinitesimal transformation is a limiting form of small transformation. There is another characterization used for W′, namely that it consists of vectors w in W that have an open stabilizer in G, or again have a finite orbit. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.
Then the Tate conjecture states that W′ is also the subspace of W generated by the cohomology classes of algebraic cycles of codimension i on V. In Mathematics, an algebraic cycle on an Algebraic variety V is roughly speaking a Homology class on V that is represented by a In Mathematics, codimension is a basic geometric idea that applies to Subspaces in Vector spaces and more generally to Submanifolds in Manifolds
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 maps for the Tate modules. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety In Mathematics, a Tate module is a Galois module constructed from an Abelian variety A over a field K. This is also known as the Tate conjecture, and several results have been proved towards it.
The same paper also contains related conjectures on L-functions. The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory.
References
- John Tate, Algebraic Cycles and Poles of Zeta Functions, Arithmetical Algebraic Geometry (1965) edited by O. F. G. Schilling
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