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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Weil conjectures ： ウィキペディア英語版 Weil conjectures In mathematics, the Weil conjectures were some highly influential proposals by on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. A variety ''V'' over a finite field with ''q'' elements has a finite number of rational points, as well as points over every finite field with ''q''''k'' elements containing that field. The generating function has coefficients derived from the numbers ''N''''k'' of points over the (essentially unique) field with ''q''''k'' elements. Weil conjectured that such ''zeta-functions'' should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by , the functional equation by , and the analogue of the Riemann hypothesis was proved by . ==Background and history== The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his ''Disquisitiones Arithmeticae'' , concerned with roots of unity and Gaussian periods. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that ''p'' is a prime number such that is divisible by 3. Then there is a cyclic cubic field inside the cyclotomic field of ''p''th roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group (Z/''p''Z)× of non-zero residues modulo ''p'' under multiplication and its unique subgroup of index three. Gauss lets $\backslash mathfrak$, $\backslash mathfrak\text{'}$, and $\backslash mathfrak\text{'}\text{'}$ be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2π''i''/''p''), he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, $(\backslash mathfrak\backslash mathfrak)$ equal to the number of elements of Z/''p''Z which are in $\backslash mathfrak$ and which, after being increased by one, are also in $\backslash mathfrak$. He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if α and are both in $\backslash mathfrak$, then there exist ''x'' and ''y'' in Z/''p''Z such that ''x''3 = α and ''y''3 = α + 1; consequently, ''x''3 + 1 = ''y''3. Therefore $(\backslash mathfrak\backslash mathfrak)$ is the number of solutions to ''x''3 + 1 = ''y''3 in the finite field Z/''p''Z. The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis. The Weil conjectures in the special case of algebraic curves were conjectured by . The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory: they implied upper bounds for exponential sums, a basic concern in analytic number theory. What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are ''discrete'' in nature, and topology speaks only about the ''continuous'', the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre. The rationality part of the conjectures was proved first by , using ''p''-adic methods. and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Artin for attacking the Weil conjectures, as outlined in . Of the four conjectures the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles . However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by , using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument. found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. 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「Weil conjectures」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.