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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 Hasse–Weil zeta function ： ウィキペディア英語版 Hasse–Weil zeta function In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is one of the two most important types of L-function. Such ''L''-functions are called 'global', in that they are defined as Euler products in terms of local zeta functions. They form one of the two major classes of global ''L''-functions, the other being the ''L''-functions associated to automorphic representations. Conjecturally there is just one essential type of global ''L''-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of the Taniyama–Shimura conjecture, itself a very deep and recent result () in number theory. The description of the Hasse–Weil zeta function ''up to finitely many factors of its Euler product'' is relatively simple. This follows the initial suggestions of Helmut Hasse and André Weil, motivated by the case in which ''V'' is a single point, and the Riemann zeta function results. Taking the case of ''K'' the rational number field Q, and ''V'' a non-singular projective variety, we can for almost all prime numbers ''p'' consider the reduction of ''V'' modulo ''p'', an algebraic variety ''V''''p'' over the finite field F''p'' with ''p'' elements, just by reducing equations for ''V''. Again for almost all ''p'' it will be non-singular. We define :$Z\_(s)$ to be the Dirichlet series of the complex variable ''s'', which is the infinite product of the local zeta functions :$\backslash zeta\_\backslash left(p^\backslash right).$ Then ''Z''(''s''), according to our definition, is well-defined only up to multiplication by rational functions in a finite number of $p^$. Since the indeterminacy is relatively harmless, and has meromorphic continuation everywhere, there is a sense in which the properties of ''Z(s)'' do not essentially depend on it. In particular, while the exact form of the functional equation for ''Z''(''s''), reflecting in a vertical line in the complex plane, will definitely depend on the 'missing' factors, the existence of some such functional equation does not. A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in ramification theory, 'bad' primes carry good information (theory of the ''conductor''). This manifests itself in the étale theory in the Ogg–Néron–Shafarevich criterion for good reduction; namely that there is good reduction, in a definite sense, at all primes ''p'' for which the Galois representation ρ on the étale cohomology groups of ''V'' is ''unramified''. For those, the definition of local zeta function can be recovered in terms of the characteristic polynomial of :$\backslash rho(\backslash operatorname(p)),$ Frob(''p'') being a Frobenius element for ''p''. What happens at the ramified ''p'' is that ρ is non-trivial on the inertia group ''I''(''p'') for ''p''. At those primes the definition must be 'corrected', taking the largest quotient of the representation ρ on which the inertia group acts by the trivial representation. With this refinement, the definition of ''Z''(''s'') can be upgraded successfully from 'almost all' ''p'' to ''all'' ''p'' participating in the Euler product. The consequences for the functional equation were worked out by Serre and Deligne in the later 1960s; the functional equation itself has not been proved in general. ==Example: elliptic curve over Q== Let ''E'' be an elliptic curve over Q of conductor ''N''. Then, ''E'' has good reduction at all primes ''p'' not dividing ''N'', it has multiplicative reduction at the primes ''p'' that ''exactly'' divide ''N'' (i.e. such that ''p'' divides ''N'', but ''p''2 does not; this is written ''p'' || ''N''), and it has additive reduction elsewhere (i.e. at the primes where ''p''2 divides ''N''). The Hasse–Weil zeta function of ''E'' then takes the form :$Z\_(s)=\; \backslash frac.\; \backslash ,$ Here, ζ(''s'') is the usual Riemann zeta function and ''L''(''s'', ''E'') is called the ''L''-function of ''E''/Q, which takes the form〔Section C.16 of 〕 :$L(s,E)=\backslash prod\_pL\_p(s,E)^\backslash ,$ where, for a given prime ''p'', :$L\_p(s,E)=\backslash begin$ (1-a_pp^+p^), & \textp\nmid N \\ (1-a_pp^), & \textp\|N \\ 1, & \textp^2|N \end where, in the case of good reduction ''a''''p'' is ''p'' + 1 − (number of points of ''E'' mod ''p''), and in the case of multiplicative reduction ''a''''p'' is ±1 depending on whether ''E'' has split or non-split multiplicative reduction at ''p''. 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「Hasse–Weil zeta function」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.