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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 Artin conjecture (L-functions) ： ウィキペディア英語版 Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by automorphic forms and Langlands' philosophy. So far, only a small part of such a theory has been put on a firm basis. ==Definition== Given $\backslash rho$, a representation of $G$ on a finite-dimensional complex vector space $V$, where $G$ is the Galois group of the finite extension $L/K$ of number fields, the Artin $L$-function: $L(\backslash rho,s)$ is defined by an Euler product. For each prime ideal $\backslash mathfrak\; p$ in $K$'s ring of integers, there is an Euler factor, which is easiest to define in the case where $\backslash mathfrak\; p$ is unramified in $L$ (true for almost all $\backslash mathfrak\; p$). In that case, the Frobenius element $\backslash mathbf\; (\backslash mathfrak\; p)$ is defined as a conjugacy class in $G$. Therefore the characteristic polynomial of $\backslash rho(\; \backslash mathbf\; (\backslash mathfrak))$ is well-defined. The Euler factor for $\backslash mathfrak$ is a slight modification of the characteristic polynomial, equally well-defined, :$\backslash operatorname(\backslash rho(\backslash mathbf(\backslash mathfrak)))^$ = \operatorname \left (I - t \rho( \mathbf( \mathfrak)) \right )^, as rational function in ''t'', evaluated at $t\; =\; N\; (\backslash mathfrak)^$, with $s$ a complex variable in the usual Riemann zeta function notation. (Here ''N'' is the field norm of an ideal.) When $\backslash mathfrak$ is ramified, and ''I'' is the inertia group which is a subgroup of ''G'', a similar construction is applied, but to the subspace of ''V'' fixed (pointwise) by ''I''.〔It is arguably more correct to think instead about the coinvariants, the largest quotient space fixed by ''I'', rather than the invariants, but the result here will be the same. Cf. Hasse–Weil L-function for a similar situation.〕 The Artin L-function $L(\backslash rho,s)$ is then the infinite product over all prime ideals $\backslash mathfrak$ of these factors. As Artin reciprocity shows, when ''G'' is an abelian group these ''L''-functions have a second description (as Dirichlet ''L''-functions when ''K'' is the rational number field, and as Hecke ''L''-functions in general). Novelty comes in with non-abelian ''G'' and their representations. One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers. In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin ''L''-functions, for each irreducible representation of ''G''. For example, the simplest case is when ''G'' is the symmetric group on three letters. Since ''G'' has an irreducible representation of degree 2, an Artin ''L''-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function (for the trivial representation) and an ''L''-function of Dirichlet's type for the signature representation. 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「Artin L-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.