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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 P-adic L-function ： ウィキペディア英語版 P-adic L-function In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime number). For example, the domain could be the ''p''-adic integers Z''p'', a profinite ''p''-group, or a ''p''-adic family of Galois representations, and the image could be the ''p''-adic numbers Q''p'' or its algebraic closure. The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a ''p''-adic ''L''-function —is via the ''p''-adic interpolation of special values of ''L''-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a ''p''-adic ''L''-function, the ''p''-adic Riemann zeta function ζ''p''(''s''), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). ''p''-adic ''L''-functions arising in this fashion are typically referred to as analytic ''p''-adic ''L''-functions. The other major source of ''p''-adic ''L''-functions—first discovered by Kenkichi Iwasawa—is from the arithmetic of cyclotomic fields, or more generally, certain Galois modules over towers of cyclotomic fields or even more general towers. A ''p''-adic ''L''-function arising in this way is typically called an arithmetic ''p''-adic ''L''-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt ''p''-adic ''L''-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic ''p''-adic ''L''-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of ''L''-functions contain arithmetic information. ==Dirichlet L-functions== The Dirichlet ''L''-function is given by the analytic continuation of :$L(s,\backslash chi)\; =\; \backslash sum\_n\backslash frac\; =\; \backslash prod\_\}$ where ''B''''n'',χ is a generalized Bernoulli number defined by :$\backslash displaystyle\; \backslash sum\_^\backslash infty\; B\_\backslash frac\; =\; \backslash sum\_^f\backslash frac-1\}$ for χ a Dirichlet character with conductor ''f''. 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「P-adic 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.