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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 Poincaré series (modular form) ： ウィキペディア英語版 Poincaré series (modular form) In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they generalize classical Eisenstein series. They are named after Henri Poincaré. If Γ is a finite group acting on a domain ''D'' and ''H''(''z'') is any meromorphic function on ''D'', then one obtains an automorphic function by averaging over Γ: :$\backslash sum\_\; H(\backslash gamma(z)).$ However, if Γ is a discrete group, then additional factors must be introduced in order to assure convergence of such a series. To this end, a Poincaré series is a series of the form :$\backslash theta\_k(z)\; =\; \backslash sum\_\; (J\_\backslash gamma(z))^k\; H(\backslash gamma(z))$ where ''J''γ is the Jacobian determinant of the group element γ,〔Or a more general factor of automorphy as discussed in .〕 and the asterisk denotes that the summation takes place only over coset representatives yielding distinct terms in the series. The classical Poincaré series of weight 2''k'' of a Fuchsian group Γ is defined by the series :$\backslash theta\_k(z)\; =\; \backslash sum\_\; (cz+d)^H\backslash left(\backslash frac\backslash right)$ the summation extending over congruence classes of fractional linear transformations : belonging to Γ. Choosing ''H'' to be a character of the cyclic group of order ''n'', one obtains the so-called Poincaré series of order ''n'': :$\backslash theta\_(z)\; =\; \backslash sum\_\; (cz+d)^\backslash exp\backslash left(2\backslash pi\; i\; n\backslash frac\backslash right)$ The latter Poincaré series converges absolutely and uniformly on compact sets (in the upper halfplane), and is a modular form of weight 2''k'' for Γ. Note that, when Γ is the full modular group and ''n'' = 0, one obtains the Eisenstein series of weight 2''k''. In general, the Poincaré series is, for ''n'' ≥ 1, a cusp form. ==Notes== 〔 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「Poincaré series (modular form)」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.