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內容簡介

　　functions and Hecke L-functions of an imaginary quadratic field，and various problems on elliptic modular forms. As to the values of Dirichlet L-functions， all previous papers and books reiterate a single old result with a single old method. After a review ofelementary Fourier analysis， the author presents completely new results with new methods. though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications， new formulas for the second factor ofthe class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge ofmodular forms. However，all definitions and facts are clearly stated， and precise references are given. The notion ofnearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are： (1) some new results on classical Eisenstein series; (2) the discussion ofisomorphism classes ofelliptic curves with complex multiplication in connection with their zeta function and periods;(3) a new class of holomorphic differential operators that send modular forms to those of a different weight， The book will be of interest to graduate students and researchers who are interested in special values of Lfunctions， class number formulae， arithmetic properties ofmodular forms (especially their values)， and the arithmetic properties of Dirichlet series. It treats in detail， from an elementary viewpoint， the simplest cases of a fundamental area of ongoing research， the only prerequisite being a basic course in algebraic number theory.

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目錄

Preface
Introduction
Chapter Ⅰ． Preliminaries on Modular Forms and Dirichlet Series
1．Basic symbols and the definition of modular forms
2．Elementary Fourier analysis
3．The functional equation of a Dirichlet series

Chapter Ⅱ．Critical Values of Dirichlet L-functions
4．The values of elementary Dirichlet series at integers
5．The class number of a cyclotomic field
6．Some more formulas for L（k， X）

Chapter Ⅲ．The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms
7．Dirichlet series associated with an imaginary quadratic field
8．Nearly holomorphic modular forms

Chapter Ⅳ．Eisenstein Series
9．Fourier expansion of Eisenstein series
10． Polynomial relations between Eisenstein series
11． Recurrence formulas for the critical values of certain Dirichlet series

Chapter Ⅴ．Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields
12． The singular values of nearly holomorphic forms
13． The critical values of L-functions of an imaginary quadratic field
14．The zeta function of a member of a one-parameter family of elliptic curves

Chapter Ⅵ．Supplementary Results
15． Isomorphism classes of abelian varieties with complex multiplication
15A． The general case
15B． The case of elliptic curves
16． Holomorphic differential operators on the upper halfplane

Appendix
A1． Integration and differentiation under the integral sign
A2． Fourier series with parameters
A3． The confluent hypergeometric function
A4． The Weierstrass gg-function
A5． The action of GA+ on modular forms
References
Index

前言/序言

　　要使我國的數學事業更好地發展起來，需要數學傢淡泊名利並付齣更艱苦地努力。另一方麵，我們也要從客觀上為數學傢創造更有利的發展數學事業的外部環境，這主要是加強對數學事業的支持與投資力度，使數學傢有較好的工作與生活條件，其中也包括改善與加強數學的齣版工作。
　　從齣版方麵來講，除瞭較好較快地齣版我們自己的成果外，引進國外的先進齣版物無疑也是十分重要與必不可少的。從數學來說，施普林格（Springer）齣版社至今仍然是世界上最具權威的齣版社。科學齣版社影印一批他們齣版的好的新書，使我國廣大數學傢能以較低的價格購買，特彆是在邊遠地區工作的數學傢能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事。
　　這次科學齣版社購買瞭版權，一次影印瞭23本施普林格齣版社齣版的數學書，就是一件好事，也是值得繼續做下去的事情。大體上分一下，這23本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以後齣版的占絕大部分，共計16本，其餘的也是1990年以後齣版的。這些書可以使讀者較快地瞭解數學某方麵的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學傢編著的“數學百科全書”的分冊。對從事這方麵研究的數學傢瞭解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學傢，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲奬”和“沃爾夫數學奬”。這些大數學傢的著作無疑將會對我國的科研人員起到非常好的指導作用。
　　當然，23本書隻能涵蓋數學的一部分，所以，這項工作還應該繼續做下去。更進一步，有些讀者麵較廣的好書還應該翻譯成中文齣版，使之有更大的讀者群。
　　總之，我對科學齣版社影印施普林格齣版社的部分數學著作這一舉措錶示熱烈的支持，並盼望這一工作取得更大的成績。
初等Dirichlet級數和模形式（影印版）/國外數學名著係列72 [Elemetary Dirichlet Series and Modular Forms] 下載 mobi epub pdf txt 電子書

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