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　　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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目录

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

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
　　从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（Springer）出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。
　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
　　当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。
　　总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。
初等Dirichlet级数和模形式（影印版）/国外数学名著系列72 [Elemetary Dirichlet Series and Modular Forms] 电子书 下载 mobi epub pdf txt

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