基本信息
- 原书名：Elementary Dirichlet Series and Modular Forms
- 原出版社： Springer
- 作者： Goro Shimura
- 丛书名： 国外数学名著系列(影印版)
- 出版社：科学出版社
- ISBN：9787030313904
- 上架时间：2014-6-25
- 出版日期：2011 年6月
- 开本：16开
- 页码：146
- 版次：1-1
- 所属分类：数学 > 分析 > 数学分析
内容简介
书籍
数学书籍
The main topics of the book are the critical values of Dirichlet L-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 of elementary 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 of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However,all definitions and facts are clearly stated, and precise references are given. The notion of nearly 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 of isomorphism classes of elliptic 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 of modular 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.
数学书籍
The main topics of the book are the critical values of Dirichlet L-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 of elementary 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 of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However,all definitions and facts are clearly stated, and precise references are given. The notion of nearly 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 of isomorphism classes of elliptic 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 of modular 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.
目录
《初等Dirichlet级数和模形式(影印版)》
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
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
