基本信息
- 原书名：Modular Forms with Integral and Half-Integral Weights
内容简介
书籍
数学书籍
Modular Forms with Integral and Half-Integral Weights focuses on the fundamentaltheory of modular forms of one variable with integral and half-integral weights.The main theme of the book is the theory of Eisenstein series. It is a fundamental problem to construct a basis of the orthogonal complement of the space of cusp forms; as is well known, this space is spanned by Eisenstein series for any weight greater than or equal to 2. The book proves that the conclusion holds true for weight 3/2 by explicitly constructing a basis of the orthogonal complement of the space of cusp forms. The problem for weight 1/2, which was solved by Serre and Stark, will also be discussed in this book. The book provides readers not only basic knowledge on this topic but also a general survey of modem investigation methods of modular forms with integral, and half-integral weights. It will be of significant interest to researchers and practitioners in modular forms of mathematics.
数学书籍
Modular Forms with Integral and Half-Integral Weights focuses on the fundamentaltheory of modular forms of one variable with integral and half-integral weights.The main theme of the book is the theory of Eisenstein series. It is a fundamental problem to construct a basis of the orthogonal complement of the space of cusp forms; as is well known, this space is spanned by Eisenstein series for any weight greater than or equal to 2. The book proves that the conclusion holds true for weight 3/2 by explicitly constructing a basis of the orthogonal complement of the space of cusp forms. The problem for weight 1/2, which was solved by Serre and Stark, will also be discussed in this book. The book provides readers not only basic knowledge on this topic but also a general survey of modem investigation methods of modular forms with integral, and half-integral weights. It will be of significant interest to researchers and practitioners in modular forms of mathematics.
作译者
Dr. Xueli Wang is a Professor at South China Normal University, China. Dingyi Pei is a Professor at Guangzhou University, China.
目录
《整权与半整权模形式(英文版)》
Chapter 1 Theta Functions and Their Transformation Formulae
Chapter 2 Eisenstein Series
2.1 Eisenstein Series with Half Integral Weight
2.2 Eisenstein Series with Integral Weight
Chapter 3 The Modular Group and Its Subgroups
Chapter 4 Modular Forms with Integral Weight or Half-integral Weight
4.1 Dimension Formula for Modular Forms with Integral Weight
4.2 Dimension Formula for Modular Forms with Half-Integral Weight
References
Chapter 5 Operators on the Space of Modular Forms
5.1 Hecke Rings
5.2 A Representation of the Hecke Ring on the Space of Modular Forms
5.3 Zeta Functions of Modular Forms, Functional Equation, Weil Theorem
5.4 Hecke Operators on the Space of Modular Forms with Half-Integral Weight
References
Chapter 6 New Forms and Old Forms
6.1 New Forms with Integral Weight
6.2 New Forms with Half Integral Weight
6.3 Dimension Formulae for the Spaces of New Forms
Chapter 1 Theta Functions and Their Transformation Formulae
Chapter 2 Eisenstein Series
2.1 Eisenstein Series with Half Integral Weight
2.2 Eisenstein Series with Integral Weight
Chapter 3 The Modular Group and Its Subgroups
Chapter 4 Modular Forms with Integral Weight or Half-integral Weight
4.1 Dimension Formula for Modular Forms with Integral Weight
4.2 Dimension Formula for Modular Forms with Half-Integral Weight
References
Chapter 5 Operators on the Space of Modular Forms
5.1 Hecke Rings
5.2 A Representation of the Hecke Ring on the Space of Modular Forms
5.3 Zeta Functions of Modular Forms, Functional Equation, Weil Theorem
5.4 Hecke Operators on the Space of Modular Forms with Half-Integral Weight
References
Chapter 6 New Forms and Old Forms
6.1 New Forms with Integral Weight
6.2 New Forms with Half Integral Weight
6.3 Dimension Formulae for the Spaces of New Forms
前言
The theory of modular forms is an important subject of number theory. Also it has very important applications to other areas of number theory such as elliptic curves, quadratic forms, etc. Its contents is vast. So any book on it must necessarily make a rather limited selection from the fascinating array of possible topics. Our focus is on topics which deal with the fundamental theory of modular forms of one variable with integral and half-integral weight. Even for such a selection we have to make further limitations on the themes discussed in this book. The leading theme of the book is the development of the theory of Eisenstein series.
A fundamental problem is the construction of a basis of the space of modular forms. It is well known that, for any weight ≥ 2 and the weight 1, the orthogonal complement of the space of cusp forms is spanned by Eisenstein series. Does this conclusion hold for the half-integral weight ＜ 2? The problem for weight 1/2 was solved by J.P. Serre and H.M.Stark. Then one of the authors of this book, Dingyi Pei, proved that the conclusion holds for weight 3/2 by constructing explicitly a basis of the orthogonal complement of the space of cusp forms. To introduce this result and some of its applications is our motivation for writing this book, which is a large extension version of the book "Modular forms and ternary quadratic forms" (in Chinese) written by Dingyi Pei.
Chapter 1 can be viewed as an introduction to the themes discussed in the book.Starting from the problem of representing integers by quadratic forms we introduce the concept of modular forms. In Chapter 2, we discuss the analytic continuation of Eisenstein series with integral and half-integral weight, which prepares the construc-tion of Eisenstein series in Chapter 7.
In Chapters 3-5, some fundamental concepts, notations and results about modu-lar forms are introduced which are necessary for understanding later chapters. More specifically, we introduce in Chapter 3 the modular group and its congruence sub-groups and the Riemannian surface associated with a discrete subgroup of SL2 (R).Furthermore, the concept of cusp points for a congruence subgroup is presented. In Chapter 4, we define modular forms with integral and half-integral weight, calculate the dimension of the space of modular forms using the theorem of Riemann-Roch.Chapter 5 is dedicated to define Hecke rings and discuss some of their fundamental properties. Also in this chapter the Zeta function of a modular form with integral orhalf-integral weight is described. In particular, we deduce the functional equation of the Zeta function of a modular form, and discuss Weil's Theorem.
In Chapter 6, the definitions of new forms and old forms with integral and half-integral weight are given. In particular the Atkin-Lehner's theory and the Kohnen's theory, with respect to new forms for integral and half-integral weight, are discussed at length respectively.
In Chapter 7, we construct Eisenstein series. The first objective is to construct Eisenstein series with half-integral weight ≥ 5/2. The second objective is the con-struction of Eisenstein series with weight 1/2 according to Serre and Stark. Then the method of the construction for Eisenstein series of weight 3/2 is introduced, followed by the construction of Cohen-Eisenstein series. For completeness, the construction of Eisenstein series with integral weight, which is due to Hecke, is also given in the last section of the chapter.
The Shimura lifting is the main objective of Chapter 8 where we follow the way depicted by Shintani. Weil representation is introduced first and some elementary properties of Weil representation are discussed. Then the Shimura lifting from cusp forms with half-integral weight to ones with integral weight is constructed. Also the Shimura lifting for Eisenstein spaces is deduced in this chapter.
In Chapter 9, we discuss the Eichler-Selberg trace formula for the space of modular forms with integral and half-integral weight. The simplest case of the Eichler-Selberg trace formula on SL2(Z) is deduced in terms of Zagier's method. Then the trace formula on a Fuchsian group is obtained by Selberg's method. Finally the Niwa's and Koknen's trace formulae are obtained for the space of modular forms with half-integral weight and the group Fo(N).
In Chapter 10, some applications of modular forms and Eisenstein series to the arithmetic of quadratic forms are described. We first present the Schulze-Pillot's proof of Siegel theorem. Then some results of representation of integers by ternary quadratic forms are explained. We also give an upper bound of the minimal positive integer represented by a positive definite even quadratic form with level 1 or 2.
Although many modern results on modular forms with half-integral weight are contained in this book, it is written as elementarily as possible and it's content is self-contained. We hope it can be used as a reference book for researchers and as a textbook for graduate students.
The authors would like to thank Ms. Yuzhuo Chen for her many helps. Also many thanks should be given to Dr. Junwu Dong for his helpful suggestions and carefully typesetting the draft of this book. We especially wish to thank Dr. Wotfgang Happle Happle for carefully reading the draft of this book' and correcting some errors in the draft. The author Xueli Wang wishes to thank Prof. Dr. Gerhard Frey for stimulating discussions and providing the environment of I.E.M in Essen University, where part of the draft has been done. Xueli Wang hope to give deepest gratitude for his lovely and beautiful wife, Dr. Dongping Xu, who assumed all of the housework over the years. Finally, the author Xueli Wang would like to dedicate this book to the 80th birthday of his father.
Xueli Wang Dingyi Pei
Guangzhou
September, 2011
A fundamental problem is the construction of a basis of the space of modular forms. It is well known that, for any weight ≥ 2 and the weight 1, the orthogonal complement of the space of cusp forms is spanned by Eisenstein series. Does this conclusion hold for the half-integral weight ＜ 2? The problem for weight 1/2 was solved by J.P. Serre and H.M.Stark. Then one of the authors of this book, Dingyi Pei, proved that the conclusion holds for weight 3/2 by constructing explicitly a basis of the orthogonal complement of the space of cusp forms. To introduce this result and some of its applications is our motivation for writing this book, which is a large extension version of the book "Modular forms and ternary quadratic forms" (in Chinese) written by Dingyi Pei.
Chapter 1 can be viewed as an introduction to the themes discussed in the book.Starting from the problem of representing integers by quadratic forms we introduce the concept of modular forms. In Chapter 2, we discuss the analytic continuation of Eisenstein series with integral and half-integral weight, which prepares the construc-tion of Eisenstein series in Chapter 7.
In Chapters 3-5, some fundamental concepts, notations and results about modu-lar forms are introduced which are necessary for understanding later chapters. More specifically, we introduce in Chapter 3 the modular group and its congruence sub-groups and the Riemannian surface associated with a discrete subgroup of SL2 (R).Furthermore, the concept of cusp points for a congruence subgroup is presented. In Chapter 4, we define modular forms with integral and half-integral weight, calculate the dimension of the space of modular forms using the theorem of Riemann-Roch.Chapter 5 is dedicated to define Hecke rings and discuss some of their fundamental properties. Also in this chapter the Zeta function of a modular form with integral orhalf-integral weight is described. In particular, we deduce the functional equation of the Zeta function of a modular form, and discuss Weil's Theorem.
In Chapter 6, the definitions of new forms and old forms with integral and half-integral weight are given. In particular the Atkin-Lehner's theory and the Kohnen's theory, with respect to new forms for integral and half-integral weight, are discussed at length respectively.
In Chapter 7, we construct Eisenstein series. The first objective is to construct Eisenstein series with half-integral weight ≥ 5/2. The second objective is the con-struction of Eisenstein series with weight 1/2 according to Serre and Stark. Then the method of the construction for Eisenstein series of weight 3/2 is introduced, followed by the construction of Cohen-Eisenstein series. For completeness, the construction of Eisenstein series with integral weight, which is due to Hecke, is also given in the last section of the chapter.
The Shimura lifting is the main objective of Chapter 8 where we follow the way depicted by Shintani. Weil representation is introduced first and some elementary properties of Weil representation are discussed. Then the Shimura lifting from cusp forms with half-integral weight to ones with integral weight is constructed. Also the Shimura lifting for Eisenstein spaces is deduced in this chapter.
In Chapter 9, we discuss the Eichler-Selberg trace formula for the space of modular forms with integral and half-integral weight. The simplest case of the Eichler-Selberg trace formula on SL2(Z) is deduced in terms of Zagier's method. Then the trace formula on a Fuchsian group is obtained by Selberg's method. Finally the Niwa's and Koknen's trace formulae are obtained for the space of modular forms with half-integral weight and the group Fo(N).
In Chapter 10, some applications of modular forms and Eisenstein series to the arithmetic of quadratic forms are described. We first present the Schulze-Pillot's proof of Siegel theorem. Then some results of representation of integers by ternary quadratic forms are explained. We also give an upper bound of the minimal positive integer represented by a positive definite even quadratic form with level 1 or 2.
Although many modern results on modular forms with half-integral weight are contained in this book, it is written as elementarily as possible and it's content is self-contained. We hope it can be used as a reference book for researchers and as a textbook for graduate students.
The authors would like to thank Ms. Yuzhuo Chen for her many helps. Also many thanks should be given to Dr. Junwu Dong for his helpful suggestions and carefully typesetting the draft of this book. We especially wish to thank Dr. Wotfgang Happle Happle for carefully reading the draft of this book' and correcting some errors in the draft. The author Xueli Wang wishes to thank Prof. Dr. Gerhard Frey for stimulating discussions and providing the environment of I.E.M in Essen University, where part of the draft has been done. Xueli Wang hope to give deepest gratitude for his lovely and beautiful wife, Dr. Dongping Xu, who assumed all of the housework over the years. Finally, the author Xueli Wang would like to dedicate this book to the 80th birthday of his father.
Xueli Wang Dingyi Pei
Guangzhou
September, 2011
