模形式理论是数论的一个十分重要的分支，它在数学和物理学的许多领域有十分重要的应用。《整权与半整权模形式(精)》由Xueli Wang、Dingyi Pei所著，本书将全面介绍整权和半整权单变量模形式的基本理论和现代研究成果：低权模形式（主要是低权Eisenstein级数）的构造，整权与半整权模形式之间的联系，模形式在二次型的算术研究中的某些应用。本书的主要特点是同时介绍和研究整权与半整权模形式的理论及其应用。书中既包含了模形式的基本理论，如：模群及其同余子群，Hecke算子等，也包含了许多现代的研究成果，如：整权和半整权模形式的Zeta函数，整权和半整权的Eisenstein级数，Cohen-Eisenstein级数，半整权模形式到整权模形式的Shimura提升，整权和半整权模形式空间上Hecke算子的迹公式，以及模形式理论在二次型的某些算术问题中的应用。

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 7 Construction of Eisenstein Series

7.1 Construction of Eisenstein Series with Weight > 5/2

7.2 Construction of Eisenstein Series with Weight 1/2

7.3 Construction of Eisenstein Series with Weight 3/2

7.4 Construction of Cohen-Eisenstein Series

7.5 Construction of Eisenstein Series with Integral Weight

References

Chapter 8 Well Representation and Shimura Lifting

8.1 Weil Representation

8.2 Shimura Lifting for Cusp Forms

8.3 Shimura Lifting of Eisenstein Spaces

8.4 A Congruence Relation between Some Modular Forms

References

Chapter 9 Trace Formula

9.1 Eichler-Selberg Trace Formula on SL2(Z)

9.2 Eichler-Selberg Trace Formula on Fuchsian Groups

9.3 Trace Formula on the Space Sk+1/2(N,x)

References

Chapter 10 Integers Represented by Positive Definite Quadratic Forms

10.1 Theta Function of a Positive Definite Quadratic Form and Its Values at Cusp Points

10.2 The Minimal Integer Represented by a Positive Definite Quadratic Form

10.3 The Eligible Numbers of a Positive Definite Ternary Quadratic Form

References

Index