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Chapter 1 Theta Functions and Their Transformation Formulae 1

Chapter 2 Eisenstein Series 13

2.1 Eisenstein Series with Half Integral Weight 13

2.2 Eisenstein Series with Integral Weight 37

Chapter 3 The Modular Group and Its Subgroups 45

Chapter 4 Modular Forms with Integral Weight or Half-integral Weight 65

4.1 Dimension Formula for Modular Forms with Integral Weight 65

4.2 Dimension Formula for Modular Forms with Half-Integral Weight 81

References 88

Chapter 5 Operators on the Space of Modular Forms 89

5.1 Hecke Rings 89

5.2 A Representation of the Hecke Ring on the Space of Modular Forms 113

5.3 Zeta Functions of Modular Forms,Functional Equation,Weil Theorem 120

5.4 Hecke Operators on the Space of Modular Forms with Half-Integral Weight 134

References 152

Chapter 6 New Forms and Old Forms 153

6.1 New Forms with Integral Weight 153

6.2 New Forms with Half Integral Weight 178

6.3 Dimension Formulae for the Spaces of New Forms 200

Chapter 7 Construction of Eisenstein Series 205

7.1 Construction of Eisenstein Series with Weight≥5/2 205

7.2 Construction of Eisenstein Series with Weight 1/2 221

7.3 Construction of Eisenstein Series with Weight 3/2 232

7.4 Construction of Cohen-Eisenstein Series 246

7.5 Construction of Eisenstein Series with Integral Weight 255

References 263

Chapter 8 Weil Representation and Shimura Lifting 265

8.1 Weil Representation 265

8.2 Shimura Lifting for Cusp Forms 280

8.3 Shimura Lifting of Eisenstein Spaces 299

8.4 A Congruence Relation between Some Modular Forms 309

References 318

Chapter 9 Trace Formula 321

9.1 Eichler-Selberg Trace Formula on SL2(Z) 321

9.2 Eichler-Selberg Trace Formula on Fuchsian Groups 335

9.3 Trace Formula on the Space Sk+1/2(N,X) 348

References 362

Chapter 10 Integers Represented by Positive Definite Quadratic Forms 363

10.1 Theta Function of a Positive Definite Quadratic Form and Its Values at Cusp Points 363

10.2 The Minimal Integer Represented by a Positive Definite Quadratic Form 376

10.3 The Eligible Numbers of a Positive Definite Ternary Quadratic Form 390

References 428

Index 431