自守形式与L-函数简明六章 英文PDF电子书下载

Chapter 1 Modular forms and the Shimura-Taniyama Conjecture 1

1.1 Elliptic functions 1

1.2 Modular forms 3

1.3 Examples 5

1.4 Hecke operators and eigenforms 8

1.5 L-functions 10

1.6 Modular forms of higher level 12

1.7 Elliptic curves 13

1.8 Conjectures,and the theorem of Wiles,et al 15

References 16

Chapter 2 Periods of automorphic forms 17

2.1 Automorphic forms 17

2.2 Periods arising from Dirichlet series 19

2.3 Periods arising from elliptic curves 20

2.4 Periods arising from cohomology theory 22

2.5 Recapitulation 23

2.6 Hilbert modular forms 23

2.7 Automorphic forms with respect to a quaternion algebra 25

2.8 Adelic automorphic forms 27

2.9 Hecke operators 29

2.10 The standard L-function for Hilbert modular forms and the V periods 32

2.11 The inner product and the Q periods 35

2.12 The Eichler-Shimura Isomorphism and the P periods 36

References 42

Chapter 3 Lifting of automorphic forms 43

3.1 Dirichlet series 43

3.2 Hecke characters with respect to a real quadratic number field 44

3.3 L-series and the base change lift for characters 46

3.4 Hilbert modular forms over a real quadratic field and the Doi-Naganuma lift 48

3.5 Applications of base change 51

3.6 Shimura's period conjectures 52

3.7 Yoshida's work 54

References 55

Chapter 4 Zeros of L-functions 57

4.1 Trivial zeros 57

4.2 The Grand Riemann Hypothesis 60

4.3 Zero-free regions 62

4.4 Siegel zeros 67

4.5 Correlations of zeros 71

4.6 Random matrix theory 73

References 75

Chapter 5 Special values of L-functions 77

5.1 The Birch-Swinnerton-Dyer conjecture 77

5.2 The Gross-Zagier formula 80

5.3 The class number problem 82

5.4 The Lindel？f conjecture and subconvexity bounds 85

5.5 Discrete moment conjecture of L-functions 89

5.6 Integral moment conjectures of L-functions 94

References 97

Chapter 6 Theta lifts and periods with respect to a quadratic extension 99

6.1 Setting 99

6.2 Automorphic forms with respect to BE 100

6.3 Theta function 103

6.4 Transformation formulas 106

6.5 Theta correspondence 107

6.6 Reduction of the integral 107

6.7 Two technical lemmata 109

6.8 Main theorems on theta correspondence 110

6.9 Adelic forms on ？ 110

6.10 Embeddings and inner product 111

6.11 Theta correspondence of adelic forms 113

6.12 Hecke Operators on GL2(F) 115

6.13 The action of Hecke operators on I(z;C;r;h) 117

6.14 Hecke operators on adelic forms 118

6.15 Relation between actions of Hecke operators 120

6.16 Further Reading 121

References 122