Contents

Preface vii

Chapter 1. Basic facts 1

1.1. Congruence subgroups 1

1.2. Integer weight modular forms 3

1.3. Half-integral weight modular forms 10

1.4. Dedekind’s eta-function 16

Chapter 2. Integer weight modular forms 21

2.1. Hecke operators 21

2.2. Twists of modular forms 22

2.3. The Theta operator 23

2.4. Further operators 27

2.5. Newforms 28

2.6. Divisors of modular forms on SL2(Z) 31

2.7. Modular forms modulo 2 on SL2(Z) 32

2.8. Modular forms modulo p on SL2(Z) for p ≥ 5 35

2.9. Sturm’s Theorem 40

2.10. Theory of Serre and Swinnerton-Dyer 41

2.11. U(p)-congruences for weakly holomorphic modular forms 43

Chapter 3. Half-integral weight modular forms 49

3.1. Hecke operators 49

3.2. Further operators 50

3.3. Shimura’s correspondence 52

3.4. Kohnen’s theory 54

3.5. Congruences for coeﬃcients of half-integral weight forms 56

3.6. Nonvanishing of Fourier coeﬃcients 61

3.7. Open problems 68

Chapter 4. Product expansions of modular forms on SL2(Z) 69

4.1. Introduction 69

4.2. Borcherds’ products 69

4.3. A sequence of weight 3/2 modular forms 72

4.4. p-adic properties of infinite product exponents 74

4.5. Polynomial recursions and infinite products of forms on SL2(Z) 80

4.6. Open problems 83

Chapter 5. Partitions 85

5.1. Introduction 85

5.2. Congruences for p(n) 89

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