vi CONTENTS
5.3. Distribution of p(n) modulo M 101
5.4. Open problems 106
Chapter 6. Weierstrass points on modular curves 109
6.1. Introduction 109
6.2. Weierstrass points and supersingular points 110
6.3. The X0(p) cases 112
6.4. Open problems 118
Chapter 7. Traces of singular moduli and class equations 121
7.1. Introduction 121
7.2. A result of Gross and Zagier 121
7.3. Formulas for traces and Hecke traces of singular moduli 123
7.4. p-adic properties of traces of singular moduli 125
7.5. U(p)-congruences for class equations 130
7.6. Open problems 132
Chapter 8. Class numbers of quadratic fields 133
8.1. Introduction 133
8.2. Class numbers as coefficients of modular forms 134
8.3. Indivisibility of class numbers of imaginary quadratic fields 136
8.4. Indivisibility of class numbers of real quadratic fields 138
8.5. Divisibility of class numbers 143
8.6. Open problems 146
Chapter 9. Central values of modular L-functions and applications 149
9.1. Introduction 149
9.2. Central critical values of modular L-functions 149
9.3. Coefficients of half-integral weight modular forms 153
9.4. Nonvanishing results 155
9.5. Elements in Selmer and Shafarevich-Tate groups 157
9.6. Open problems 161
Chapter 10. Basic hypergeometric generating functions for L-values 165
10.1. Introduction 165
10.2. Basic hypergeometric series 165
10.3. Summing the tails of certain infinite products 168
10.4. Generating functions for L-values 175
10.5. Open problems 182
Chapter 11. Gaussian hypergeometric functions 183
11.1. Definitions and notation 183
11.2. Arithmetic of certain special values 186
11.3. Traces of Hecke operators 196
11.4. Beukers’ supercongruence for Ap´ ery numbers 198
11.5. Further congruences 203
11.6. Open problems 204
Bibliography 207
Index 215
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