viii Contents
§2.12. Exercises 69
Chapter 3. Modular curves 77
§3.1. Elliptic curves over C 77
§3.2. Functions on lattices and elliptic functions 82
§3.3. Elliptic curves and the upper half-plane 84
§3.4. The modular curve X(1) 87
§3.5. Congruence subgroups 90
§3.6. Modular curves 91
§3.7. Exercises 94
Chapter 4. Modular forms 99
§4.1. Modular forms for the modular group 99
§4.2. Modular forms for congruence subgroups 105
§4.3. The Petersson inner product 110
§4.4. Hecke operators acting on cusp forms 111
§4.5. Exercises 118
Chapter 5. L-functions 123
§5.1. The L-function of an elliptic curve 123
§5.2. The Birch and Swinnerton-Dyer conjecture 127
§5.3. The L-function of a modular (cusp) form 135
§5.4. The Taniyama-Shimura-Weil conjecture 137
§5.5. Fermat’s last theorem 140
§5.6. Looking back and looking forward 142
§5.7. Exercises 143
Appendix A. PARI/GP and Sage 147
§A.1. Elliptic curves 147
§A.2. Modular forms 154
§A.3. L-functions 156
§A.4. Other Sage commands 158
Appendix B. Complex analysis 159
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