Contents Preface xi Chapter 1. Introduction 1 §1.1. Elliptic curves 1 §1.2. Modular forms 7 §1.3. L-functions 11 §1.4. Exercises 15 Chapter 2. Elliptic curves 17 §2.1. Why elliptic curves? 17 §2.2. Definition 20 §2.3. Integral points 23 §2.4. The group structure on E(Q) 24 §2.5. The torsion subgroup 32 §2.6. Elliptic curves over finite fields 35 §2.7. The rank and the free part of E(Q) 43 §2.8. Linear independence of rational points 46 §2.9. Descent and the weak Mordell-Weil theorem 49 §2.10. Homogeneous spaces 59 §2.11. Selmer and Sha 66 vii
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