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Introduction 1. Curves of genus: introduction 2. p-adic numbers 3. The local-global principle for conics 4. Geometry of numbers 5. Local-global principle: conclusion of proof 6. Cubic curves 7. Non-singular cubics: the group law 8. Elliptic curves: canonical form 9. Degenerate laws 10. Reduction 11. The p-adic case 12. Global torsion 13. Finite basis theorem: strategy and comments 14. A 2-isogeny 15. The weak finite basis theorem 16. Remedial mathematics: resultants 17. Heights: finite basis theorem 18. Local-global for genus principle 19. Elements of Galois cohomology 20. Construction of the jacobian 21. Some abstract nonsense 22. Principle homogeneous spaces and Galois cohomology 23. The Tate-Shafarevich group 24. The endomorphism ring 25. Points over finite fields 26. Factorizing using elliptic curves Formulary Further reading Index.