Table of Contents

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.

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A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.

Reviews

'… an excellent introduction … written with humour.' Monatshefte für Mathematik