Lectures on elliptic curves

Includes index

Includes bibliography and (p. 135) references.

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

The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.