Elliptic Curves

v1.01; August 21, 1996; first version on the web; 158 pages.

Contents

1. Review of Plane Curves
2. Rational Points on Plane Curves
3. The Group Law on a Cubic Curve
4. Functions on Algebraic Curves and the Riemann-Roch Theorem
5. Definition of an Elliptic Curve
6. Reduction of an Elliptic Curve Modulo p
7. Elliptic Curves over Q_p
8. Torsion Points
9. Neron Models
10. Elliptic Curves over the Complex Numbers
11. The Mordell-Weil Theorem: Statement and Strategy
12. Group Cohomology
13. The Selmer and Tate-Shafarevich Groups
14. The Finiteness of the Selmer Group
15. Heights
16. Completion of the Proof of the Mordell-Weil Theorem, and Further Remarks
17. Geometric Interpretation of the Cohomology Groups; Jacobians
18. The Tate-Shafarevich Group; Failure of the Hasse Principle
19. Elliptic Curves over Finite Fields
20. The Conjecture of Birch and Swinnerton-Dyer
21. Elliptic Curves and Sphere Packings
22. Algorithms for Elliptic Curves
23. The Riemann Surfaces X₀(N)
24. X₀ as an Algebraic Curve over Q
25. Modular Forms
26. Modular Forms and the L-series of Elliptic Curves
27. Statements of the Main Theorems
28. How to get an Elliptic Curve from a Cusp Form
29. Why the L-Series of E Agrees with the L-series of f
30. Wiles's Proof
31. Fermat, At Last

Errata