Kursinformation

On Monday, 16-18h, 

In this seminar, we will study basic properties of elliptic curves (over complex numbers, over p-adic fields, over finite fields), their arithmetic properties, the Selmer groups and Tate-Shafarevich groups and prove the Mordell-Weil theorem.

Prerequisite: algebra, analysis and topology on advanced undergraduate level.

Tentative program

1. Introduction and distribution of talks;
2. [M] I.1-2, Bezout's theorem and rational points on plane curves;
3. [M] I.3-4, group law on a cubic curve and Riemann-Roch theorem;
4. [M] II.1-2, definition of elliptic curves and Weierstrass equations;
5. [M] II.3-4, elliptic curves over p-adic fields and reduction modulo p;
6. [M] II.5-6, torsion points on elliptic curves and Neron models;
7. [M] III.1-2, doubly periodic functions;
8. [M] III.3, elliptic curves as Riemann surfaces;
9. [M] IV.1-2, group cohomology, Selmer groups and Tate-Shafarevich groups;
10. [M] IV.3, the finiteness of Selmer groups;
11. [M] IV.4, heights;
12. [M] IV.5-6, the rank of Mordell-Weil group;
13. [M] IV.7+I.5, cohomology groups and Jacobians;
14. [M] IV.9, elliptic curves over finite fields;
15. [M] IV.10, the conjecture of Birch and Swinnerton-Dyer

References:

1. [M] Milne, Elliptic curves (here is a link to the pdf);
2. Silverman, The arithmetic of elliptic curves;
3. Silverman and Tate, Rational points on elliptic curves.