Elliptic curves course

This is the schedule for the "Elliptic curves" course held at Institute of Science and Technology Austria in the spring of 2024. Every class will be held in Central Bldg / O1 / Mondi 3 (I01.O1.010).

1. Definition of elliptic curves, Weierstrass equation, group law, isogenies, heights. Ref: Chap III 1-4, Chap VIII 5-6, AEC. Date: 23/04/24. Time: 13:15/14:30.

2. Weak Mordell-Weil theorem, Kummer pairing, 2-descent, Mordell-Weil theorem. Ref: Chap VIII 1-4, AEC. Date: 24/04/24. Time: 10:15/ 11:30.

3. Tate module, Elliptic curves over finite fields, Weil conjectures. Ref: Chap V 1-2, AEC. Date: 07/05/24. Time: 13:15/14:30.

4. Elliptic curves over the field of complex numbers. Ref: Chap VI, AEC. Class held by Tim Browning. Date: 08/05/24. Time: 10:15/ 11:30.

5. Elliptic curves over local fields. Ref: Chap VII, AEC. Class held by Tim Browning. Date: 14/05/24. Time: 13:15/14:30.

6. Tate-Shafarevic group, Selmer group. Ref: Chap X 4, AEC. Class held by Tim Browning. Date: 15/05/24. Time: 10:15/ 11:30.

7. Birch and Swinnerton-Dyer conjecture. Ref: Chap C.16, AEC. Class held by Tim Browning. Date: 21/05/24. Time: 13:15/14:30.

8. Introduction to diophantine approximation and Roth theorem. Ref: Chap IX 1&8, AEC. Date: 22/05/24. Time: 10:15/ 11:30.

9. Siegel's theorem. Ref: Chap IX 2-3, AEC. Date: 28/05/24. Time: 13:15/14:30.

10. S-unit equation and effective methods for integral points on elliptic curves. Ref: Chap IX 4-5, AEC. Date: 29/05/24. Time: 10:15/ 11:30.

11. Elliptic surfaces. Ref: Chap III, ATAEC. Date: 04/06/24. Time: 13:15/14:30.

12. Modell-Weil theorem for elliptic curves over function fields. Ref: Chap III 6, ATAEC. Date: 05/06/24. Time: 10:15/ 11:30.

Bibliography: 

• AEC: The Arithmetic of Elliptic Curves, Joseph H. Silverman.

• ATAEC: Advanced topic in the Arithmetic of Elliptic Curves, Joseph H. Silverman

Exercise sheets:

Exercise sheet 1