Seminar Elliptic Curves

The seminar gives an introduction to the basic theory of elliptic curves.

We first study the basic properties of elliptic curves, such as the group law. Then we will proceed to study elliptic curves over the rationals and the question whether they have rational or integral points. One of the main goals of the seminar is the proof of the Mordell-Weil theorem, which states that the set of rational points of a rational elliptic curve is a finitely generated abelian group. Using the theory of elliptic functions we will show that an elliptic curve over the complex numbers can be viewed as a torus. As an outlook, we will sketch several deep results and conjectures about elliptic curves, such as Wiles' Modularity Theorem, which played an important role in the proof of Fermat's Last Theorem, and such as the Birch and Swinnerton-Dyer Conjecture.

General informations

The talks should take between 90 - 100 minutes. Two students share a talk. A script in LaTex is required.

The seminar takes place Tuesdays from 10:15 - 12:00 in ML H 41.1, from 15.09. to 15.12.

You can find some useful information here:

List of topics

Overview talk

Here a two LaTex templates that you can use for your script and your talk:

Template for your script

Template for your talk

Talks

1. Cubic curves Slides Handout

2. Points of finite order Slides Handout

3. Heights Slides Handout

4. Mordell's Theorem Slides Handout

5. Cubic curves over finite fields Handout

6. Integral points on elliptic curves Handout

7. Elliptic functions Handout

8. Complex elliptic curves Handout

9. Complex multiplication Handout

10. Modular forms Handout

11. Fermat's Last Theorem Handout

12. The Birch and Swinnerton-Dyer Conjecture Handout