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 MordellWeil 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 SwinnertonDyer Conjecture.
General informations
The talks should take between 80  90 minutes. Two students share a talk. A script in LaTex is required.
The seminar takes place Tuesdays from 14:15  16:00 in CHN D 42, from 03.10. to 19.12. (12 talks).
You can find some useful information here:
Here a two LaTex templates that you can use for your script and your talk:
Here you can find the handouts of the talks. See also the website of the Elliptic Curves Seminar 2020 for the old handouts.

Cubic curves Handout

Points of finite order Handout

Heights Handout

Mordell's Theorem Handout

Cubic curves over finite fields Handout

Integer points on cubic curves Handout

Elliptic functions Handout

Complex elliptic curves Handout

Complex multiplication Handout

Modular forms Handout

Fermat's Last Theorem Handout

The Birch and SwinnertonDyer Conjecture Handout