Seminar on the foundations of the theory of Elliptic Curves.
Learning objective
The participants learn the basics about elliptic curves, which will enable them to write a Bachelor's or Master's thesis in number theory. In addition to a talk, the writing of a short manuscript in latex will be required.
Content
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 it has rational or integral points. One of the main goal 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.
Literature
Knapp: Elliptic Curves Koecher, Krieg: Elliptische Funktionen und Modulformen Milne: Elliptic Curves Silverman: The Arithmetic of Elliptic Curves Silverman, Tate: Rational Points on Elliptic Curves
Prerequisites / Notice
Basic knowledge of Algebra and Complex Analysis will be helpful.
Performance assessment
Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course