# Lecture on Elliptic Functions

Elliptic functions are doubly periodic meromorphic functions which historically emerged from the study of elliptic integrals. The most basic and at the same time most important example is the Weierstrass p-function, which can in fact be used to describe every elliptic function. Moreover, using the Weierstrass p-function we will classify complex elliptic curves in terms of lattices. Looking at the Laurent expansion of the p-function, one is naturally led to Eisenstein series, the discriminant function, and the j-invariant of elliptic curves.

We will also study elliptic curves over the rational numbers. They are plane curves defined by the zero sets of cubic polynomials. A key fact is that they also have the structure of abelian groups, that is, one can add points on an elliptic curve in a natural way. The main result that we will prove in the lecture is the Theorem of Mordell-Weil, which states that the group of rational points of every rational elliptic curve is finitely generated. As an outlook at the end of the lecture we will discuss the famous Birch and Swinnerton-Dyer conjecture.

## General informations

The lecture and exercise classes take place at the following times:

Lecture on **Monday 16-18 in HG F 26.5** (starting on 21.02., until 30.05.)

Exercise on **Thursday 12-14 in HG G 26.5** (biweekly, 03.03.; 17.03.; 31.03.; 14.04.; 05.05.; 19.05.)

The script for the lecture (which will be written during the semester and will be updated perpetually) and the recordings can be found on polybox.