Elliptic Functions

Abstract

Lecture about the basic theory of elliptic functions and elliptic curves

Objective

The participants will learn about the classical theory of elliptic functions and elliptic curves over the rationals and the complex numbers. The students will learn how to work with the basic objects from the theory, such as the Weierstrass p-function or elliptic curves, and how to prove their basic properties. The results presented in the lecture will also be useful for further studies in number theory, e.g. in a lecture on modular forms.

Content

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.

Resources