Special sessions

Mini-course on Elliptic Curves
A mini course by Herwig Hauser, Universities of Innsbruck and Vienna

This series of lectures is designed for Master and PhD students, and also researchers. It aims at highlighting five central features of the theory of elliptic curves. We start with the classical concept of ellipses and some of their (quite surprising) properties, e.g., the behaviour of billiards played on them, or mechanical devices allowing to draw them. Also, osculating circles and the computation of the length of an arc on the ellipse will be discussed.

The second lecture turns to complex analysis and doubly periodic complex meromorphic functions. These are the complex analogon of real trigonometric functions, and deploy a beautiful internal structure culminating in the construction of the Weierstrass p-function p(z).

We then proceed to aspects from algebraic geometry, defining now explicitly elliptic curves as the zero-set E of a bivariate cubic polynomial in the projective plane. The pair consisting of p(z) and its derivative p′(z) will be shown to establish a diffeomorphism with the real torus. Further, a simple version of Bézout’s theorem allows us to construct a magic addition on E, which thus becomes an abelian group.

This leads us to lecture four where group theoretic properties of E are discussed. Most prominently, we look at E defined over the integers or the rational numbers and describe the theorems of Nagel-Lutz and Mordell about the finite generation of E(Q).

The last lecture picks up number theoretic questions, for instance, the existence of rational points on E(Q), the theory of congruent numbers, and the notorious conjectures about the rank of an elliptic curve.

No special prerequisites will be required, but it will be good to refresh forehand the theory of holomorphic and meromorphic functions (in one variable), the basics of algebraic geometry (tangents, intersection numbers, Bézout’s theorem), and the structure theorem about finitely generated abelian groups.

We will provide reading material and accompany the course by many exercises you may be tempted to solve on your own.

Further information at www.hh.hauser.cc.

Permanent link to this information: https://geolis.math.tecnico.ulisboa.pt/lecture_series?sgid=132