Pauli Lectures 2003

The first lecture is suitable for a general public and follows to a large extent the historical development of prime number theory, culminating with the introduction of the zeta function. The lecture will also touch upon topics which have recently reached the newspapers, such as the Agrawal - Kayal - Saxena new deterministic primality test. There will also be a live computer demonstration of factoring large integers and computing very large primes.

The second lecture is meant for a public with first level basic training in mathematics, in particular algebra and geometry and should be at the level of university students. The topics treated include a survey of diophantine equations, congruences and the associated geometry over finite fields, a discussion of the Weil conjectures at an elementary level, and a presentation of recent themes in arithmetic geometry.

The third lecture deals with speculations about the future and presents evidence for what is still a completely hypothetical picture of the theory of general zeta and L-functions. The title "The Rosetta stone of L-functions" is borrowed from Weil's realization that the main three lines of approaching the theory of L-functions really should be three different ways of looking at the same objects, much as the three texts in the Rosetta stone were a single text written in three different languages. The lecture will include a presentation of today's knowledge about L-functions of elliptic curves, including the Birch and Swinnerton-Dyer conjecture and the Taniyama - Shimura -Weil conjecture, recently proved with the introduction, by Andrew Wiles, of new methods.