Cahit Arf Lecture 2002 by Don B. Zagier

Don B. Zagier

Taylor Coefficients of Modular Forms

Date: November 11, 2002
at 15:00

Cahit Arf Auditorium
Department of Mathematics
Middle East Technical University
Ankara, Turkey

Supported by


Modular forms are a special class of holomorphic functions with an infinite group of symmetries and with many important arithmetic properties. They play a crucial role in much of modern number Theory, the most spectacular example being in the proof of Fermat's Last Theorem by Andrew Wiles a few years ago. Usually one derives arithmetic information from modular forms by looking at their Fourier expansions or "Taylor coefficients at infinity", but it turns out that their Taylor expansions at suitably chosen finite points also have beautiful arithmetic properties and many applications (in particular to Diophantine equations such as the question: "which prime numbers can be written as the sum of two perfect cubes?"). Moreover, these Taylor coefficients, unlike the more familiar Fourier coefficients, can be computed by means of a quite elementary algorithm. The lecture will treat several aspects of the theory, examples, and applications.