Math 296

Representations of Finite Groups and Applications to

Number Theory

**Description**: The theory of representations of finite groups is basic to many areas of mathematics, especially number theory. Unfortunately, in number theory courses we usually (and rather disastrously for the student) rush directly to the important applications of this theory, such as Artin L-functions, omitting a detailed treatment of the representation theory itself.

In this 9 week course we will develop the theory of representations of finite groups with a view to setting up the theory of Artin L-functions. In particular, we will spend about 6 weeks developing the basics of the theory, and about 3 weeks establishing in detail the "Artin Formalism" and some of its uses. The first part of the course has no real prerequisite, but the second part will assume the basics of a 1 quarter course in algebraic number theory such as Math 205.

To get an idea of the variety of topics-from classical number theory to mathematical physics- in which Artin L-functions today play a role, just Google search "Artin L-functions"!

**Texts**:

1. Linear Representations of Finite Groups (Graduate Texts in

Mathematics: Vol 42) by Jean Pierre Serre.

2. Chapter on Representations of Finite Groups in Algebra by Serge Lang

3. H. Heilbronn, Zeta-functions and L-functions (pp. 204--230) in Algebraic number theory. Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1--17, 1965. Edited by J. W.S. Cassels and A. Fröhlich. Reprint of the 1967 original. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. xviii+366 pp. \$27.00. ISBN 0-12-163251-2.

4. E. de Shalit's lecture notes at http://www.ma.huji.ac.il/~deshalit/frame-lecture.html

5. Chapter 12 of Serge Lang's Algebraic Number Theory.

**Time**: Tuesdays 4:30-6

**Room**: MS 5217.

**Organizational Meeting**: Tuesday, January 13 in MS 5217.