Past lectures

Abstract

An arithmetic group, roughly speaking, is a group of integral matrices defined by polynomial equations.For example, a subgroup of finite index in the special linear group of (nxn) - matrices with entries in the ring of integers of a given algebraic number field is an arithmetic group. The theory of arithmetic groups has its origins in number theory,especially the theory of quadratic forms. Reduction theory is the most powerful technique to derive some group theoretical properties shared by all arithmetic groups.

Geometry, number theory and group theory meet each other in the theory of arithmetic groups, and, in turn, there are close connections with the theory of automorphic forms.

In these lectures I shall try to give an impression of some of the geometric aspects of the theory of arithmetic groups from a reasonably modern standpoint. The following topics will be dealt with:

1. homogeneous spaces and discrete groups

2. first examples of arithmetic groups: units of orders in division algebras, congruence subgroups in classical groups

3. arithmetic quotients (as locally symmetric spaces): (non-)compactness criterion and examples

4. reduction theory, fundamental domains

5. volume of the fundamental domain of the unimodular group, values of zeta functions of integers (Minkowski, Siegel, Weil)

6. cycles on arithmetic quotients

7. geometric construction of homology for arithmetic groups

A general introduction into the subject will be given in the first lecture.