250090 VO Selcted topics in Number Theory and Algebra (2015S)

Theory of arithmetic groups -- 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 (n x n)--matrices with entries in the ring of integers of an algebraic number field k is an arithmetic group. Such groups arise in a wide variety of contexts: number theory, Fourier analysis, quadratic forms, discrete subgroups of Lie groups, locally symmetric spaces, hyperbolic manifolds, automorphic forms etc. In this course I attempt to develop in an elementary way several of the underlying themes, illustrated by
specific groups to be considered. While no special knowledge of Lie groups or algebraic groups is needed to appreciate these particular examples, I will emphasize methods which carry over to a more general setting.

Topics: special linear groups over k and its arithmetic subgroups, or, more generally, groups originating with orders in division algebras [e.g. quaternion algebras] over k; homogenous spaces and discrete groups; reduction theory and fundamental domains; geometric cycles.