The group of rational points of an algebraic group.

This colloquium is a part of the Yuktibhasa seminar series organized by the Department of Mathematics at IIT Palakkad. 

Abstract: Roughly speaking, a linear algebraic group is a group of matrices that can be defined by a bunch of simultaneous polynomial equations. Standard examples are SL(n), Sp(2n), O(n). We say that such a group is defined over a field k if the defining polynomials live over k. If G is such a group and L is any field extension of k, then the simultaneous solutions of the equations of G over L, denoted by G(L), is a group in the usual sense. In this talk, we will discuss some algebraic, geometric and arithmetic aspects of the structure of the group G(k), where G is a (linear) algebraic group defined over a field k, while introducing some important problems in the subject. The talk is intended to be expository.