Brian Conrad

Abstract:   The Tamagawa number of a linear algebraic group (over a number field or over the function field of a curve over a finite field) is a certain canonical volume that arises in settings as varied as the arithmetic of quadratic forms and counting connected components of certain moduli spaces of bundles over curves over finite fields. The finiteness of such volumes for all reductive groups was proved by Borel and Harder. The finiteness in general can then be easily deduced in characteristic zero, but the case of positive characteristic needs completely different ideas and was settled only very recently. We discuss some of the highlights of this story.