On the Euler characteristic of the discrete spectrum

This paper, which is largely expository in nature, seeks to illustrate some of the advances that have been made on the trace formula in the past 15 years. We review the basic theory of the trace formula, then introduce some ideas of Arthur and Kottwitz that allow one to calculate the Euler characteristic of the S-cohomology of the discrete spectrum. This Euler characteristic is first expressed as a trace of a certain test function on the space of automorphic forms, and then, by the stable trace formula, is converted into a sum of orbital integrals. A result on global measures allows us to calculate these integrals in terms of the values of certain Artin L-functions at negative integers. Our intention is to show how advances in the theory have allowed one to render such calculations completely explicit. As a byproduct of this calculation, we obtain the existence of automorphic representations with certain local behavior at the places in S.