2 The Trace Formula and Hecke Operators

This lecture is intended as a general introduction to the trace formula. We shall describe a formula that is a natural generalization of the Selberg trace formula for compact quotient. Selberg also established trace formulas for noncompact quotients of rank 1, and our formula can be regarded as an analogue for general rank of these. As an application, we shall look at the "finite case" of the trace formula. We shall describe a finite closed formula for the traces of Hecke operators on certain eingenspaces. A short introduction of this nature will by necessity be rather superficial. The details of the trace formula are in [l(e)] (and the references there), while the formula for the traces ofHecke operators is proved in [l(f)]. There are also other survey articles [1(c)], [l(d)], [5], and [l(g)], where some of the topics in this paper are discussed in more detail and others are treated from a different point of view.