Summary: Contemporary Mathematics
Report on the Trace Formula
This paper is dedicated to Steve Gelbart on the occasion of his sixtieth birthday.
Abstract. We report briefly on the present state of the trace formula and
some of its applications.
This article is a summary of the two hour presentation/discussion on the trace
formula. The proposed topic was very broad. It included a recapitulation of the
trace formula, past and present, as well as an outlook for its future. The article
will treat these matters in only the most concise terms.
I include just two references [A] and [L]. The first of these is a general (and
detailed) introduction to the trace formula and related topics. It contains refer-
ences to just about everything discussed in this article. The second is a review by
Langlands of his ideas for possible application of the trace formula to the general
principle of functoriality. We shall discuss this topic at the end of the article.
1. Invariant trace formula
Let G be a connected reductive algebraic group over a global field F of charac-
teristic 0. Then G(F) embeds as a discrete subgroup of the locally compact adelic
group G(A). We write R for the unitary representation of G(A) on L2