 
Summary: A (very brief) History of the Trace Formula
James Arthur
This note is a short summary of a lecture in the series celebrating
the tenth anniversary of PIMS. The lecture itself was an attempt to
introduce the trace formula through its historical origins. I thank Bill
Casselman for suggesting the topic. I would also like to thank Peter
Sarnak for sharing his historical insights with me. I hope I have not
distorted them too grievously.
As it is presently understood, the trace formula is a general identity
(GTF) {geometric terms} = {spectral terms}.
The spectral terms contain arithmetic information of a fundamental
nature. However, they are highly inaccessible, "spectral" actually, in
the nonmathematical meaning of the word. The geometric terms are
quite explicit, but they have the drawback of being very complicated.
There are simple analogues of the trace formula, "toy models" one
could say, which are familiar to all. For example, suppose that A =
(aij) is a complex (n × n)matrix, with diagonal entries {ui} = {aii}
and eigenvalues {j}. By evaluating its trace in two different ways, we
obtain an identity
n
