## A History of the Trace Formula

- Date: 11/06/2006

James Arthur (University of Toronto)

University of Toronto

The trace formula is a far reaching generalization of the Poisson

summation formula. It relates spectral data of deep arithmetic

significance to explicit but complicated geometric data. With its

applications to the Langlands programme, some already realized and

others still far away, the trace formula represents a mathematical

equation of great power.

In trying to give some sense of its history, we will begin with

Selberg's original discovery of a formula that gave remarkable

relations between the spectral and geometric properties of Riemann

surfaces. We shall then describe Langlands' ideas for using this

formula to establish reciprocity laws between different kinds of

arithmetic quantities. Finally, we will say something about the present

state trace formula, as it applies to spaces and groups of arbitrary

dimension.

10th Anniversary Speaker Series 2006