A History of the Trace Formula

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.