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

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