Journal d’Analyse Mathématique
, Volume 50, Issue 1, pp 189200
First online:
A simple trace formula
 Yuval Z. FlickerAffiliated withDepartment of Mathematics, Harvard University, Science Center
 , David A. KazhdanAffiliated withDepartment of Mathematics, Harvard University, Science Center
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The Selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In principal it is a simple and natural formula, generalizing the Poisson summation formula, relating traces of convolution operators with orbital integrals. This paper is motivated by the belief that such a fundamental and natural relation should admit asimple and short proof. This is accomplished here for test functions with a single supercuspcomponent, and another component which is spherical and “sufficientlyadmissible” with respect to the other components. The resulting trace formula is then use to sharpen and extend the metaplectic correspondence, and the simple algebras correspondence, of automorphic representations, to the context of automorphic forms with asingle supercuspidal component, over any global field. It will be interesting to extend these theorems to the context of all automorphic forms by means of a simple proof. Previously a simple form of the trace formula was known for test functions with two supercusp components; this was used to establish these correspondences for automorphic forms with two supercuspidal components. The notion of “sufficientlyadmissible” spherical functions has its origins in Drinfeld's study of the reciprocity law for GL(2) over a function field, and our form of the trace formula is analogous to Deligne's conjecture on the fixed point formula in étale cohomology, for a correspondence which is multiplied by by a sufficiently high power of the Frobenius, on a separated scheme of finite type over a finite field. Our trace formula can be used (see [FK′]) to prove the Ramanujan conjecture for automorphic forms with a supercuspidal component on GL(n) over a function field, and to reduce the reciprocity law for such forms to Deligne's conjecture. Similar techniques are used in ['t'F] to establish base change for GL (n) in the context of automorphic forms with a single supercuspidal component. They can be used to give short and simple proofs of rank one lifting theorems forarbitrary automorphic forms; see [″F] for base change for GL(2), [F′] for base change forU(3), and [′F′] for the symmetric square lifting from SL(2) to PGL(3).
 Title
 A simple trace formula
 Journal

Journal d’Analyse Mathématique
Volume 50, Issue 1 , pp 189200
 Cover Date
 198812
 DOI
 10.1007/BF02796122
 Print ISSN
 00217670
 Online ISSN
 15658538
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 Yuval Z. Flicker ^{(1)}
 David A. Kazhdan ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Harvard University, Science Center, One Oxford Street, 02138, Cambridge, MA, USA