Measures, orbital integrals, and counting points.

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Broadly speaking, this proposal aims to contribute to our understanding of the Langlands programme, and more specifically, the geometric side of Arthur's Trace Formula,  using a uniform geometric approach to measures that arise on p-adic manifolds.  Until recently, my research has been largely motivated by a long-term project of applications of motivic integration to the representation theory of p--adic groups. Motivic integration is a theory based initially on algebraic geometry and, more recently, on formal logic and model theory, that allows one to do integration on p--adic fields (and more generally, on the set of points of a variety over a p--adic field) in a uniform, p--independent, way. At its source is the observation that integration over the set of points of a variety over a local field can be reduced to point--counting over the residue field and summation of geometric series with base 1/p. The same observation powers another classical idea in number theory - computation of local densities, as in the Minkowski--Siegel mass formula. In a recently completed work with Jeff Achter, Ali Altug and Luis Garcia, we have used this observation to re-express the formula by Langlands and Kottwitz for the cardinality of the isogeny class of a principally polarized ordinary abelian variety over a finite field in terms of a product of local densities, Siegel--style (the Langlands--Kottwitz formula expresses this cardinality as an adelic orbital integral). Surprisingly, some of the technical steps we had to implement (e.g., careful tracking of the normalization of measures on orbits of semisimple elements in the symplectic group) turned out to be very similar to the first steps one has to take to follow the Langlands--Frenkel--Ngo approach to the `Beyond endoscopy' proposal of Langlands. My current proposal has three complementary directions that stem from these ideas: 1. Resolving  some persistent open questions remaining  in the program of making harmonic analysis on p-adic groups `motivic', that was started by my Ph.D. advisor, T.C. Hales, in 1999. 2. Further advances in the project with J. Achter on Siegel--style formulas and relationships between orbital integrals and certain local densities, and 3. Trying to understand the so--called basic functions and their orbital integrals using the ideas of Igusa. This direction is still in a speculative stage, and is largely informed by conversations with W. Casselman.