Uniform estimates and asymptotics for p-adic orbital integrals and characters

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This proposal is centered around the main theme of fine understanding of the measures that arise in the various contexts associated with the local Langlands correspondence. Our objects of study are algebraic groups and Lie algebras over non-Archimedean local fields, and the distributions that arise in their harmonic analysis. Harmonic analysis usually starts with a Haar measure on a given (locally compact) group G, which is unique up to a constant multiple. A lot of my work stems from two very basic considerations: first, the nature of the Haar measure itself, and second, the interplay between different normalizations of Haar measures that have some arithmetic significance. The present proposal is built around two projects united by these common themes.***I. MOTIVIC HARMONIC ANALYSIS.  This is a continuation of the long-term project, initiated by T. Hales in 1999, of making harmonic analysis on p-adic groups `independent of p' by means of  replacing the usual integration with respect to Haar measure by `motivic integration'. This allows one to treat non-Archimedean local  fields and functions on them  uniformly and field-independently. Initially, the main application of this approach was transfer of  integral identities, such as the Fundamental Lemma,  between local fields of characteristic zero and of sufficiently large positive characteristic (as in [11]).  In the recent work with R. Cluckers and I. Halupczok, we proved new transfer principles for analytic properties, e.g., integrability and boundedness. This method also yields uniform in p estimates for orbital integrals, which turned out to be very useful in Number theory (cf. [34]). Proposed research aims to expand these new  estimates to a larger class of functions and distributions arising in harmonic analysis on p-adic groups, including Harish-Chandra characters, leading to new applications. ***II. SIZES OF ISOGENY CLASSES. In 2003, E.-U. Gekeler studied the question "how likely is a given elliptic curve over a prime finite field to have a given number of rational points?". He gave an explicit answer based on a probabilistic heuristic that was too strong to be literally true, which appeared somewhat mysterious. In this project with J. Achter, we aim to provide an explanation for Gekeler's formula by making an explicit and very natural connection with Langlands-Kottwitz formula that expresses the size of an isogeny class of principally polarized abelian varieties in terms of an adelic orbital integral. Then we plan to extend Gekeler's computations from elliptic curves to abelian varieties. The connection of probability distributions analogous to the ones studied by Gekeler with orbital integrals allows one to ask a number of questions about their asymptotic behaviour. We hope to be able to formulate and answer some of these asymptotic questions using uniform estimates for orbital integrals from Part I of the proposal.********