Julia Gordon : Uniform in p estimates for orbital integrals

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

Feb. 10, 2015, 11 a.m.

In

SEO 427

Abstract

It is a well-known theorem of Harish-Chandra that the orbital integrals, normalized by the square root of the discriminant, are bounded (for a fixed test function). However, it is not easy to see how this bound behaves if we let the $p$-adic field vary (for example, if the group $G$ is defined over a number field $F$, and we consider the family of groups $G_v=G(F_v)$, as $v$ runs over the set of finite places of $F$), and how it varies for a family of test functions. Using a method based on model theory and motivic integration, we prove that for a fixed test function, the bound on orbital integrals can be taken to be a fixed power (depending on $G$) of the cardinality of the residue field, and also obtain a uniform bound for the family of generators of the spherical Hecke algebra playing the role of the test functions. This statement has an application to the recent work of S.-W. Shin and N. Templier on counting zeroes of L-functions. This project is joint work with R. Cluckers and I. Halupczok.