Number Theory Seminar
Julia Gordon
University of British Columbia
Uniform in p estimates for orbital integrals
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.
Tuesday February 10, 2015 at 11:00 AM in SEO 427