Collaborative Number Theory Seminar at the CUNY Graduate CenterCo-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin,
Maria Sabitova,
Lucien Szpiro. Spring 2015 Schedule:
May 15:
Julia Gordon (University of British Columbia) Title: 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
low-lying zeroes of L-functions. This project is
joint work with R. Cluckers and I. Halupczok.
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