# Integral zeta values and the number of automorphic representations

*Algebra/Geometry/Topology Seminar*

*by Benedict Gross*

*Institution:*Harvard

*Date: Tue 10th February 2009*

*Time: 2:30 PM*

*Location: JH Michell Theatre*

*Abstract*: Let \zeta^*(s):= (1 2^(1 s))\zeta(s). Euler proved that the values of \zeta^*(s) at negative integers are elements of the ring Z[1/2]. Cassou-Nogues and Deligne/Ribet generalized this to an integrality result for the values of arbitrary partial zeta functions at negative integers. I will review their results, and show how these special values can be used to compute the number of irreducible automorphic representations of G with prescribed local behavior, where G is a simple group over a global field k. Via the global Langlands correspondence for k = F(t), I will compare this result with work of Katz and Deligne on Kloosterman sheaves. This is joint work with Mark Reeder.

*For More Information:* S.Tillmann@ms.unimelb.edu.au