Integral zeta values and the number of automorphic representations

Mardi 30 septembre 2008 16:00-17:00 - Gross Benedict - Harvard

Résumé : Let $zeta^*(s)=(1-2^1-s)zeta(s) (=1-1/2^s+1/3^s-1/4^s+... for Re(s)>1)$.
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.

Lieu : bât. 425 - 113-115

Integral zeta values and the number of automorphic representations  Version PDF