Date & Time: Tuesday, 4 December 2012, 4 p.m.
Venue: Ramanujan Hall
Title: Automorphy and the Sato-Tate conjecture
Speaker: Prof. M. Ram Murthy, Queens University, Canada
Abstract:Let tau(n) be the Ramanujan tau-function. The Sato-Tate conjecture predicts that the numbers tau(p)/2p^{11/2} is uniformly distributed in [-1.1] with respect to the "semi-circular" measure of Wigner. In the 1970's, Langlands conjectured that if the automorphy of symmetric power L-functions attached to tau is known, then this conjecture would follow. However, this conjecture was recently proved by Barnet-Lamb, Geraghty, Harris, and Taylor without establishing the conjectured automorphy.
We will show that a special case of the Langlands functoriality conjecture leads to the automorphy, which in turn, provides excellent error terms for the Sato-Tate distribution.
It is interesting that our proof makes essential use of an identity of Ramanujan. This is joint work with V. Kumar Murty