Date & Time: Tuesday, August 13, 2013, 14:00-15:30.
Venue: Ramanujan Hall
Title: Zeta Functions of Simple Algebras - Tate's Thesis
Speaker: Ravi Raghunathan, IIT Bombay
Abstract: Zeta functions of simple algebras are generalistions of the Riemann zeta-function and the Dirichlet $L$-series. We propose to have a series of lectures on the subject starting with the case of fields (Tate's pioneering thesis) and going on to the zeta functions of division algebras (in Weil's Basic Number Theory, for example), before giving the more general constructions of Godement-Jacquet and the doubling method of Rallis-Piatetski-Shapiro for general classical reductive groups. The main point is this. While the L-functions are not hard to define, their properties are difficult to establish. The template for progress was laid down by Tate whose reformulation of the classical theory of Hecke in terms of adeles allowed the systematic use of Harmonic analysis (the Fourier transform, the Poisson summaton formula etc.) to prove theorems in number theory.
The first lecture will be relatively elementary and will not presuppose any knowledge of number theory on the part of the audience beyond the definition of a $p$-adic field.