**Date & Time:** Tuesday, August 6, 2013, 15:30-17:30

**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.