Ian Petrow: Counting characters on algebraic tori according to their Langlands L-functions

Abstract:

Given a connected reductive group \(G\) over a global field, Langlands introduced the automorphic L-function \(L(s, \pi, r)\) of a cuspidal automorphic representation \(\pi\) of \(G\) and a complex representation \(r\) of the L-group of \(G\). While in general very little is known about Langlands L-functions, if \(G = T\) is a torus the properties of these L-functions can be obtained from class field theory and one can attempt to study analytic problems pertaining to them. In this talk I will describe some analytic results on automorphic characters of tori with respect to the analytic conductor of \(L(s, \pi, r)\), attempting to focus on the interplay of analytic and algebraic ideas that arise in the proofs.