Title: Petersson's formula and related topics
Abstract: A crucial ingredient in the ILS paper is an estimate for a sum of Hecke eigenvalues. This is handled using a variant of the trace formula due to Petersson, but there are some obstacles, especially the fact that Petersson's formula includes the contribution of old forms. I will briefly outline the ILS strategy for isolating the newform contribution when N is square-free. I will then propose an alternate strategy coming from representation theory. Along the way, I will give a very basic introduction to the trace formula, and explain how one can use it to derive various identities of interest in classical analytic number theory. These include:
The trace of a Hecke operator;
The Petersson and Kuznetsov formulas;
Formulas for averages/moments of L-functions;
Vertical Sato-Tate laws for Hecke eigenvalues.
If there is time, I will describe work in progress with Charles Li in which we use the simple supercuspidal representations discovered recently by Gross and Reeder to spectrally isolate newforms (holomorphic or Maass) of level N^3, where N is square-free. One consequence is a simple Petersson formula for such newforms.