I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to mention the classical Ramanujan-Petersson conjecture. From my point of view what's really interesting is its generalization to automorphic forms, which, as I understand it, says that the local components of a globally generic cuspidal automorphic representation are tempered. But for a talk it's nice to connect something abstract to something classical.
My problem is that I haven't been able to find a clear statement of the Ramanujan-Petersson conjectures anywhere. Here are some of the places I've looked.
- The Wikipedia article clearly states Ramanujan's conjecture for the modular discriminant function, but doesn't state the generalizations clearly.
- The 1930 paper of Petersson states his generalization of the conjecture, but I don't know German well enough to muddle my way through it.
- The 1965 Boulder paper of Satake, where he explains how to reformulate the conjecture using automorphic forms for GL$_2$, probably has the statement somewhere, but I'm not comfortable enough with the dictionary between modular forms and automorphic forms to extract it.
- A paper of Blomer and Brumley states the conjecture for Maass forms in Section 3.1. This makes me wonder if perhaps there are separate statements for Maass forms and holomorphic forms.
So I was hoping that the conjecture was written down clearly and simply somewhere, and that perhaps someone on this site could point me to a reference. But barring that, it would be great if someone could just state the conjecture -- I'm hoping that it's ultimately a simple estimate on the Hecke eigenvalues, which is what I've been led to believe.
I understand that the conjecture is known for holomorphic forms thanks to the work of Deligne, and that the conjecture for Maass forms is still open. Are these the only essential cases?
(Part of my problem is that I haven't thought as much as I should about classical modular forms, and it is not clear to me what general functions, if any, encompass both holomorphic modular forms and Maass forms. I would assume the conjecture says something about such functions.)