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.)

  • $\begingroup$ Indeed: The R-P conjecture for holomorphic forms is known since Deligne, but is still conjectural for Maass forms, the best bounds are due to Iwaniec I believe. $\endgroup$ – Henri Cohen Feb 25 at 16:47
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    $\begingroup$ For classical modular forms and Maass forms, you can look at Iwaniec-Kowalski, Analytic number theory, section 5.11. The statement of the R-P conjecture is given on page 95. $\endgroup$ – François Brunault Feb 25 at 21:39
  • $\begingroup$ @FrançoisBrunault Thank you for the reference; it has exactly what I was looking for. The book is excellent: I finally have a good source for big-picture ideas on L-functions! $\endgroup$ – David Schwein Feb 25 at 22:24
  • $\begingroup$ @FrançoisBrunault Maybe you can make your comment into an answer? $\endgroup$ – Timothy Chow Feb 27 at 15:54
  • $\begingroup$ (I would certainly accept that answer.) $\endgroup$ – David Schwein Feb 27 at 17:27

For classical modular forms and Maass forms, you can find the definition of the associated $L$-function in Section 5.11 of the book Analytic number theory by Iwaniec-Kowalski. The statement of the Ramanujan-Petersson conjecture is given at the bottom of page 95 (a definition of general $L$-functions is also given on pages 93-94).


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