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At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in the light of global adelic automorphic representations. I would like to understand how far this is true. I recall the different versions, say in $GL(2)$.

Ramanujan-Petersson conjecture. Let $f$ be a modular form (we could state it for a Maass form too) of weight $k$ and level $1$, if the $a_n$ denotes its Fourier coefficients then $$a_n \ll n^{(k-1)/2}.$$

Selberg conjecture. Let $f$ be a Maass form and $\lambda_1$ its smallest nonzero eigenvalue for the Laplacian. Then $$\lambda > \frac 14.$$

Now, there is a "more modern" version in terms of automorphic representations.

"Automorphic" Ramanujan conjecture. Let $\pi$ be an automorphic cuspidal representation of $GL(2, \mathbb{A})$. It decomposes by Flath's theorem as $\pi \simeq \bigotimes_v \pi_v$. Then $\pi_v$ is tempered for all place $v$.

Are these three formulation equivalent, or only in a mild sense? More precisely,

  • does the automorphic version imply the two others?
  • do the two "local" versions imply the automorphic one?
  • are the two "local" version equivalent? (or: does one of them imply the automorphic version, by a kind of rigidity of the automorphic representations)

(I haven't found proper answer to these questions and would be glad to know if I missed papers or notes about it.)

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Let $f$ be an automorphic form corresponding to an automorphic representation $\pi =\otimes_v \pi_v$ of $GL_2(\mathbb A_{\mathbb Q})$.

For an unramified prime $p$, the following are equivalent (for $f$ holomorphic of weight $k$):

  • $|a_p| \leq 2p ^{(k-1)/2 }$.
  • For all $n$, $|a_{p^n} |\leq (n+1) p^{n (k-1)/2} $.
  • For all $n$, $|a_{p^n} |\ll p^{n ((k-1)/2 +\epsilon) } $.
  • $\pi_p$ is tempered.

There is an analogous local equivalence for Maass forms, where for the standard normalization you should take $k=1$. There is also, a more complicated, statement about ramified primes.

For the place $\infty$, the following are equivalent (for $f$ Mass)

  • $\lambda \geq \frac{1}{4}$
  • $\pi_{\infty}$ is tempered.

For holomorphic forms, $\pi_{\infty}$ is automatically tempered.

Putting this together, we see that $\pi$ is tempered at all places if and only if $a_n \ll n^{ (k-1)/2+ \epsilon}$ for all $n$, plus some additional conditions at the ramified primes, and (if $f$ is Maass) $\lambda_1 \geq 1/4$.

Because all cuspidal automorphic representations of $GL_2(\mathbb A_{\mathbb Q})$ correspond to cuspidal holomorphic or Maass forms, the Ramanujan conjecture for $GL_2(\mathbb A_{\mathbb Q})$ is equivalent to these statements for all holomorphic or Maass forms simultaneously.

In summary, the automorphic one implies the two local ones, the two local ones imply the automorphic (at least away from ramified primes), but the local ones are not equivalent in any meaningful sense.

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  • $\begingroup$ An additional comment: No rigidity of the kind you suggest is expected to be provable over number fields as far as I know, but over function fields, one knows from work of V. Lafforgue and P. Deligne a very strong rigidity: A cuspidal automorphic form tempered at one unramified place is tempered at every unramified place. $\endgroup$ – Will Sawin Apr 24 at 17:14

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