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.)