Early experiences with classical (holomorphic) cusp forms,
which initially started with the Ramanujan t-function, and later extended to even
Maass cusp forms (cf. [70];[78], last paragraph) on the upper half plane, suggested
that their Fourier coefficients ap at a prime p must be bounded by 2p(k-1)/2, where k
is the weight (cf. [25, 96]). This is what is classically called the Ramanujan-Petersson
conjecture. Its archimedean counterpart, the Selberg conjecture [79], states that the
positive eigenvalues of the hyperbolic Laplacian on the space of cuspidal functions
(functions vanishing at all the cusps) on a hyperbolic Riemann surface parametrized
by a congruence subgroup must all be at least 1/4 (cf. [76, 79, 94]). While for the
holomorphic modular cusp forms, this is a theorem ([25], also see [8, 12]), the case of
Maass forms is far from resolved and both conjectures are yet unsettled and out of
reach.
Satake [78] was the first to observe that both conjectures can be uniformly formulated.
More precisely, if one considers the global cuspidal representation attached
to a given cuspidal eigenfunction, then all its local components must be tempered.
This means that their matrix coefficients must all belong to $L2+(PGL2(Qp))$ for all
$e > 0$ and every prime p of Q. We note that here we are allowing p = 8 and letting
$Q_{/inf} = R$.
It is now generally believed that the conjecture in its general form should be
valid for GLn over number fields to the effect that all the local components of an
irreducible (unitary) cuspidal representation of GLn(AF ) must be tempered (modulo
center).