66 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
conjecture (nv tempered) for primes where 1fv is ramified. We say no more, referring
the reader to [25b].
Now the number field - if we consider CL( n, AF) -and the cuspidal represen-
tation are arbitrary. In that case the first non-trivial result is due to J acquet and
Shalika ([32], 1981):
Theorem 3.1. - For all i ,
(3.2)
This is the assertion for F = <Ql; for a number field we get of course, with usual
notations,
(3.3)
For n = 2 this is trivial - it is essentially Hecke's estimate for classical forms,
and in general it just says that 1fp is non-trivial if n is cuspidal. In general the
Hecke matrix of t he trivial representation is
p-2-n-^3so the gain given by (3.2) is important.
The original proof of Jacquet-Shalika relied on the Rankin-Selberg £-functions
and Landau's lemma. A different proof follows from t he fact that local components
of cuspidal representations are generic representations ([Piatetsky-Shapiro, Shalika
[52]) and from t he classification of generic, unitary representations by Tadic [55].
This gives analogous estimates for the ramified primes.
These estimates are often useful. In fact, they are used in t he proof of the
Ramanujan conjecture for cohomological representations: t he point is that if we
have (3.2) and if we know that t he ti are eigenvalues of Frobenius, so itil = pw/Z for
w E Z (cf. §2.3), then we must have w = 0 and itil = l. This is used to separate
parts of the cohomology of Shimura varieties of different degrees (cf. [15b]: in the
case of function fields this was used before by Laumon).
The estimate (3.2), however, is too weak for certain arithmetic questions. It
has been improved by Serre in 1981 , and then again by Luo, Rudnick and Sarnak
[43a,b]. They obtain the following estimate:
Theorem 3.2. - For all i, q;~+c(n) ::::; iti l ::::; qJ-c(n) with
(3.4)1
c(n) = n2+ 1I have stated the theorem for an arbitrary number field; for <Ql it had already
been obtained by Serre. For n = 2, one gets an improvement of t over the trivial
estimate -not as good as the Gelbar t-Jacquet estimate, see§ 3 .7; but by combining
this with the symmetric square lift one gets q:;;^115 < ltil ::::; q~^15 (see [43a, Thm.3].