44 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
those for a subgroup H, supposed better known. This is explained, with proofs, in
Lecture 3. Several applications to spectral bounds are given in Lecture 4.
While writing these notes I found that the Burger-Sarnak method, combined
with Arthur's conjectures, had remarkable consequ ences in the representation the-
ory of groups over local fields - only p-adic fields are considered here. These are
explained in § 3.4. I hope they will interest students of local representation theory.
Here is a more explicit description of the contents.
Lecture 1 is introductory and devoted to the conjectural spectra of the Lapla-
cian and Hecke operators for SL(2) or GL(2). This includes Selberg's conjecture
and the Ramanujan conjecture for Maass forms. I have stated the spectacular
new results of Kim, Shahidi and Sarnak (§ 1.5). Some proofs are sketched, for ex-
ample of the Ramanujan conjecture for classical forms of weight 2 (§ 1.4). I also
explain Langlands's approach and Ramakrishnan's recent work on the density of
Ramanujan primes(§ 1.6, 1.7).
Lecture 2 is devoted to Arthur's conjectures for finite primes (=Hecke oper-
ators), with the simplest formulation I could find. See Conjectures 2 (§ 2.3) and 2A
(§ 2.4). In § 2.5 I introduce part of the formalism for the global conjectures (para-
metrization of automorphic forms), but only what is relevant to the local cases; see
Conjectures 4 and 5.
Lecture 3 states what is known of the Ramanujan conjecture for GL(n), in
the "geometric" cases - where the conjecture is known - and for general forms,
where a useful approximation has been obtained by Luo, Rudnick and Sarnak.
I then describe the method of Burger, Li and Sarnak, extended by Ullmo and
myself to the S-arithmetic case (§ 3 .3). Since this relies crucially on (serious) local
representation theory, I have recalled the basic representation-theoretic facts in
§ 3.2. The consequences for (semi-) local representation theory are explained in
§3.4. In §3.5 I show that they are approximately true for SL(n) over a local non-
Archimedian field of characteristic 0, and (thanks to the work of L. Lafforgue !)
true in finite characteristic. Finally, in § 3.6 I give a totally new proof, relying
on these methods, of the Gelbart-Jacquet bound for Hecke eigenvalues on Maass
forms.
Finally, Lecture 4 reviews common work with H. Oh and E. Ullmo on spectral
bounds for Hecke operators (§ 4.1), and the application of the methods in Lecture 3
to the T-conjecture (§ 4.2). The last section (§ 4.3) concerns the problem of ex-
tending Selberg's eigenvalue conjecture from functions to differential forms. This
is common work with N. Bergeron.
In this final version I have not tried completely to repress the informal style
of the lectures. Thus the definitions are not always given in full generality - for
instance, I mostly consider split groups over Ql -but I feel free to state some results
for cases which have not been completely defined. I hope this actu ally makes the
reading easier; the careful reader will of course have to consult the references, in
particular the Corvallis volumes or Arthur's papers. Some proofs, which are only
sketched, will also send the reader looking at the original papers.