1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. MOSTLY SL(2) 51

This implies that '{Jp, tr.pp and Tp commute as endomorphisms of H^1. Moreover 'Pp

verifies the equation

(1.12)

If now ap (seen as a n element of((lJe) is an eigenvalue ofTp, and a; a n eigenvalue

of 'Pp in the corresponding eigenspace, (1.12) yields ap =a;+ p(a;)-^1.

Viewed as a complex number, a; verifies la;, I = p^112 by Weil's proof of the

Riemann hypothesis; thus lapl ::::; 2p^112.

1.5. Results for Maass forms
Assume f is a cuspidal Maass eigenform with eigenvalues Ap (p f N). As in
§ 1.3 we write >.P = p^112 (ap + {Jp). Then p-^1 /^2 < lapl < p^1 /^2 and lap{Jpl = 1. We
measure the (known) approximation to the Ramanujan Conjecture by B E [O, l].
We assume that for all eigenforms on all congruence subgroups of SL(2, Z) (and
all pf N)

Thus B = 1 is Hecke' s trivial estimate

B = 0 is the Ramanujan hypothesis.

At the Archimedian prime write >. = ~ -s2 for the eigenvalue of 6.. Thus >. ~ ~
if s E ilR. and the "exceptional" eigenvalues (conjectured not to exist) correspond
to s E] - ~' H We define analogously B 00 by s E [-~, ~] for all exceptional
eigenvalues on all congruence subgroups (and B 00 minimal).

Theorem 1.2 (2000-02). -

(i) (Kim-Shahidi) B, B 00 ::::; ~

(ii) (Kim-Sarnak) B, Boo ::::; J 2

For the proofs see [34a, b, 35]. This is very hard. The reader may also consult
Henniart's expository paper [27]. We will give in§ 3.7 a n easy proof of the estimate
B ::::; ~' due to Gelbart and Jacquet. See Shahidi's lectures in this volume for an
outline of the proof.

1.6. Langlands's approach
The approach which has given the spectacular results in Theorem 1.2 is due to
Langlands and relies on the (conjectured) existence of t he higher symmetric powers
associated to a modular form.
Assume f is an eigenform of the Hecke operators (Maass form or classical
form). Anticipating on § 1.8, recall that f is associated to a cuspidal representation
of GL(2, A.) where A. denotes the rational adeles. This decomposes as a tensor
product:

7f = Q9 1fv

v=oo,p

over all primes of Q, 1fv being an irreducible unitary representation of GL(2, Qv)·
(For this see Gelbart [23]). For pf N, 1fp is unramified, i.e., it has a vector fixed