1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 81

For Sp(g) this approach yields the following result. We work in fact with the

group G = GSp(g) of symplectic similitudes relative to the form (lg -lg), so

we have the natural Hecke operator

p

p

1

1

analogous to the standard Tp for GL(2).

Theorem 4.1 ([19a]). -Assume B is the constant expressing the (known) approx-

imation to the Ramanujan conjecture for GL(2) (Lecture 1). If 7r P is an unramified

automorphic representation of GSp(g, Qp), and is not an Abelian character, the

eigenvalue of Tp relative to 7r P verifies

l>-pl::::; 2g pg(gil)_~(l-0)_

Note that the degree of Tp is (cf. [19a, § 6.3])

( )( ) (

g) g(g+l}

1 + p 1 + p2... 1 + p "-' p 2 '

so the improvement on the trivial bound is essentially in ~(1 - B), very close now

(in 2002) to ~.

This depends, as we explained, on the control of nplHP where H is SL(2)g

naturally embedded in Sp(g). Simultaneously with [19a], Oh studied the restriction

to SL(2)g of a non-trivial unitary representation of Sp(g) and found that it was

always tempered [46]. The consequence is that we can essentially assume, in the

previous argument, that the Ramanujan conjecture is true!

The outcome is the following (we only state the result for automorphic forms

on G(Z)\G(JR)):

Theorem 4.2 ([17] (Same assumptions). - For any c > 0, there exists C(c) > 0

such the

for any p.

N.B. The constant C(c) is determined by the growth of the Harish-Chandra 3-

function for SL(2) [17] and could be made explicit. For higher level the constant is

larger.

We have only studied the symplectic groups, but the paper by Oh, Ullmo and

the author contains many results concerning other groups. Arthur's description of

the spectrum shows that Theorem 4.2 cannot be substantially improved, and this

is also proved in [17, Thm. 3.3]. The results are also completely explicit for GL(n),

cf. [19a, 17].

One interesting consequence of these results is equidistribution. Namely, as-
sume for simplicity G simple, simply connected and split over Q, assumer c G(Q)