1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. RAMANUJAN GRAPHS 409

Exercise. Show that the eigenvalues of Cay(JF q2, N _ 1 ) are as stated, and prove

Proposition 1.5.

Consequently, the generalized Kloosterman sums are eigenvalues of the Hecke

operator Tt at place t. It is natural to find automorphic forms on the quaternion

group which are eigenfunctions of Tt with eigenvalues given by these generalized

Kloosterman sums. In view of the Jacquet-Langlands correspondence [15], there

should exist automorphic forms for G L 2 (AK) which have the same property. In a

joint paper with Chai, [5], we proved that in fact there are automorphic forms for

G L2 (AK) which are eigenfunction for the Hecke operators Tw at almost all places

w of K with eigenvalues given by generalized Kloosterman sums which arise from

global elements in K by reduction at w. Hereafter, Nw, the norm of the place w ,

denotes the cardinality of the residue field of Kw.

Theorem 1.6. Kloosterman sum conjecture over function fields [Chai-Li

[5]] Let K be a function field with the field of constants lF a finite field. Given a

nonzero element a EK, there exists an automorphic form fa of GL2(AK) which is

an eigenfunction of the Hecke operator Tw at all places w of K, which is neither a

zero nor a pole of a, with eigenvalue -Kl(lFw, a) (= kl(lFw, a)). Here Fw denotes the

residue field of Kw. In other words, outside finitely many bad places, the L-function

L(s,fa) is given by

1

II 1 + Kl(lFw, a) Nw-s + Nwl-2s ·

w good

It is worth pointing out that the similar statement over Q is open. Using an idea
of Sarnak, Booker in [2] showed that if such an automorphic form for GL 2 (AQ) were
to exist for the case a = 1, then it would be a Maass wave form with the product of
level and the eigenvalue of Laplacian very large. He also showed that it is impossible
to disprove the conjecture numerically.

In the case of norm graphs, we can compute the components at bad places to
get

Theorem 1.7. [Chai-Li [5]] Let K = lFq(t) and a E JF~. There is an automorphic
form fa on D(K)\D(AK )/ D(K 00 ) (1 + pt_ 1 ) Tiw#oo,t-l D(Ow) whose associated
L-function is
1 1
L(s, fa) = II 1 + Kl(JF g) Nw-s + N w^1 -^28 1 - Noo-s
w#t-1,oo W> t-1
Here D is the quaternion group as in Morgenstern graphs.

The third construction was given by Terras. As before, let lF q be a finite field
with odd characteristic. Fix a nonsquare o in lF q. The multiplicative group of

lF q ( v'o) can be imbedded in GL 2 (JF q) as

Ko = { ( ~ bao) : a, b E lF q}.

The coset space G L 2 (JF q) / K 0 may be represented by

H={(6 ~):yElF;,xElFq}·