1549380232-Automorphic_Forms_and_Applications__Sarnak_

144 J.W. COGDELL, £-FUNCTIONS FOR GLn

A simple generalization of the method used in [56] gives the analogous estimate at

the archimedean places over any number field [54]. On the other hand, [53] and

its appendix gives that one can replace ~ with 674 at all places, but only for k = Q.

For some applications, the notion of weakly Ramanujan [10] can replace know-

ing the full GRC.

Definition. A cuspidal representation 7r of GLn(A) is called weakly Ramanujan

if for every E > 0 there is a constant Ce > 0 and an infinite sequence of places { vm}

with the property that each 7rv,,, is unramified and the Satake parameters μv,,,,i

satisfy

For example, if we knew that all cuspidal representations on GLn(A) were

weakly Ramanujan, then we would know that under Langlands liftings between

general linear groups, the property of occurrence in the spectral decomposition is

preserved [ 10].

For n = 2, 3 our techniques let us show the following.

Proposition 4.1. For n = 2 or n = 3 all cuspidal representations are weakly

Ramanujan.

Proof: First, let 7r be a cuspidal representation or GLn(A). Recall that from the

absolute convergence of the Euler product for L(s, 7r x w) we know that the series

ltr(Ad )12

LL d ;; is absolutely co nvergent for Re(s) > 1, so that in particular we

vrf_S d qv

have that L I tr(A;Jl

2

is absolutely convergent for Re(s) > 1. Thus, for a set of

vrf_S qv

places of positive density, we have the estimate I tr(A7rJl^2 < q~ for each E. Since

A1rv = A;v^1 for components of cuspidal representations, we have the same estimate

for 1tr(A;v^1 )1.

In the case of n = 2 and n = 3, these estimates and the fact that I <let A1rv I =

lwv(wv)I = 1 give us estimates on the coefficients of the characteristic polynomial

for A1rv. For example, if n = 3 and the characteristic polynomial of A1rv is X^3 +

aX^2 +bX +c then we know !al= I tr(A7rJI < q~^12 , lbl =I tr(A;v^1 ) det(A7rJI < q~^12 ,

and lcl = I det(A7rJ I = 1. Then an application of Rouche's theorem gives that the

roots of this polynomial all lie in the circle of radius q~ as long as qv > 3. Applying

this to both A1rv and A;v^1 we find that for our set primes of positive density above

we have the estimate q;;e < lμv,,,,il < q~. Thus we find that for n = 2, 3 cuspidal

representations of GLn are weakly Ramanujan. D

4.6. The Generalized Riemann Hypothesis (GRH)

This is one of the most important conjectures in the analytic theory of £-functions.
Simply stated, it is

Conjecture (GRH). For any cuspidal representation 7r, all the zeros of the £-
function L( s, 7r) lie on the line Re( s) = ~.