1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. GLOBAL £-FUNCTIONS 143

Conjecture (GRC). Let 7r be a (unitary) cuspidal representation of GLn(A) with

decomposition 7r-::::: ©'nv. Then the local components 7rv are tempered representa-

tions.

However, it has an interesting interpretation in terms of £-functions which is

more in keeping with the origins of the conjecture. If 7r is cuspidal, then at every

finite place v whe(r:v:v is unramifi)ed we have associated a semisimple conjugacy

class, say A1rv = ·. so that

μv,n

n

i=l

If v is an archimedean place where n v is unramified, then we can similarly write

where

n

L(s,n) = ITrv(s+μv,i)

i=l

if kv '::::'. JR.

if kv '::::'. C

Then the statement of the GRC in these terms is

Conjecture (GRC for £-functions). If n is a cuspidal representation of GLn(A)

and if v is a place where nv is unramified, then lμv,il = 1 for v non-archimedean

and Re(μv,i) = 0 for v archimedean.

Note that by including the archimedean places, this conjecture encompasses

not only the classical Ramanujan conjectures but also the various versions of the

Selberg eigenvalue conjecture [36].

Recall that by the Corollary to Theorem 3.3 we have the bounds q;;^1!^2 <

lμv,i I < q;/^2 for v non-archimedean, and a similar local analysis for v archimedean

would give I Re(μv,i)I < ~· The best bound for general GLn over a number field is

due to Luo, Rudnick, and Sarnak [63]. They are the uniform bounds

and

if v is non-archimedean

1 1

I Re(μ v,i ·)I < - - - 2 --n (^2) + 1 for v archimedean.
Their techniques are global and rely on the theory of Rankin-Selberg £-functions
as presented here, a technique of persistence of zeros in families of £-functions, and
a positivity argument.
For function fields over a finite field, the Ramanujan Conjecture for GLn follows
from Lafforgue's proof of the Global Langlands Correspondence [59].
For GL 2 over a number field there has been much recent progress. The best gen-
eral estimates at present are due to Kim and Shahidi [56], who use the holomorphy
of the symmetric ninth power £-function for Re(s) > 1 to obtain
1 1
q-;;-^9 < lμv,il < qJ for i = 1, 2, and v non-archimedean.