1549380232-Automorphic_Forms_and_Applications__Sarnak_

52 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

by GL(2, Zp)· To it is associated a Hecke matrix, that is, a diagonal matrix in

GL(2, C), which we denote by

The diagonal coefficients are the ones introduced previously in a more direct

way.

The same applies to GL(n, A): there is a notion of cuspidal representation of

GL(n, A) - occurring in the cusp forms, a subspace of L^2 (1R~ GL(n, Q)\ GL(n, A))

• and they determine Hecke matrices of dimension n, for p outside a finite set
  of ramification. There is a notion of representation of GL( n , A) "induced from
  cuspidal", i.e. induced from a parabolic subgroup of GL(n), starting from unitary
  cuspidal representations of the blocks (see Cogdell's lectures).
  Conjecture (Langlands 1970 [42a]). -For all n , there exists a representation ITn
  of GL(n, A), induced from cuspidal, such that

tp(ITn) = 3n- 1(tp) = (a;-1 a;-2/3p ).

13;-1

If the conjecture is true (for given n) this implies that the "higher £-functions"

associated to 7r have a meromorphic continuation and a functional equation. See

Shahidi's talks.

At any rate, Jacquet-Shalika have proved for a cuspidal (or induced from cus-

pidal) representation of GL(n, A) the bounds

P-1/2 < /a/ < p1/ 2

on the eigenvalues (see§ 3). Thus, assuming the Conjecture for all n, we get:

P-1; 2 < fa;-1! < p1/ 2

whence /ap[±l < p^2 cn'-^1 i , whence /ap[ = 1!

Without assuming the full conjecture, we see that the proof for any new value

of n yields a new bound one.

1. 
   
   
   
   7. Ramakrishnan's approach
      Fix a Maass eigenform f, and define
      S = S(j) ={pf N: p-^1!^2 /,X.p[:::; 2}.
      Thus Sis a subset of the set P of prime numbers, and p E S ~ the Ramanujan
      conjecture is true at p.
   
   
   
   

Recall that lim lL(P-s ) = 1. (s is real > 1 in the following arguments).
s->l + - og s - 1
If S C P define the lower Dirichlet density of S by

I>-s

~(S) =^1 im.. m f ---,..--pES -...,...

s->l+ - log(s - 1)