1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3

method Lecture 3. Known bounds for the cuspidal spectrum and the Burger-Sarnak

Burger-Sarnak method

3.1. We first return to the original Ramanujan conjecture, now (wisely)

limited to GL(n) (§ 2.2). So let 7r = Q9 'lrv be a representation of GL(n, A) occuring

v

in Acusp (for a fixed character x of Qx \Ax). If pis a prime where 'lrp is unramified,

we have the Hecke matrix (§ 2.2):

t7r,p =

(

tl.. tn)

The conjecture is:

(?3.1) ltil = 1 , i = 1,... , n

There has been much progress about this question in recent years, in very dif-

ferent directions. The first is arithmetic, and concerns representations 7r associated

to the cohomology of algebraic varieties (Shimura varieties, generalizing the modu-

lar curves). This is the analogue of the classical modular forms of weight 2:: 2. The

precise conditions are the following:

(a) 7r is a cuspidal representation of GL(n,A<QI) or GL(n,AF) where Fifa

totally real field or a CM field (totally imaginary quadratic extension of a totally

real field).

[15b]).

(b) 7r 00 (i .e. Q9 7r v when F =f. Q) is a cohomological representation.

vloo

(c) 7r is self-dual if F is totally real; similar condition if F is CM (see

(d) At some finite prime v, 'lrv belongs to the discrete series.

Then (?3.1) was proved by the author in 1991 at almost all primes [15b]. (A
gap in the proof was later corrected in [16].) In their proof of the local Langlands
conjecture, Harris and Taylor extend the result at all primes - even the analogous

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