1549380232-Automorphic_Forms_and_Applications__Sarnak_

140 J.W. COGDELL, £-FUNCTIONS FOR GLn

seems to be no obstruction to carrying this out, and then we obtain boundedness

in vertical strips for L( s , 7r x 7r^1 ) in general.

We should point out that if one approaches these £-function by the method

of constant terms and Fourier coefficients of Eisenstein series, then Gelbart and

Shahidi have shown a wide class of automorphic £-functions, including ours, to b e

bounded in vertical strips [25].

4.2. Poles of £-functions

Let us determine where the global £-functions can have poles. The poles of the

£-functions will be related to the poles of t he global integrals. Recall from Lecture

2 that in the case of m < n we have that the global integrals I(s; cp, cp') are entire

and that when m = n then I(s;cp,cp',if?) can h ave at most simple poles and they

occur at s = -iu and s = 1-iu for u real when n :::::: if'® I det lia. As we have noted

above, the global integrals and global £-functions are related, for appropriate cp,

cp', and if?, by

I(s; cp, cp') = (rr ev(s; W'Pv, W~J) L(s, n x 7r

1

)

vES

or

I(s;cp,cp

1

,if?) = (rr ev(s;W'Pv• W~~,if?v)) L(s,n x n').

vES

On the other hand, we have seen that for any s 0 E re and any v there is a

choice of local Wv, W~, and if?v such that the local factors ev(s 0 ; Wv, W~) -1- 0

or ev(so; Wv, W~, if?v) -1-0. So as we vary cp, cp' and if? at the places v E S we see

that division by these factors can introduce no extraneous poles in L(s, 7r x 7r^1 ) ,

that is, in keeping with the local characterization of the £-factor in terms of poles

of local integrals, globally the poles of L(s, 7r x n') are precisely the poles of the

family of global integrals {I(s; cp, cp')} or {I(s; cp, cp', if?)}. Hence from Theorems 2.1

and 2.2 we have.

Theorem 4.2. If m < n then L(s, 7r x 7r^1 ) is entire. If m = n, then L(s, n x n')

h as at most simple poles and they occur iff 7r :::::: if'® I det lia with u real and are

then at s = -iu and s = 1 - iu.

If we apply this with n' = if we obtain the following useful corollary.

Corollary. L(s, 7r x if) has simple poles at s = 0 ands= l.

4.3. Strong Multiplicity One

Let us return to the Strong Multiplicity One Theorem for cuspidal representations.
First, recall the statement:

Theorem (Strong Multiplicity One). Let (n, V1r) and (n', V1r1) be two cuspidal
representations of GLn(A). Suppose there is a finite set of places S such that for
all v tf. S we have nv:::::: n~. Then 7r = n'.

We will now present Jacquet and Shalika's proof of this statement via £-
functions [45]. First note the following observation, which follows from our analysis
of the location of the poles of the £-functions.