1549380232-Automorphic_Forms_and_Applications__Sarnak_

148 J.W. COGDELL, L-FUNCTIONS FOR GLn

5.1. The results

Once again, let k be a global field, A its adele ring, and 'l/; a fixed non-trivial

continuous additive character of A which is trivial on k. We will take n 2: 3 to be

an integer.

To state these Converse Theorems, we begin with an irreducible admissible

representation II of GLn(A). In keeping with the conventions of these notes, we

will assume that II is unitary and generic, but this is not necessary. It has a decom-

position II = @'IIv, where IIv is an irreducible admissible generic representation

of GLn(kv)· By the local theory of Lecture 3, to each IIv is associated a local

£-function L(s, IIv) and a local €-factor c(s, IIv, 'l/Jv)· Hence formally we can form

L(s, II)= IT L(s, IIv) and c(s,II,'l/;) =IT c(s,IIv,'l/Jv)·

v v

We will always assume the following two things about II:

(1) L( s, II) converges in some half plane Re( s) > > 0,

(2) the central character wrr of II is automorphic, that is, invariant under kx.

Under these assumptions, c( s , II, 'l/;) = c( s, II) is independent of our choice of 'l/; [9].

Our Converse Theorems will involve twists by cuspidal automorphic represen-

tations of GLm(A) for certain m. Let 7r^1 = Q9^1 7r^1 v be a cuspidal representation of

GLm(A) with m < n. Then again we can formally define

L(s,II x 7r^1 ) =IT L(s,IIv x 7r^1 v) and c(s,II X 7r^1 ) =IT c(s,IIv X 7r^1 v,'l/Jv)

v v

since again the local factors make sense whether II is automorphic or not. A

consequence of (1) and (2) above and the cuspidality of 7r^1 is that both L(s, II x 7r^1 )

and L(s,IT x ;') converge absolutely for Re(s) >> 0, where IT and;, are the

contragredient representations, and that c(s, II x 7r^1 ) is independent of the choice

of 'lj;.

We say that L( s, II x 7r^1 ) is nice if it satisfies the same analytic properties it

would if II were cuspidal, i.e.,

(1) L( s, II x 7r^1 ) and L( s, IT x ;') have analytic continuations to entire functions

of s,

(2) thes~ entire continuations are bounded in vertical strips of finite width,

(3) they satisfy the standard functional equation

L(s,II x 7r^1 ) = c(s,II x 7r')L(l-s,IT x ;').
For convenience, let us set A(m) to be the set of automorphic representations
of GLm(A), Ao(m) the set of cuspidal representations of GLm(A), and T(m) =
m
IJ Ao(d). If we fix a finite set of S of finite places, then let T^8 (m) denote the
d=l
subset of T(m) consisting of representations that are unramified at all places v E S.
The basic Converse Theorem for GLn is the following.

Theorem 5.1. Let II be an irreducible admissible representation of GLn(A) as
above. Let S be a finite set of finite places of k. Suppose that L(s, II x 7r^1 ) is nice
for all 7r^1 E T^8 (n - 1).

(1) If S = 0 then II is a cuspidal automorphic representation.