1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. GLOBAL £-FUNCTIONS 141

Observation. For n and n' cuspidal representations of GLn(A), L(s, n x n') has

a pole at s = 1 iff 7r '.::::' n'.

Thus the £-function gives us an analytic method of testing when two cuspidal

representations are isomorphic, and so by the Multiplicity One Theorem, the same.

Proof: If we take 7r and n' as in the statement of Strong Multiplicity One, we have

that 7r v '.::::' n~ for v tf-S and hence

L^8 (s,n x n) =II L(s,'Trv x nv) =II L(s,'Trv x n~) = L^8 (s,n x n')

v~S v~S

Since the local £-factors never vanish and for unitary representations they have no

poles in Re(s) ~ 1 (since the local integrals have no poles in this region) we see

that for s = 1 that L(s,n x 7r') has a pole at s = 1 iff L^8 (s,n x n') does. Hence

we have that since L(s,n x n) has a pole at s = 1, so does L^8 (s,n x n). But

L^8 (s,n x n) = L^8 (s, n x n'), so that both L^8 (s,n x n') and then L(s,n x n') have

poles at s = 1. But then the £-function criterion above gives that n '.::::' n'. Now

apply Multiplicity One. D

In fact, Jacquet and Shalika push this method much further. If n is an ir-

reducible automorphic representation of GLn(A), but not necessarily cuspidal,

then it is a theorem of Langlands [61] that there are cuspidal representations,

say Ti, ... , Tr of GLn 1 , .•• , GLnr with n = ni + · · · + n,., such that 7r is a constituent

of Ind(T 1 © · · · © T,.). Similarly, n' is a constituent of Ind(T{ © · · · © T;,). Then
the generalized version of the Strong Multiplicity One theorem that Jacquet and
Shalika establish in [45] is the following.

Theorem (Generalized Strong Multiplicity One). Given 7r and n' irreducible
automorphic representations of GLn(A) as above, suppose that there is a finite set
of places S such that 'Trv '.::::''Tr~ for all v tf-S. Then r = r' and there is a permutation
<7 of the set {1, ... , r} such that ni = n~(i) and Ti = T~(i).

Note, the cuspidal representations Ti and Tf need not be unitary in this state-
ment.

4.4. Non-vanishing results

Of interest for questions from analytic number theory, for example questions of
equidistribution, are results on the non-vanishing of £-functions. The fundamental
non-vanishing result for GLn is the following theorem of Jacquet and Shalika [44]
and Shahidi [75, 76].

Theorem 4.3. Let 7r and n' be cuspidal representations of GLn(A) and GLm(A).
Then the £-function L(s, 7r x n') is non-vanishing for Re(s) ~ 1.

The proof of non-vanishing for Re(s) > 1 is in keeping with the spirit of these
notes [45, I, Theorem 5.3]. Since the local £-functions are never zero, to establish
the non-vanishing of the Euler product for Re(s) > 1 it suffices to show that the
Euler product is absolutely convergent for Re(s) > 1, and for this it is sufficient to