322 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHODWe refer to [Kl,K2,K3] concerning the progress made on this assumption. It
should be mentioned that it is a result of Yuanli Zhang [Z] that, if Conjecture 3.10
is valid for the pair ( G, M), then the non- vanishing of N ( s , n v, w 0 ) for Re( s) ~ 1 /2
follows from its holomorphy over the same range.
Arguments given in [CKPSS1,K4,KS2], then proceed under the validity of As-
sumption 4.1 for (G, M) as well as for all other related lower rank pairs (that come
into the multiplicativity), which consequently are verified in each of the cases in
[CKPSS1,K4,KS2].
The main issue with this argument is that one cannot always get such a con-
tradiction and rule out the pole. In fact, there are many unitary duals whose
complementary series extend all the way to Re( s) = 1, making the results far from
general.
On the other hand, if one considers a highly ramified twist n,,, (see below) of n,
then it can be shown quite generally that every L(s, n,,,, ri) is entire (cf. [Sh8] for its
local analogue). In fact, if rJ is highly ramified, then by checking central characters,
wo(n,,,) i'-n,,,, whose negation is a necessary condition for M(s, n,,,) to have a pole,
a basic fact from Langlands spectral theory of Eisenstein series (Lemma 7.5 of
[La2]). This lemma was used by Kim in [K2] and in view of the present powerful
converse theorems of Cogdell and Piatetski- Shapiro [CPSl,2,3], that is all one needs
to prove our cases of functoriality [CKPSS1,K4,KS2]. We formalize this by quoting
the following (Proposition 2.1) from [KS2].Theorem 4.2. Assume the validity of Assumption 4.1 for all the relevant pairs,
e.g., n,,, below (Section 2 of [KS2}). Then there exists a rational character~ E
X(M)F = X(M), with the following property. Let S be a non-empty finite set
of finite places of F. For every globally generic cuspidal representation 7r of M =
M(A.p ), there exist non- negative integers fv , v ES, such that for every
grossencharacter rJ = 0vl/v of F for which the conductor of l/v, v E S, is larger
than or equal to fv, every L -function L(s, n,,,, ri), i = 1, ... , m, is entire, where
7r,,, = 7r 0 ( rJ · (). The rational character ~ can be simply taken to be ~ ( m) =
det(Ad(m)ln), m EM, where n is the Lie algebra of N.The last ingredient in applying converse theorems is that of boundedness of each
L(s, n, ri) in every vertical strip of finite width, away from its finite number of poles.
The finiteness of poles is again a consequence of the finiteness of the poles of M ( s, n)
for Re(s) ~ 0 and the functional equation satisfied by each L(s, n, ri), but under
the validity of Assumption 4.1 (cf. [GSl]). In this full generality, the boundedness
in finite vertical strips, away from their poles, were proved by Gelbart- Shahidi
in [GSl], again using our method. The main difficulty in proving this result is
having to deal with reciprocals of each L(s, n, ri), 2:::; i :::; m, near and on the line
Re(s) = 1, the edge of critical strip, whenever m ~ 2, which is unfortunately the
case for each of the £ - functions appearing in our cases of functoriality. We handle
this by appealing to equation (3.5) (Theorem 3.2) and estimating the non-constant
term (3.2) by means of Langlands [HC,La2] and Muller [Mu], and a non-trivial
result from complex function theory (Matsaev's theorem). Here is the statement of
the main result of [GSl] as formulated for n,,, to avoid the issue of poles.
Theorem 4.3. Under the validity of Assumption 4.1 in each of relevant cases,
let ~ and l/ be as in Theorem 4.2. Assume l/ is sufficiently ramified so that each