LECTURE 4
Holomorphy and boundedness; applications
4.1. Twists by Highly Ramified Characters, Holomorphy and
Boundedness
Since our aim is to establish those analytic properties of £-functions from our
method which are crucial in proving the striking new cases of functoriality, we
will limit our discussion on the issue of holomorphy of £ - functions only to twists
by highly ramified characters. In fact, as we explained in earlier sections, the
functional equations for £ - functions within our method are proved quite generally
and multiplicativity and the related machinary necessary for applying converse
theorems to our £ - functions are in perfect shape.
Nothing that general can be said about the holomorphy and possible poles of
these £ - functions. On the other hand, there has recently been some remarkable
new progress on the question of holomorphy of these £ - functions, mainly due to
Kim [K2,K3,KS1]. They rely on reducing the existence of the poles to that of
existence of certain unitary automorphic forms, which in turn point to the existence
of certain local unitary representations. One then disposes of these representations,
and therefore the pole, by checking the corresponding unitary dual of the local
group. In view of the functional equation, this needs to be checked only for Re( s) :'.'.'.
1/2, if a certain local assumption on normalized local intertwining operators is valid.
To explain, let A(s, nv, w 0 ) be the local intertwining operator attached by equation
(2.10) to our inducing representation 1fv· We recall that we are dealing with a pair
(G, M) and ax-generic cuspidal representation 7r = ©v1rv of M = M(AF ). Let, for
each i, 1 :::; i:::; m, L(s, nv, ri) and c:(s, nv, ri, 'l/Jv) be the corresponding £-function
and root number specified earlier. We define a normalized operator N(s, nv, wo) by
(4.1) N(s, 1fv, wo) = r(s, 1fv, 'l/Jv)A(s, 1fv, wo),
where the normalizing factor is defined as ([Shl])
m
(4.2) r(s,nv,'l/Jv) = IJ c:(is,nv,ri,7j}v)L(l +is,nv,ri)/L(is,nv,ri)·
i=l
Assumption 4.1. The operator N(s, 1fv, wo) is holomorphic and non-zero for
Re(s) :'.'.'. 1/2.
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