1549380232-Automorphic_Forms_and_Applications__Sarnak_

316 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHOD

In fact, if W1Jgv) = AxJs, 7rv)(Iv(gv)fv) is the Whittaker function attached
to fv, then (3.3) can be written as

Theorem 3.2. One has
m
(3.5) Ex(s, if>s, e, f) = II W1Jev) II Ls(l +is, 7r, 7)-^1.
vES i=l

Remark. As opposed to intertwining operators, Whittaker functions are by no
means multiplicative and therefore proof of Theorem 3.1 cannot be reduced to rank
one calculations by means multiplicativity (cocycle relations). It should be pointed
out that in the case of SL 2 , Theorem 3.1 is an easy exercise.

Corollary 3.3 [Sh3]. The product

m

(3.6) II Ls(l +is, 7r, ri) -# 0

i=l

for Re( s) = 0. In particular, if 7r and 7r^1 are cusp forms on G Ln (AF) and G Lt (AF),
then

(3.7) L(l, 7r x 7r^1 ) -# 0,

where local L-functions at every place are corresponding Artin ones ((HT}, {He1}}.

Proof. Modulo non-vanishing of W 1 Je) for Re(s) = 1, which is highly non-trivial
if v = oo (cf. Casselman- Wallach [Ca2,W]), this follows from unitarity (and there-
fore holomorphy) of M(s, 7r) for Re(s) = 0, and Theorem 3.2. Observe that the
integration in (3.2) is over a compact set. 0

Now, computing the non-constant terms from the two sides of the functional
equation (2.17), Lemma 2.1 and Theorems 3.1 and 3.2, together with Definition
(3.1) implies:

Theorem 3.4 (Crude Functional Equation [Sh3,Sh6)). We have

m m

(3.8)

i=l vES i=l

We just point out that by Lemma 2.1, Theorem 3.1 and Definition (3.1)

m

(3.9) CxJs, 1i-v) =II L(l - is, 7rv,i\)j L(is, 7rv, ri),

i=l

whenever 'lrv is unramified.

3.2. The Main Induction, Functional Equations and Multiplica-
tivity

To prove the individual functional equations, i.e., for each L(s, 7r, ri) with precise
root numbers and L-functions , we appeal to the following induction statement
(cf. [Shl ,Sh2]). It is crucial in all the results that we prove from now on.