1549380232-Automorphic_Forms_and_Applications__Sarnak_

324 FREYDOON SHAHIDI, LANGLANDS-SHAHIDI METHOD

for almost all v. Similarly for root numbers. We in fact prove these equalities for

all v.

This is immediate from t he fact that t he local components of t he weak transfer

established through the converse t heorem (Theorem 3.8 of [KS2]) is in fact 7riv~7r 2 v

for each v (Theorem 5.1 of [KS2]). Proof of this is quite delicate and beside the

techniques already discusse d (functional equations, multiplicativity, ... ), relies on

several other techniques such as base change, both normal [AC] and non-normal

[ JPSS2], as well as certain results from the theory of types [BH].

Next, let 7r = ®v1fv be a cusp form on G L2(Ap). Let Ad(7r) be its Gelbart-

J acquet transfer [GJ]. Then

7r ~ Ad(7r) = Sym^3 (7r) ® w;^1 ffi 7r

implies that Sym^3 (7r) is an automorphic representation of GL 4 (Ap).

More precisely, for every v let Pv: W.k _, GL2(C) be the two dimensional

representation of the Deligne-Weil group W.k attached to 1fv (cf. [Ku]).

Consider:

Sym^3 · Pv = Sym^3 pv: W.k _, GL4(C).

Let Sym^3 7rv be the irreducible admissible representation of GL 4 (Fv) attached to

Sym^3 Pv· Set

Sym^3 7r = ®vSym^3 1fv ·

Theorem 4.5 (Kim-Shahidi [KS2)). Sym^3 7r is an automorphic representation

of GL4(Ap). It is cuspidal unless 7r is of dihedral or tetrahedral type.

Applications: (towards Ramanujan and Selberg Conjectures): For almost

all v, 1fv is given by tv = (~v JJ E GL2(C). Then Sym^3 7r is a cusp form on

GL4(Ap), unless 7r is dihedral or tetrahedral in which case the full Ramanujan is

valid.

Now, we apply the estimates of Luo- Rudnick-Sarnak [LRS] for GL 4 (Ap) to

get

and

Thus

q;;^5!^34 ~ lavl and l.Bvl ~ q~/^34.

This already breaks 1/ 6 and leads to a number of definitive results ([KS2]). Similar

bounds hold towards Selberg Conjecture.

Next, let IT= ®v ITv be a cuspidal r epresentation of GL 4 (Ap). Let

A^2 : GL4(C) _, GL5(C)

b e the exterior square map. Let l.{Jv: W.k _, GL 4 (C) parametrize ITv for all v
[HT,Hel]. Then A^2 cpv: w;,.v ---) GL5(C) parametrizes A^2 IT v' an irreducible admis-
sible representation of GL 6 (Fv)·

Theorem 4.6 (Kim [K4)). There exists an automorphic representation IT'
®v IT~ of GL5(Ap) such that IT~ =A^2 7rv, unless possibly when v l2 or vl3.