A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 491
CONJECTURE 1
Let π denote an irreducible generic cuspidal representation of the group H (A), and let
ɛ denote an irreducible cuspidal representation of H 2 (A) according to Section 6.1(1) –
(5). Assume that τ = τ(ɛ) is a cuspidal representation. Then the following are
equivalent.
(1) The partial tensor L-function L S (π × ɛ, s) = L S (π × τ(ɛ),s) has a simple
pole at s = 1.HereS is a finite set of places, including the archimedean ones,
such that outside of S, all data is unramified.
(2) There is a choice of data such that the period integral P(π, τ(ɛ)) is not zero
for some choice of data.
(3) There is a generic cuspidal representation σ of H 1 (A) such that π is the weak
endoscopic lift from σ and ɛ.
Two parts of the conjecture are, in fact, a theorem. The implication that (1) implies
(2) follows from the usual Rankin-Selberg theory. Indeed, it follows from the above
references that when we unfold the global integrals, we represent the above tensor
product L-functions. It also follows from the above references that for any finite
place, data can be chosen so that the integral is not zero. From this, it follows that
if the partial L-function L S (π × τ(ɛ),s) has a simple pole at s = 1, the period
integral P(π, τ(ɛ)) is not zero for some choice of data. The implication that (3)
implies (1) follows from the definition of the weak lift. Indeed, if we assume (3), then
L S (π × ɛ, s) = L S (σ × ɛ, s)L S (ɛ × ɛ, s). Since all data are generic, we know from
[CKPS] that all representations have a lift to an automorphic representation of GL.
By the result of [Sh], we know that the tensor product L-function of two automorphic
representations does not vanish at s = 1. From this, it follows that L S (π × ɛ, s) has a
simple pole at s = 1.
We note that the implication that (2) implies (1) in Conjecture 1 should, in
principle, follow also from the Rankin-Selberg integral representations given above.
In this part, we study the implication that (2) implies (3). We use the lifting studied
in the previous sections to prove this implication in one case. The other four cases
stated in Conjecture 1 are different. The main problem is that the representations ɛ
involve a representation µ(ɛ) defined on a classical group. Therefore, another step
is required. This step requires the study of the descent of a certain residue of an
Eisenstein series and to prove that this descent is by itself a residue. At this point, it is
not clear how to prove this.
THEOREM 6
Let H and H i be as in Definition 2(5). In other words, suppose that H = SO 2(n+m)+1 ,
that H 1 = SO 2n+1 , and that H 2 = SO 2m+1 . Then Conjecture 1 holds.