A NULLSTELLENSATZ FOR AMOEBAS

500 DAVID GINZBURG
is GL 2m+1 × SO 2(m+1) . Then we attach to the corresponding induced representation
an Eisenstein series whose residue we denote by ′ ɛ
. All other notation is as in
Section 2.3(c). Arguing as in [GRS1, pages 114 – 115], by choosing suitable Schwartz
functions in the representation θ ψ Sp 4(m+4)
, the integral (34) converges absolutely. We now
prove the following theorem.
THEOREM 10
Conjecture 2(2) implies Conjecture 2(3).
Proof
The case when m = 1 was shown in [GH2] in complete detail. For general values
of m, the proof is similar and follows similar steps as in the proof of Theorem 6.
Therefore, we sketch only the main steps.
The key ingredient is to use the construction of the lifting from the group Sp 6 ×
Sp 2m to the group SO 2(m+4) . This lifting was introduced in Section 2.3(c), where
we now take n = 3. LetM = SO 14m+8 ,andletH = SO 2(m+4) . Starting with the
representations π and ɛ as above, we construct an automorphic representation σ ′ ,
defined on the group Sp 6 (A), as the space generated by all functions
∫ ∫
f (g) =
ϕ π (h)θ ψ ( ) ( )
Sp 12(m+4)
l(u)(g, h) θτ(ɛ) u(g, h) ψU (u) dudh.
H (F )\H (A) U(F )\U(A)
As in Section 3, we can prove that σ ′ is a cuspidal representation of the group Sp 6 (A).
In fact, the computations are very similar to those in [GH2, Lemma 8], where the case
m = 1 was studied in detail. If nonzero, then it follows from Section 5 that if σ is
any nonzero irreducible summand of σ ′ , π is an endoscopic lift from σ and ɛ. Hence
we need to know when σ ′ is nonzero and when it is a lift from the exceptional group
G 2 (A). For that, we use the result from [GJ], described as follows. Let σ denote an
irreducible cuspidal representation of Sp 6 (A). Suppose that for some vector ϕ σ in the
space of σ , the period integral
⎛⎛
⎞ ⎛ ⎞⎞
∫ ∫ I 1 X Y k 1
ϕ ψ σ (g) =
ϕ σ
⎝⎝
I 2 X ∗ ⎠ ⎝ k 1
⎠⎠ ψ(tr X) dXdY dk 1
SL 2 (F )\SL 2 (A) (F \A) I 7 2 k 1
is nonzero. Here X ∈ Mat 2×2 ,andY ∈ Mat 0 2×2
. Then it follows from [GJ] that there
exists a cuspidal generic representation ν of the exceptional group G 2 (A) such that σ
is the weak functorial lift from ν.
Performing steps similar to those in the proof of Theorem 6, we deduce that the
above integral is nonzero if and only if Q(π, ɛ) is nonzero.