A NULLSTELLENSATZ FOR AMOEBAS

ENDOSCOPIC LIFTING 449
In the last section, Section 6, we apply our construction to relate liftings with
period integrals and poles of L-functions. We consider two cases. In the first case, we
characterize the existence of a simple pole to standard tensor L-functions. (The main
statement is Conjecture 1, together with Theorem 6.) More precisely, we prove the
following theorem.
MAIN THEOREM
Let π denote an irreducible generic cuspidal representation of the orthogonal group
SO 2(n+m)+1 (A), and let ɛ denote an irreducible generic cuspidal representation of
SO 2m+1 (A). Letτ(ɛ) denote the lift of ɛ to GL 2m (A) whose existence was proved
in [CKPS]. Assume that 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)), defined in
the proof of Theorem 6, is not zero for some choice of data.
(3) There is a generic cuspidal representation σ of SO 2n+1 (A) such that π is the
weak endoscopic lift from σ and ɛ.
Two implications of this theorem follow from other references (see the first paragraph
following Conjecture 1). The implication that statement (2) implies statement (3) is
proved in Theorem 6 using our construction of the lifting.
The second application that we discuss in Section 6 is more conjectural. Motivated
by the low-rank cases that we describe in that section, we give a conjecture as to when
the L-function associated to the tensor product of two Spin representations can have
poles at certain values. Although the main conjecture is stated in Conjecture 3, we
have more evidence in a special case of that conjecture stated in this form.
CONJECTURE 2
Let π and ɛ denote irreducible cuspidal generic representations of the groups
GSO 2(m+4) (A) and GSp 2m (A). The following statements are equivalent.
(1) The partial L-function L S (π × ɛ, Spin 2(m+4) × Spin 2m+1 ,s) has a simple pole
at s = 1.
(2) The period integral Q(π, ɛ), described in Section 6, is not zero for some choice
of data.
(3) There exists a cuspidal generic representation ν of the exceptional group G 2 (A)
such that π is the weak functorial lift from ν and from ɛ.
In other words, this conjecture ties the poles of the L-function of a tensor product
of Spin representations, with a certain period integral and with a lifting related to
the exceptional group G 2 . Indeed, Conjecture 2 is related to the following lifting.