AN APPROACCH TO A CONJECTURE OF JACQUET

DIHUA JIANG

1. Introduction

In this talk, we shall provide an approach to a conjecture of Jacquet, by means of periods of residual representations from certain Eisenstein series. First we recall Jacquet’s conjecture and some relevant work by D. Prasad and by M. Harris and S. Kudla.

1.1. Jacquet’s Conjecture. Let F be a number field and 𝔸 = 𝔸_{F} the ring of adeles of F. Let (π_{i}, V_{i}) (i = 1, 2, 3) be irreducible cuspidal automorphic representations of GL_{2}(𝔸) with ∏_{i=1}^{3} ω_{π_{i}} = 1, where ω_{π_{i}} is the central character of π_{i}. Then one has the restricted tensor product decomposition for π_{i} (i = 1, 2, 3)

π_{i} = ⊗′_{v∈Ω_{F}} π_{i,v}.

We need following notations:

S := {v ∈ Ω_{F} | π_{i,v}, i = 1, 2, 3, belong to discrete series}

and

D_{S} := {D | quaternion algebras over F, which is ramified only at v ∈ S}.

By the global Jacquet-Langlands correspondence, for each quaternion algebra D ∈ D_{S}, there is an irreducible automorphic representation π_{i}^{D} of D^{×}(𝔸) for i = 1, 2, 3, which corresponds to π_{i}, respectively.

Now, for given three irreducible cuspidal automorphic representations (π_{i}, V_{i}) (i = 1, 2, 3) of GL_{2}(𝔸) with ∏_{i=1}^{3} ω_{π_{i}} = 1, one can have the triple product L-function L(s, π_{1} ⊗ π_{2} ⊗ π_{3}). The integral representation for the triple product L-functions was first given by P. Garrett in [G] and more generally by I. Piatetski-Shapiro and Rallis in [PSR].

Conjecture 1 (Jacquet). The triple product L-function L(s, π_{1} ⊗ π_{2} ⊗ π_{3}) does not vanish at s = \frac{1}{2} if and only if there is at least one quaternion algebra D ∈ D_{S} such that for some f_{i}^{D} ∈ π_{i}^{D}, i = 1, 2, 3, the following period integral

∫_{𝔸_{F}^{×}D(F)\D(𝔸)} f_{1}^{D}(g)f_{2}^{D}(g)f_{3}^{D}(g)dg

does not vanish.

This is the summary for my talk at Seattle meeting, July, 1997.

Remark 1. Since the pair (P GL_{2}×P GL_{2}, P GL_{2}) can be viewed as the pair (SO(4), SO(3)), the Jacquet conjecture can be related to the more general conjecture of Gross and Prasad for the pair (SO(n), SO(n − 1)) [GP].

1.2. The Work of D. Prasad. Let (π_{i}, V_{i}) (i = 1, 2, 3) be irreduccible admissible representations of GL_{2}(F_{v}) with ∏_{i=1}^{3} ω_{π_{i}} = 1. D. Prasad proved in [Pr] the following theorem.

Theorem 1 (Prasad). If at least one of the representations (π_{i}, V_{i}) (i = 1, 2, 3) does not belong to the discrete series of representations of GL_{2}(F_{v}), then

dim_{ℂ} Hom_{GL_{2}(F_{v})}(V_{1} ⊗ V_{2} ⊗ V_{3}, ℂ) = 1.

If all of the representations (π_{i}, V_{i}) (i = 1, 2, 3) belong to the discrete series of representations of GL_{2}(F_{v}), then, by the local Jacquet-Langlands correspondence, for each i, there is an irreducible repressentation (π_{i}^{D}, V_{i}^{D}) of D^{×}(F_{v}), where D is the nonsplit quaternion algebra over F_{v}, such that

dim_{ℂ} Hom_{D^{×}(F_{v})}(V_{1}^{D} ⊗ V_{2}^{D} ⊗ V_{3}^{D}, ℂ) = 1.

Remark 2. (1) When F_{v} = ℝ, it is assumed in [Pr] that at least one of the representations (π_{i}, V_{i}) (i = 1, 2, 3) belongs to the discrete series of representations of GL_{2}(F_{v}).

(2) When all of the representations (π_{i}, V_{i}) (i = 1, 2, 3) are supercuspidal, then the existence of the GL_{2}(F_{v}) (or D^{×}(F_{v})) invariant trilinear functionals can be determined by the corresponding local ϵ-factors. This is the local evidence for the Jacquet conjecture.

1.3. The Work of M. Harris and S. Kudla. Based on the work of P. Garrett and the work of I. Piatetski-Shapiro and S. Rallis on the integral representation for the triple product L-function, Harris and Kudla established in [HK] the ‘Main Identity’, which relates the central critical value of the triple product L-function and the trilinear period integrals, by means of Seesaw duality, Siegel-Weil formula, and a theorem of Shimizu. The Jacquet conjecture follows from the ‘Main Identity’ if one knows explicitly the analytic properties of the relevant local zeta integrals, which was determined in [HK] for the cases when the Ramanujan conjecture is valid.

2. Approach by means of Periods of Residual Representations

We shall provide an approach by means of constant terms of Eisentein series and the periods of residual representations.

2.1. G_{2}-periods of Residual Representations. The study of periods of residual representations associated to Eisenstein series was initiated by Jacquet and Rallis in

[JR] for the case of (GL(2n), Sp(2n)), which was motivated by the devolopment of various trace formulas for symmetric spaces. In [Jng], some cases of non-symmetric space, spherical type was computed explicitly, essentially following the idea in [JR]. The following is one of the cases from [Jng].

Let G = P GO(4, 4) and P = M_{2}N_{2} the standard maximal parabolic subgroup of G with M_{2} = GL_{2} × P GO(2, 2). Let π_{i}, i = 1, 2, 3, be irreducible cuspidal automorphic representations of GL_{2}(𝔸) with ω_{π_{i}} = 1. Then the tensor product σ = π_{1} ⊗ π_{2} ⊗ π_{3} can be viewed as an irreducible cuspidal automorphic representation of M_{2}(𝔸). From the cuspidal data (M_{2}, σ), one can form an Eisenstein series as follows:

E(g; Φ_{σ,s}, s) := ∑_{γ∈P \G} Φ_{σ,s}(γg).

It is known from Langlands theory of Eisenstein series that the Eisenstein series E(g; Φ_{σ,s}, s) attached to the cuspidall data (M_{2}, σ) defines a meromorphic function in s over ℂ and has finitely many possible simple poles when the real part of s is non-negative.

Let H = G_{2} be the split group of type G_{2}, which can be embedded into G via triality. We are interested in the following period integral

P_{G_{2}}(res_{s=s_{0}}E(·, Φ_{σ,s}, s)) := ∫_{G_{2}(F)\G_{2}(𝔸)} res_{s=s_{0}}E(h; Φ_{σ,s}, s)dh.

Theorem 2 ([Jng]). With notations as above, we have

(1) If there is a Φ_{σ,s} such that P_{G_{2}}(res_{s=s_{0}}E(·, Φ_{σ,s}, s)) ≠ 0, then s_{0} = \frac{1}{2}.

(2) There is a Φ_{σ,s} such that P_{G_{2}}(res_{s= \frac{1}{2}} E(·, Φ_{σ,s}, s)) ≠ 0 if and only if there exists ϕ_{π_{i}} ∈ V_{i} for i = 1, 2, 3, such that the following period integral

∫_{P GL_{2}(F)\P GL_{2}(𝔸)} ϕ_{π_{1}}(g)ϕ_{π_{2}}(g)ϕ_{π_{3}}(g)dg ≠ 0.

Corollary 1. Let π_{i}, i = 1, 2, 3, be irreducible cuspidal automorphic representations of GL_{2}(𝔸) with trivial central character, such that the residue at s = \frac{1}{2} of the Eisenstein series E(g; Φ_{σ,s}, s) has non-zero G_{2}-period P_{G_{2}}(res_{s= \frac{1}{2}} E(·, Φ_{σ,s}, s)). Then, for each local place v of the number field F, at least one of the three local components at v, π_{1,v}, π_{2,v} and π_{3,v}, of π_{1}, π_{2} and π_{3}, respectively, does not belong to the discrete series of representations of GL_{2}(F_{v}).

Proof. By the second part of the Theorem, the period gives rise to a nontrivial global GL_{2}(𝔸)-invariant trilinear functional over the space V_{1} ⊗V_{2} ⊗V_{3}. Then, after restricting to each local place v, we have a nontrivial GL_{2}(F_{v})-invariant trilinear functional over the space V_{1,v} ⊗ V_{2,v} ⊗ V_{3,v}. Finally, the claimed result follows from the theorem of Prasad.

2.2. Constant Term of Eisenstein Series. By Langlands theory of constant terms of Eisenstein series [L], we can relate our nonvanishing result of the G_{2}-period of the residual representations to the nonvanishing of the central critical value of the triple product L-function L(s, π_{1} ⊗ π_{2} ⊗ π_{3}).

We consider the constant term along the maximal parabolic subgroup P_{2} of the Eisentein series E(g; Φ_{σ,s}, s), which is given by

E_{P_{2}}(g; Φ_{σ,s}, s) = Φ_{σ,s}(g) + M_{σ,s}(Φ_{σ,s})(g),

where M_{σ,s}(Φ_{σ,s}) is the intertwining operator associated the longest Weyl group element in this situation. Then the nonvanishing of the residue at s = \frac{1}{2} of the Eisenstein series E_{P_{2}}(g; Φ_{σ,s}, s) is equivalent to the nonvanishing of the residue at s = \frac{1}{2} of M_{σ,s}(Φ_{σ,s})(g).

Following the theory of Langlands-Shahidi [Sh], the local c-function at each unramified local place v of the local component of the intertwining operator M_{σ,s}(Φ_{σ,s}) is

\frac{L(s, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v})ζ_{v}(2s)}{L(s + 1, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v})ζ_{v}(2s + 1)}.

Then, by proving a theorem of Casselman-Shahidi type [CS] on the normalization of the local intertwining operators, we obtain that the nonvanishing of the residue at s = \frac{1}{2} of M_{σ,s}(Φ_{σ,s})(g) is equivalent to the nonvanishing of the value at s = \frac{1}{2} of the triple product L-function L(s, π_{1} ⊗ π_{2} ⊗ π_{3}).

Combining with the theorem in [Jng], we will have the following theorem.

Theorem 3 ([Jng1]). The nonvanishing of the period integral

∫_{P GL_{2}(F)\P GL_{2}(𝔸)} ϕ_{π_{1}}(g)ϕ_{π_{2}}(g)ϕ_{π_{3}}(g)dg

for some ϕ_{π_{i}} ∈ V_{i} for i = 1, 2, 3 implies that

(1) for each local place v of the number field F, at least one of the three local components at v, π_{1,v}, π_{2,v} and π_{3,v}, of π_{1}, π_{2} and π_{3}, respectively, does not belong to the discrete series of representations of GL_{2}(F_{v});

(2) the nonvanishing of the value at s = \frac{1}{2} of the triple product L-function L(s, π_{1} ⊗ π_{2} ⊗ π_{3}).

Remark 3. This theorem can be viewed as one direction of the conjecture of Jacquet when S is an empty set. The other direction is more involved and the work is in progress. It will be very interesting to formulate a similar approach which is able to deal with the general case when S is not empty.

References

[CS] W. Casselman and F. Shahidi, On irreducibility of standard modules for generic representations, preprint, 1997.

[G] P. B. Garrett, Decomposition of Eisenstein series: Rankin triple products. Ann. Math., 125(1987), 209–235.

[GP] B. Gross and D. Prasad, On the decomposition of a representation of SO(n) when restricted to SO(n − 1), Canad. J. Math. 44(1992)(5), 974–1002.

[HK] M. Harris and S. Kudla, The central criticaal value of a triple L-function, Ann. of Math., 133(1991)(3), 605–672.

[JR] H. Jacquet and S. Rallis, Symplectic periods, J. reine angew. Math., 423(1992), 175–197.

[Jng] D. Jiang, G_{2}-periods and Residual Representations, preprint (1997).

[Jng1] D. Jiang, Non-vanishing of the central critical value of the triple product L-functions, in preparation.

[L] R. Langlands, On the functional equation satisfied by Eisenstein series, Lecture Notes in Math., Vol. 544, Springer-Verlag, 1976.

[PSR] I. Piatetski-Shapiro and S. Rallis, Rankin triple L-functions, Compositio Mathematica 64 (1987),31–115.

[Pr] D. Prasad, Trilinear forms for representations of GL(2) and local ϵ-factors, Compos. Math., 75(1990(1), 1–46.

[Sh] F. Shahidi, On the Ramanujan conjecture and the finiteness of poles of certain Lfunctions, Ann. of Math. (2)(1988)(127), 547–584.

Department of Mathematics, Yale University, New Haven, CT 06520-8283