On a conjecture of Jacquet by Michael Harris and Stephen S. Kudla

For Joe Shalika, with our admiration and appreciation.

Introduction.

Let k be a number field and let π_{i}, i = 1, 2, 3 be cuspidal automorphic representations of GL_{2}(𝔸) such that the product of their central characters is trivial. Jacquet then conjectured that the central value L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) of the triple product L–function is nonzero if and only if there exists a quaternion algebra B over k and automorphic forms f_{i}^{B} ∈ π_{i}^{B} such that the integral

(0.1)  I(f_{1}^{B}, f_{2}^{B}, f_{3}^{B}) = ∫_{Z(𝔸)B^{×}(k)\B^{×}(𝔸)} f_{1}^{B}(b)f_{2}^{B}(b)f_{3}^{B}(b) d^{×}b ≠ 0,

where π_{i}^{B} is the representation of B^{×}(𝔸) corresponding to π_{i} via the Jacquet-Langlands correspondence.

In a previous paper [4], we proved this conjecture in the special case where k = ℚ and the π_{i}’s correspond to a triple of holomorphic newforms. Our method was based on a combination of the Garrett, Piatetski-Shapiro, Rallis integral representation of the triple product L-function with the extended Siegel–Weil formula and the seesaw identity. The restriction to holomorphic newforms over ℚ arose from (i) the need to invoke the Ramanujan Conjecture to control the poles of some bad local factors and (ii) the use of a version of the Siegel–Weil formula for similitudes. In this note, we show that, thanks to the recent improvement on the Ramanujan bound due to Kim–Shahidi [11], together with a slight variation in the setup of (ii), our method yields Jacquet’s conjecture in general.

Since the exposition in [4] was specialized from the start to the case of interest for certain arithmetic applications, we will briefly sketch the method in general in the first few sections. We then prove the facts required about the extended Siegel–Weil formula.

Several authors have considered interpretations of the vanishing of the central value L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}). Here we mention only the work of Dihua Jiang, [9], who gave an intriguing relation with a period of an Eisenstein series on G_{2}.

The authors would like to thank the IHP in Paris where this project was realized during the special program on ‘geometric aspects of automorphic forms’ in June of 2000. We also thank the referee for helpful comments, and in particular for

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reminding us that the completion of the proof of Proposition 5.2 made implicit use of recent results of Loke [16].

§1. The integral representation of the triple product L–function.

Let G = GSp_{6} be the group of similitudes of the standard 6 dimensional symplectic vector space over k, and let P = MN be the Siegel parabolic subgroup of G. For a ∈ GL_{3}, b ∈ Sym_{3} and ν a scalar, let

(1.1) m(a) = ( \begin{matrix}a \\ & ^{t}a^{−1} \end{matrix} ) , n(b) = \binom{1 b}{1} , and d(ν) = ( \begin{matrix}1 \\ & ν \end{matrix} ) ∈ G.

Let K_{G} = K_{G,∞} · K_{G,f} be the standard maximal compact subgroup of G(𝔸). For s ∈ ℂ, let λ_{s} be the character of P (𝔸) defined by

(1.2)  λ_{s}(d(ν)n(b)m(a)) = |ν|^{−3s} | det(a)|^{2s}.

Let I(s) = I_{P}^{G}(λ_{s}) be the normalized induced representation of G(𝔸), consisting of all smooth K_{G}–finite functions Φ_{s} on G(𝔸) such that

(1.3)  Φ_{s}(d(ν)n(b)m(a)g, s) = |ν|^{−3s−3} | det(a)|^{2s+2} Φ_{s}(g).

The Eisenstein series associated to a section Φ_{s} ∈ I(s) is defined for Re(s) > 2 by

(1.4)  E(g, s, Φ_{s}) = ∑_{γ∈P (k)\G(k)} Φ_{s}(γg),

and the normalized Eisenstein series is

(1.5)  E^{∗}(g, s, Φ_{s}) = b_{G}(s) · E(g, s, Φ_{s}),

where b_{G}(s) = ζk(2s + 2) ζk(4s + 2), as in [17]. Note that the central character of E(g, s, Φ_{s}) is trivial. These functions have meromorphic analytic continuations to the whole s–plane and have no poles on the unitary axis Re(s) = 0. In particular, the map

(1.6)  E^{∗}(0) : I(0) ⟶ A(G),  Φ_{0} ↦ (g ↦ E^{∗}(g, 0, Φ_{s}))

gives a (𝔤_{∞}, K_{G,∞})×G(𝔸_{f})–intertwining map from the induced representation I(0) at s = 0 to the space of automorphic forms on G with trivial central character.

Let

(1.7)  G = (GL_{2} × GL_{2} × GL_{2})_{0}
= {(g_{1}, g_{2}, g_{3}) ∈ (GL_{2})^{3} | det(g_{1}) = det(g_{2}) = det(g_{3})}.

This group embeds diagonally in G = GSp_{6}. For automorphic forms f_{i} ∈ π_{i}, i = 1, 2, 3, let F = f_{1} ⊗ f_{2} ⊗ f_{3} be the corresponding function on G(𝔸). The global zeta integral [17] is given by

(1.8)  Z(s, F, Φ_{s}) = ∫_{Z_{G}(𝔸)G(k)\G(𝔸)} E^{∗}(g, s, Φ_{s}) F (g) dg.

Suppose that the automorphic forms f_{i} ∈ π_{i} have factorizable Whittaker functions W_{i}^{ψ} = ⊗_{v}W_{i,v}^{ψ} and that the section Φ_{s} is factorizable. Let S be a finite set of places of k, including all archimedean places, such that, for v ∉ S,

(i) the fixed additive character ψ of 𝔸/k has conductor O_{k,v} at v.

(ii) π_{i,v} is unramified, f_{i} is fixed under K_{v} = GL_{2}(O_{k,v}), and W_{i,v}^{ψ}(e) = 1. (iii) Φ_{s,v} is right invariant under G(O_{k,v}) = K_{G,v} and Φ_{s,v}(e) = 1,

Then

(1.9)  Z(s, F, Φ_{s}) = L^{S}(s + \frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) · ∏_{v∈S} Z_{v}(s, W_{v}^{ψ}, Φ_{s,v}),

for local zeta integrals Z_{v}(s, W_{v}^{ψ}, Φ_{s,v}), where W_{v}^{ψ} = W_{1,v}^{ψ} ⊗ W_{2,v}^{ψ} ⊗ W_{3,v}^{ψ}. Here

(1.10)  Z(s, W_{v}^{ψ}, Φ_{s,v}) = ∫_{Z_{G}(k_{v})M(k_{v})\G(k_{v})} Φ_{s,v}(δ g) W_{v}^{ψ}(g) dg,

where δ ∈ G(k) is a representative for the open orbit of G in P \G, cf. for example [2], and

(1.11)  M = \{(\binom{1 x_{1}}{1} , \binom{1 x_{2}}{1} , \binom{1 x_{3}}{1}) ∈ G | x_{1} + x_{2} + x_{3} = 0\}.

Here L^{S}(s, π_{1} ⊗ π_{2} ⊗ π_{3}) is the triple product L-functions with the factors for v ∈ S omitted.

§2. Local zeta integrals.

In this section, we record some consequences of recent results of Kim and Shahidi [11] on the Ramanujan estimate for the π_{i}’s. We begin by recalling relevant aspects of the local theory of the triple product, as recently completed by Ikeda and Ramakrishnan. In the following proposition by “local Euler factor” at a finite place v of k we mean a function of the form P (q_{v}^{−s})^{−1}, where P is a polynomial, P (0) = 1, and q_{v} is the order of the residue field; at an archimedean field we mean a finite product of Tate’s local Euler factors for GL(1).

Proposition 2.1. Let v be a place of k and let π_{i,v}, i = 1, 2, 3, be a triple of admissible irreducible representations of GL(2, k_{v}) that arise as local components at v of cuspidal automorphic representations π_{i}.

(i) There exists a local Euler factor L(s, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v}) such that, for any local data (W_{v}^{ψ}, Φ_{s,v}), the quotient

Z̃_{v}(s, W_{v}^{ψ}, Φ_{s,v}) = Z_{v}(s, W_{v}^{ψ}, Φ_{s,v}) · L(s + \frac{1}{2}, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v})^{−1}

is entire as a function of s.

(ii) Let σ_{i,v}, i = 1, 2, 3, be the representations of the Weil-Deligne group of k_{v} associated to π_{i,v} by the local Langlands correspondence. Then

L(s, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v}) = L(s, σ_{1,v} ⊗ σ_{2,v} ⊗ σ_{3,v}).

(iii) For any finite place v, there is a local section Φ_{s,v} and a Whittaker function W_{v}^{ψ} = W_{1,v}^{ψ} ⊗ W_{2,v}^{ψ} ⊗ W_{3,v}^{ψ}, such that

Z(s, Φ_{s,v}, W_{v}^{ψ}) ≡ 1.

(iv) For any archimedean place v, there exists a finite collection of Whittaker functions W_{v}^{ψ,j} and of sections Φ_{s,v}^{j}, holomorphic in a neighborhood of s = 0 such that

∑_{j} Z(0, Φ_{s,v}^{j}, W_{v}^{ψ,j}) = 1.

Proof. For v non-archimedean, assertion (i) is proved in §3, Appendix 3, of [17]; see [7], p. 227 for a concise statement. For v real or complex, (i) and (ii) were proved in several steps by Ikeda, of which the crucial one is [8], Theorem 1.10. Assertion (ii) in general is due to Ramakrishnan, [21], Theorem 4.4.1. For the moment, the hypothesis that the π_{i,v} embed in global cuspidal representations seems to be necessary.

Assertions (iii) and (iv) are contained in Proposition 3.3 of [17]. □

Proposition 2.2. (i) For any triple π_{i} of cusp forms for GL_{2} over k, and for any place v, the local Langlands L–factor L(s, π_{1,v} ⊗ π_{2,v} ⊗ π_{3,v}) is holomorphic at s = \frac{1}{2}. (ii) For any place v, for any triple of Whittaker functions W_{i,v}^{ψ} in the Whittaker spaces of π_{i,v}, and for any section Φ_{s,v} ∈ I_{v}(s), holomorphic in a neighborhood of s = 0, the local zeta integral Z(s, W_{v}^{ψ}, P_{s,v}) is holomorphic in a neighborhood of s = 0.

Proof. This follows from the results of Kim and Shahidi. We sketch the simple argument, quoting the proof of Proposition 3.3.2 of [21]. Let σ_{i,v} correspond to π_{i,v} as in the previous proposition. For the present purposes we can assume each π_{i} to be unitary. Indeed, this can be arranged by twisting π_{i} by a (unique) character of the form | · |^{a_{i}}, where | · | is the idèle norm and a_{i} ∈ ℂ. Since the product of the central characters of π_{i} is trivial, we have a_{1} + a_{2} + a_{3} = 0, so the triple product L-factor is left unaffected.

To each π_{i,v}, necessarily generic and now assumed unitary, we can assign an index λ_{i,v} which measures the failure of π_{i,v} to be tempered; we have λ_{i,v} = t if π_{i,v} is a complementary series attached to (μ| · |^{t}, μ| · |^{−t}) with t > 0 and μ unitary, λ_{i,v} = 0 otherwise. Then, according to [21], (3.3.10),

(2.1) L(s, π_{1,v}⊗π_{2,v}⊗π_{3,v}) is holomophic for Re(s) > λ(π_{1,v})+λ(π_{2,v})+λ(π_{3,v}).

Now (i) follows from (2.1) and the Kim-Shahidi estimate λ(π_{i,v}) < \frac{5}{34} for all i and all v [11], whereas (ii) follows from (i), and Proposition 2.1 (i) and (ii). □

By (1.8), (1.9), and the holomorphy of E^{∗}(g, s, Φ) on the unitary axis, the expression

(2.2)  L^{S}(s + \frac{1}{2},π_{1} ⊗ π_{2} ⊗ π_{3}) · ∏_{v∈S} Z_{v}(s, F, Φ_{s,v})

= ∫_{Z_{G}(𝔸)G(k)\G(𝔸)} E^{∗}(g, s, Φ_{s}) F (g) dg.

is holomorphic at s = 0 for all choices of data F and Φ_{s}. By varying the data for places in S and applying (iii) and (iv) of Proposition 2.1, it follows that the partial Euler product L^{S}(s + \frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) is holomorphic at s = 0. By (i) of Proposition 2.2, the Euler product L(s + \frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) over all finite places is holomorphic at s = 0, and we obtain the identity

(2.3)  L(\frac{1}{2},π_{1} ⊗ π_{2} ⊗ π_{3}) · ∏_{v∈S} Z_{v}^{∗}(0, W_{v}^{ψ}, Φ_{s,v})

= ∫_{Z_{G}(𝔸)G(k)\G(𝔸)} E^{∗}(g, 0, Φ_{s}) F (g) dg.

where

(2.4)  Z_{v}^{∗}(s, W_{v}^{ψ}, Φ_{s,v}) = \{ \begin{matrix}Z̃_{v}(s, W_{v}^{ψ}, Φ_{s,v}) & if v ∈ S_{f}, \\ Z_{v}(s, W_{v}^{ψ}, Φ_{s,v}) & if v ∈ S_{∞}. \end{matrix}

Corollary 2.3. L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) = 0 if and only if

∫_{Z_{G}(𝔸)G(k)\G(𝔸)} E^{∗}(g, 0, Φ_{s}) F (g) dg = 0,

for all choices of F ∈ Π = π_{1} ⊗ π_{2} ⊗ π_{3} and Φ_{s} ∈ I(s).

Of course, relation (2.3) gives a formula for L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) for a suitable choice of F and Φ_{s}.

§3. The Weil representation for similitudes.

The material of this section is a slight variation on that of section 5 of [3]. We consider only the case of the dual pair (GO(V), GSp_{6}) where the space V has square discriminant.

Let B be a quaternion algebra over k (including the possibility B = M_{2}(k)), and let V = B be a 4 dimensional quadratic space over k where the quadratic form is given by Q(x) = α ν_{B}(x), where ν_{B} is the reduced norm on B and α ∈ k^{×}. Note that the isomorphism class of V is determined by B and the sign of α at the set Σ_{∞}(B) of real archimedean places of k at which B is division. Let H = GO(V) and let H_{1} = O(V) be the kernel of the scale map ν : H → 𝔾_{m}. Let G = GSp_{6}, and let G_{1} = Sp_{6} be the kernel of the scale map ν : G → 𝔾_{m}. We want to extend the standard Weil representation ω = ω_{ψ} of H_{1}(𝔸) × G_{1}(𝔸) on the Schwartz space S(V (𝔸)^{3}). First, there is a natural action of H(𝔸) on S(V (𝔸)^{3}) given by

(3.1)  L(h)φ(x) = |ν(h)|^{−3} φ(h^{−1}x).

For g_{1} ∈ G_{1}(𝔸) one has

(3.2)  L(h)ω(g_{1})L(h)^{−1} = ω(d(ν)g_{1}d(ν)^{−1}),

where ν = ν(h), and d(ν) is as in section 1. Therefore, one obtains a representation of the semidirect product H(𝔸) ⋉ G_{1}(𝔸) on S(V (𝔸)^{3}). Let

(3.3)  R = {(h, g) ∈ H × G | ν(h) = ν(g)}.

Then there is an isomorphism

(3.4)  R ⟶ H ⋉ G_{1},  (h, g) ↦ (h, d(ν(g))^{−1}g) = (h, g_{1}),

(this defines a map g ↦ g_{1}) and a representation of R(𝔸) on S(V (𝔸)^{3}) given by

(3.5)  ω(h, g)φ(x) = (L(h)ω(g_{1})φ)(x) = |ν(h)|^{−3}(ω(g_{1})φ)(h^{−1}x).

The theta distribution Θ on S(V (𝔸)^{3}) is invariant under R(k), since, for (h, g) ∈ R(k),

Θ(ω(h, g)φ) = ∑_{x∈V (k)^{3}} |ν(h)|^{−3}(ω(g_{1})φ)(h^{−1}x)

= ∑_{x∈V (k)^{3}}(ω(g_{1})φ)(x) (3.6)

= Θ(ω(g_{1})φ) = Θ(φ),

since g_{1} ∈ G_{1}(k). The theta kernel, defined for (h, g) ∈ R(𝔸) by

(3.7)  θ(h, g; φ) = ∑_{x∈V (k)^{3}} ω(h, g)φ(x),

is thus left R(k) invariant.

Remark 3.1. Asside from a shift in notation, the convention here is essentially the same as in section 5 of [3] and section 3 of [4], except that we take pairs (h, g) here versus (g, h) there. Compare (3.5) above with (5.1.5) of [3]. It turns out that this seemingly slight shift in convention will be crucial for the extension of the Siegel–Weil formula to similitudes, as we will see below.

Note that the set of archimedean places Σ_{∞}(B) introduced above is the set of all real archimedean places of k at which V is definite (positive or negative). Then,

(3.8)  G(𝔸)^{+} := {g ∈ G(𝔸) | ν(g) ∈ ν(H(𝔸))}
= {g ∈ G(𝔸) | ν(g)_{v} > 0, ∀v ∈ Σ_{∞}(V)}.

For g ∈ G(𝔸)^{+}, and φ ∈ S(V (𝔸)^{3}), and for V anisotropic over k, i.e., for B ≠ M_{2}(k), the theta integral is defined by

(3.9)  I(g, φ) = ∫_{H_{1}(k)\H_{1}(𝔸)} θ(h_{1}h, g; φ) dh_{1},

where h ∈ H(𝔸) with ν(h) = ν(g). It does not depend on the choice of h.

In the case B = M_{2}(k), the theta integral must be defined by regularization. If k has a real place, the procedure outlined on p.621 of [4], [15], using a certain differential operator to kill support, can be applied. An analogous procedure using an element of the Bernstein center can be applied at a nonarchimedean place, [25]. We omit the details.

Lemma 3.2. (i) (Eichler’s norm Theorem) If α ∈ ν(H(𝔸))∩k^{×}, then there exists an element h ∈ H(k) with ν(h) = α. (ii) The theta integral is left invariant under G(𝔸)^{+} ∩ G(k). (iii) The theta integral has trivial central character, i.e., for z ∈ Z_{G}(𝔸) ⊂ G(𝔸)^{+}, I(zg, φ) = I(g, φ).

Proof. (i) is a standard characterization of ν(H(k)) in the present case. To check (ii), given γ ∈ G(𝔸)^{+} ∩ G(k), choose γ′ ∈ H(k) with ν(γ′) = ν(γ). Then

I(γg, φ) = ∫_{H_{1}(k)\H_{1}(𝔸)} θ(h_{1}γ′h, γg; φ) dh_{1}

= ∫_{H_{1}(k)\H_{1}(𝔸)} θ(γ′h_{1}h, γg; φ) dh_{1} (3.10)

= I(g, φ),

via the left invariance of the theta kernel under (γ′, γ) ∈ R(k). Here, in the next to last step, we have conjugated the domain of integration H_{1}(k)\H_{1}(𝔸) by the element γ′ ∈ H(k).

Finally, the proof of (iii) is just like that of Lemma 5.1.9 (ii) in [3]. □

Since G(𝔸) = G(k)G(𝔸)^{+}, it follows that I(g, φ) has a unique extension to a left G(k)–invariant function on G(𝔸). Moreover, for any g_{0} ∈ G(𝔸)^{+}, we have

(3.11)  I(gg_{0}, φ) = I(g, ω(h_{0}, g_{0})φ),

where h_{0} ∈ H(𝔸) with ν(h_{0}) = ν(g_{0}). In particular, if h_{1} ∈ H_{1}(𝔸), then

(3.12)  I(g, ω(h_{1})φ) = I(g, φ).

§4. The Siegel–Weil formula for (GO(V), GSp_{6}).

First we recall the Siegel–Weil formula for (O(V), Sp_{6}). The results of [15] on the regularized Siegel–Weil formula were formulated over a totally real number field, since, at a number of points, we needed facts about degenerate principal series, intertwining operators, etc. which had not been checked for complex places. The proof in the case of the central value of the Siegel–Eisenstein series is simpler than the general case, and the additional facts needed at complex places are easy to check. In the rest of this section, we will state the results for an arbitrary number field k. A sketch of the proof of Theorem 4.1 below for such a field k will be given in the Appendix below.

Let I_{1}(s) = I_{P_{1}}^{G_{1}}(λ_{s}) be the global induced representation of G_{1}(𝔸) = Sp_{6}(𝔸) induced from the restriction of the character λ_{s} of P (𝔸) to P_{1}(𝔸) = P (𝔸) ∩ G_{1}(𝔸).

For a global quadratic space V of dimension 4 over k associated to a quaternion algebra B, as in the previous section, there is a (𝔤_{1,∞}, K_{G_{1},∞})×G_{1}(𝔸_{f})–equivariant map

(4.1)  S(V (𝔸)^{3}) ⟶ I_{1}(0),  φ ↦ [φ],

where

(4.2)  [φ](g_{1}) = (ω(g_{1})φ)(0).

The image, Π_{1}(V), is an irreducible summand of the unitarizable induced representation I_{1}(0). By the results of Rallis [19], Kudla–Rallis [14], [13], and the appendix,

(4.3)  Π_{1}(V) ≃ S(V (𝔸)^{3})_{O(V)(𝔸)},

the space of H_{1}(𝔸) = O(V)(𝔸)–coinvariants. One then has a decomposition

(4.4)  I_{1}(0) = ( ⊕_{V} Π_{1}(V)) ⊕ ( ⊕_{V} Π_{1}(V)),

into irreducible representations of G_{1}(𝔸), as V runs over the isomorphism classes of such spaces and as V runs over the incoherent collections, obtained by switching one local component of a Π_{1}(V), cf. [12].

The Siegel–Weil formula of [4], asserts the following in the present case.

Theorem 4.1. (i) The (𝔤_{1,∞}, K_{G_{1},∞}) × G_{1}(𝔸_{f})–intertwining map

E_{1}(0) : I_{1}(0) ⟶ A(G_{1}), Φ_{0} ↦ (g_{1} ↦ E(g_{1}, 0, Φ_{s}))

has kernel ⊕_{V}Π_{1}(V). (ii) For a section Φ_{s} ∈ I_{1}(s) with Φ_{0} = [φ] for some φ ∈ S(V (𝔸)^{3}),

(SW)  E(g_{1}, 0, Φ_{s}) = 2 I(g_{1}, φ),

for the theta integral as defined in §3.

As explained in the previous section, the theta integral can be extended to an automorphic form on G(𝔸). We will see presently that it coincides with an Eisenstein series on G(𝔸).

Restriction of functions from G(𝔸) to G_{1}(𝔸) yields an isomorphism I(s) ⟶^{∼} I_{1}(s), which is intertwining for the right action of G_{1}(𝔸). Here I(s) is the induced representation of G(𝔸) defined in section 1. The inverse map is given by Φ_{s} ↦ Φ_{s}^{∼} where

(4.5)  Φ_{s}^{∼}(g) = |ν(g)|^{−3s−3}Φ_{s}(g_{1}),

for g_{1} = d(ν(g))^{−1}g, as above. The decomposition (4.4) into G_{1}(𝔸)–irreducibles yields a decomposition

(4.6)  I(0) = ( ⊕_{B} Π(B)) ⊕ ( ⊕_{B} Π(B)),

into irreducible representations of G(𝔸), where, for a global quaternion algebra B over k,

(4.7)  Π(B) = ⊕_{V} Π(V)

where V runs over the non-isomorphic spaces associated to B (i.e., different multiples of the norm form) and Π(V) denotes the image of Π_{1}(V) under the inverse of the restriction isomorphism. Note that there are 2^{|Σ_{∞}(B)|} such V ’s. In effect, at a real archimedean place v, the local induced representation I_{1}(0)_{v} has a decomposition into irreducible (𝔤_{1,v}, K_{G_{1},v})–modules

(4.8)  I(0)_{v} = Π(4, 0)_{v} ⊕ Π(2, 2)_{v} ⊕ Π(0, 4)_{v}

according to signatures. The space Π(2, 2)_{v} is actually stable under (𝔤_{v}, K_{G,v}), as is the sum Π(4, 0)_{v} ⊕ Π(0, 4)_{v}, and

(4.9)  Π(B)_{v} = \{ \begin{matrix}Π(2, 2)_{v} & if B_{v} ≃ M_{2}(ℝ). \\ Π(4, 0)_{v} ⊕ Π(0, 4)_{v} & if B_{v} is division. \end{matrix}

The sumands Π(B) are defined similarly.

Theorem 4.2. (i) The (𝔤_{∞}, K_{G,∞}) × G(𝔸_{f})–intertwining map

E(0) : I(0) ⟶ A(G), Φ_{s} ↦ (g ↦ E(g, 0, Φ_{s}))

has kernel ⊕_{B}Π(B). (ii) For a section Φ_{s} ∈ I(s) with Φ_{0} ∈ Π(V) so that Φ_{0} = [φ]^{∼} for some φ ∈ S(V (𝔸)^{3}),

(GSW)  E(g, 0, Φ_{s}) = 2 I(g, φ),

for the theta integral as defined in §3.

Proof. For g_{0} ∈ G(𝔸_{f}), we have

(4.10)  E(gg_{0}, s, Φ_{s}) = E(g, s, r_{s}(g_{0})Φ_{s}),

where r_{s} denotes the action in the induced representation I(s) by right translation. Taking the value at s = 0, we obtain

(4.11)  E(gg_{0}, 0, Φ_{s}) = E(g, 0, r_{s}(g_{0})Φ_{s}).

Note that this value depends only on Φ_{0} and r_{0}(g_{0})Φ_{0}.

Lemma 4.3. For φ ∈ S(V (𝔸)^{3}), let [φ] ∈ I_{1}(0) be defined by (4.2) and let [φ]^{∼} be the corresponding function in I(0) under the inverse of the restriction isomorphism.

(i) For g ∈ G(𝔸)^{+},

[φ]^{∼}(g) = (ω(h, g)φ)(0),

where h ∈ GO(V)(𝔸) with ν(h) = ν(g). (ii) For g_{0} ∈ G(𝔸_{f}),

r_{0}(g_{0})[φ]^{∼} = [ω(h_{0}, g_{0})φ]^{∼},

where h_{0} ∈ GO(V)(𝔸_{f}) with ν(h_{0}) = ν(g_{0}). Proof. For (i), we have

[φ]^{∼}(g) = [φ]^{∼}(d(ν)g_{1})
= |ν|^{−3}[φ](g_{1})

= |ν|^{−3}(ω(g_{1})φ)(0) (4.12)

= (L(h)ω(g_{1})φ)(0)
= (ω(h, g)φ)(0).

For (ii),

(r_{0}(g_{0})[φ]^{∼})(g) = |ν|^{−3}[φ]^{∼}(g_{1}g_{0})
= |ν|^{−3}(ω(h_{0}, g_{1}g_{0})φ)(0)

= |ν|^{−3}(ω(g_{1})ω(h_{0}, g_{0})φ)(0) (4.13)

= |ν|^{−3}[ω(h_{0}, g_{0})φ](g_{1})
= [ω(h_{0}, g_{0})φ]^{∼}(g).

□ Thus, if Φ_{0} = [φ]^{∼}, then r_{0}(g_{0})Φ_{0} = [ω(h_{0}, g_{0})φ]^{∼}. Since G(𝔸) = G(k)Z_{G}(𝔸)G_{1}(𝔸)G(𝔸_{f}), we have, by (4.11),

(4.14)  E(g, 0, Φ_{s}) = E(γzg_{1}g_{0}, 0, Φ_{s})
= E(g_{1}, 0, r_{s}(g_{0})Φ_{s}),

the value at g_{1} of the Siegel–Eisenstein series attached to ω(h_{0}, g_{0})φ ∈ S(V (𝔸)^{3}). On the other hand, by (3.11),

(4.15)  I(g, φ) = I(γzg_{1}g_{0}, φ)
= I(g_{1}, ω(h_{0}, g_{0})φ).

Thus (GSW) follows from (SW). □

§5. Proof of Jacquet’s conjecture.

Applying the Siegel–Weil formula for similitudes to the basic identity (2.3), we obtain

L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) · Z^{∗}(F, Φ)

(5.1) = ∫_{Z_{G}(𝔸)G(k)\G(𝔸)} E^{∗}(g, 0, Φ_{s}) F (g) dg

= 2ζk(2)^{2} ∑_{V} ∫_{Z_{G}(𝔸)G(k)\G(𝔸)} I(g, φ^{V}) F (g) dg.

where

(5.2)  Z^{∗}(F, Φ) = ∏_{v∈S} Z_{v}^{∗}(0, W_{v}^{ψ}, Φ_{s,v}),

and where φ^{V} ∈ S(V (𝔸)^{3}), and, in fact, only a finite set of V ’s occurs in the sum. More precisely, in the decomposition (4.6),

(5.3)  Φ_{0} = ∑_{V} [φ^{V}]^{∼} + terms in the Π(B)’s ∈ I(0),

where the quaternion algebras B associated to V ’s are split outside the set S, due to condition (iii) in the definition of S in section 1. We thus have the following reformulation of Corollary 2.3, generalizing Proposition 5.6 of [4]:

Corollary 5.1. L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) = 0 if and only if

∫_{Z_{G}(𝔸)G(k)\G(𝔸)} I(g, φ^{V}) F (g) dg

vanishes for all choices of F ∈ Π = π_{1} ⊗ π_{2} ⊗ π_{3}, all choices of quadratic spaces V attached to quaternion algebras B over k, and all choices of φ^{V} ∈ S(V (𝔸)^{3}).

Now consider the integral in the last line of (5.1) for a fixed φ = φ^{V}. To apply the seesaw identity, we set

H = GO(V)
(5.4)  H = {(h_{1}, h_{2}, h_{3}) ∈ H^{3} | ν(h_{1}) = ν(h_{2}) = ν(h_{3})},
R = {(h, g) ∈ H × G | ν(h) = ν(g)}
R_{0} = {(h, g) ∈ H × G | ν(h) = ν(g)},

and hence have the seesaw pair:

(5.5)  \begin{aligned}I(·, φ; F) & (GO(V)^{3})_{0} = & H & & & G & = GSp_{6} & I(·, φ) \\ & & ↑ & ↖ & ↗ & ↑ \\ & & | & ↙ & ↘ & | \\ 11 & GO(V) = & H & & & G & = (GL_{2}^{3})_{0} & F \end{aligned}

There are representations of both R(𝔸) and R(𝔸) on S(V (𝔸)^{3}), and the restriction of these representations to the common subgroup R_{0}(𝔸) coincide.

For F a cuspidal automorphic form on G(𝔸) and for h ∈ H(𝔸), let

(5.6)  I(h, φ; F) = ∫_{G_{1}(k)\G_{1}(𝔸)} θ(h, g_{1}g; φ) F (g_{1}g) dg_{1},

where g ∈ G(𝔸) with ν(g) = ν(h).

Lemma 5.2. (Seesaw identity)

∫_{Z_{G}(𝔸)G(k)\G(𝔸)} I(g, φ) F (g) dg = ∫_{Z_{H}(𝔸)H(k)\H(𝔸)} I(h, φ; F) dh.

Proof. Note that Z_{G}(𝔸)G(k)\G(𝔸) ≃ Z_{G}(𝔸)G(k)^{+}\G(𝔸)^{+}, and that

(5.7) Z_{G}(𝔸)G(k)^{+}G_{1}(𝔸)\G(𝔸)^{+} ≃ Z_{H}(𝔸)H(k)\H(𝔸) ≃ 𝔸^{×,2}k^{×,+}\𝔸^{×,+} =: C,

is compact, where 𝔸^{×,+} = ν(H(𝔸)) and k^{×,+} = ν(H(k)). Fixing a Haar measure dc giving C volume 1, we have

∫_{Z_{G}(𝔸)G(k)\G(𝔸)} I(g, φ) F (g) dg

(5.8) = ∫_{C} ∫_{G_{1}(k)\G_{1}(𝔸)} ∫_{H_{1}(k)\H_{1}(𝔸)} θ(h_{1}h(c), g_{1}g(c); φ) F (g_{1}g(c)) dh_{1} dg_{1} dc
= ∫_{Z_{H}(𝔸)H(k)\H(𝔸)} I(h, φ; F) dh,

generalizing the proof of Proposition 7.1.4 of [3]. □

To apply the seesaw identity to the restriction to G(𝔸) of a function F ∈ Π = π_{1} ⊗ π_{2} ⊗ π_{3}, we recall the description, from sections 7 and 8 of [4], of the corresponding space of functions Θ(Π) on H(𝔸) spanned by the I(h, φ; F)’s for F ∈ Π and φ ∈ S(V (𝔸)^{3}). Note that one obtains the same space by fixing a nonzero F and only varying φ [5].

The action of B^{×} × B^{×} on V = B, ρ(b_{1}, b_{2})x = b_{1}xb_{2}^{−1} determines an extension

(5.9)  1 ⟶ 𝔾_{m} ⟶ ( B^{×} × B^{×} ) ⋊ ⟨t⟩ ⟶ H = GO(V) ⟶ 1

where the involution t acts on V by ρ(t)(x) = x^{ι} and on B^{×} × B^{×} by (b_{1}, b_{2}) ↦ (b_{2}^{ι}, b_{1}^{ι})^{−1}. Write

(5.10)  H̃ = ( B^{×} × B^{×} ) ⋊ ⟨t⟩  and  H̃^{0} = B^{×} × B^{×},

and let H̃ and H̃^{0} be the analogous groups for H = (GO(V)^{3})_{0}. Thus, we have the diagram

(5.11)  \begin{aligned}& & Θ̃(Π) & & Θ(Π) \\ H̃^{0} & ↪ & H̃ & ⟶ & H \\ ↑ & & ↑ & & ↑ \\ H̃^{0} & ↪ & H̃ & ⟶ & H \end{aligned}

For an irreducible cuspidal automorphic representation π of GL_{2}(𝔸), let π^{B} be the associated automorphic representation of B^{×}(𝔸) under the Jacquet-Langlands correspondence. We take π^{B} to be zero if π does not correspond to a representation of B^{×}(𝔸). Similarly, let Π^{B} = π_{1}^{B} ⊗ π_{2}^{B} ⊗ π_{3}^{B} be the corresponding representation of B^{×}(𝔸)^{3}, or zero if some factor does not exist. Note that the central character of Π^{B} is trivial, and so, (Π^{B})^{∨} ≃ Π^{B}, where (Π^{B})^{∨} is the contragradient of Π^{B}. Thus we can view the space of of functions Π^{B} on B^{×}(𝔸)^{3} as the automorphic realization of both Π^{B} and its contragredient.

The following result is proved in [4], sections 7 and 8, based on the work of Shimizu and Prasad.

Proposition 5.3. (i) Θ(Π) is either zero or a cuspidal automorphic representation of H(𝔸) and is nonzero if and only if Π^{B} is nonzero. (ii) As spaces of functions on H̃^{0}(𝔸),

Θ̃(Π)|_{H̃^{0}(𝔸)} = (Π^{B} ⊗ (Π^{B})^{∨} )|_{H̃^{0}(𝔸)}.

For fixed F ∈ Π and φ ∈ S(V (𝔸)^{3}), we let Ĩ(·, φ; F) denote the pullback of I(·, φ; F) to H̃(𝔸), and, via (ii) of Proposition 5.3, we write the restriction of this function to H̃^{0}(A) as

(5.12)  Ĩ((b_{1}, b_{2}), φ; F) = ∑_{r} I^{1,r}(b_{1}, φ; F) I^{2,r}(b_{2}, φ; F)

for functions I^{i,r}(·, φ; F) ∈ Π^{B} and b_{i} ∈ B^{×}(𝔸)^{3}. The seesaw then gives

∫_{Z_{G}(𝔸)G(k)\G(𝔸)} I(g, φ) F (g) dg

= ∫_{Z_{H}(𝔸)H(k)\H(𝔸)} I(h, φ; F) dh

= ∫_{ZH̃^{0} (𝔸)H̃(k)\H̃(𝔸)} Ĩ(h, φ; F) dh (5.13)

= ∫_{ZH̃^{0} (𝔸)H̃^{0}(k)\H̃^{0}(𝔸)} Ĩ(h, φ; F) dh

= ∑_{r} ∫_{𝔸^{×}B^{×}(k)\B^{×}(𝔸)} I^{1,r}(b_{1}, φ; F) db_{1} · ∫_{𝔸^{×}B^{×}(k)\B^{×}(𝔸)} I^{2,r}(b_{2}, φ; F) db_{2}

The fact that the integral over ZH̃^{0}(𝔸)H̃(k)\H̃(𝔸) in the third line can be replaced by the integral over ZH̃^{0}(𝔸)H̃^{0}(k)\H̃^{0}(𝔸) is (7.3.2), p.632 of [4]. Its proof in section 8.6, p.636 of [4] depends on Prasad’s uniqueness theorem [18] for invariant trilinear forms. This theorem was recently completed by H. Y. Loke [16], who treated general triples of admissible irreducible representations of GL(2, ℝ) and GL(2, ℂ).^{1} Thus the calculation in (5.13) is valid for all number fields and for all triples of cuspidal automorphic representations.

Finally, we observe that the integrals in the last line of (5.13) are finite linear combinations of the quantities I(f_{1}^{B}, f_{2}^{B}, f_{3}^{B}) of (0.1). By (ii) of Proposition 5.3 , every such quantity can be obtained as an integral ∫_{𝔸^{×}B^{×}(k)\B^{×}(𝔸)} I^{1,r}(b_{1}, φ; F) db_{1} for some φ, F and r.

Jacquet’s conjecture now follows upon combining this observation with Corollary 5.1.

Remark 5.4. In fact, by Prasad’s uniqueness theorem, if the root number

(5.14)  ϵ(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) = 1,

then there is a unique B for which Π^{B} ≠ 0 and for which the space of global invariant trilinear forms on Π^{B} has dimension 1. The automorphic trilinear form is given by integration over 𝔸^{×}B^{×}(k)\B^{×}(𝔸) is then non-zero if and only if L(\frac{1}{2}, π_{1}⊗π_{2}⊗π_{3}) ≠ 0. Choose f_{i}^{B} ∈ π_{i}^{B}, i = 1, 2, 3, such that

(5.15)  I(f_{1}^{B}, f_{2}^{B}, f_{3}^{B}) ≠ 0.

1This is the only place in [4] where Prasad’s uniqueness theorem is used, although it was an important motivation for the article as well as for Jacquet’s conjecture.

For any nonzero F ∈ Π, we can choose φ ∈ S(V (𝔸))^{3} such that

(5.16)  Ĩ((b, b′), φ; F) = f_{1}^{B}(b_{1})f_{2}^{B}(b_{2})f_{3}^{B}(b_{3}) f_{1}^{B}(b′_{1})f_{2}^{B}(b′_{2})f_{3}^{B}(b′_{3}),

where b = (b_{1}, b_{2}, b_{3}) and b′ = (b′_{1}, b′_{2}, b′_{3}). We then obtain

(5.17)  L(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) · Z^{∗}(F, Φ) = 2ζk(2)^{2} I(f_{1}^{B}, f_{2}^{B}, f_{3}^{B})^{2}.

where, Φ is determined by φ, and Z^{∗}(F, Φ) ≠ 0. Of course, this identity is only useful when one has sufficient information about the function φ and the product of local zeta integrals Z^{∗}(F, Φ). This was a main concern in [4].

On the other hand, when the root number ϵ(\frac{1}{2}, π_{1} ⊗ π_{2} ⊗ π_{3}) = −1, then there is no Π^{B} which supports an invariant trilinear form, and the central value of the triple product L-function vanishes due to the sign in the functional equation.

Appendix: The Siegel–Weil formula for general k.

In this appendix, we will sketch the proof of Theorem 4.1 for an arbitrary number field k, indicating the additional facts which are needed when k has complex places.

First, suppose that v is a complex place of k and consider the local degenerate principal series representation I_{1,v}(0) of G_{1,v} = Sp_{3}(ℂ) and the Weil representation of G_{1,v} on S(V_{v}^{3}), where V_{v} ≃ M_{2}(ℂ) with Q(x) = det(x).

Lemma A.1. (i) I_{1,v}(0) is an irreducible unitarizable representation of G_{1,v}. (ii) (Coinvariants) The map S(V_{v}^{3}) → I_{1,v}(0), φ ↦ [φ], analogous to (4.1) induces an isomorphism

S(V_{v}^{3})_{H_{1,v}} ⟶^{∼} I_{1,v}(0).

Here H_{1} = O(V).

Remark: Statement (i) is in Sahi’s paper, [23], Theorem 3A. The proof of (ii) was explained to us by Chen-bo Zhu^{2} [27], and is based on the method of [26].

In the case B = M_{2}(k), the theta integral must be defined by regularization, – cf. the remarks before Lemma 3.2 above. We write

(A.1)  I_{reg}(g_{1}, φ) = \{ \begin{matrix}I(g_{1}, φ) & if V is anisotropic, \\ B_{−1}(g_{1}, φ) & if V is isotropic, \end{matrix}

where B_{−1} is as in (5.5.24) of [15], except that we normalize the auxillary Eisenstein series E(h, s) to have residue 1 at s′_{0}. The key facts which we need are the following.

2He also directed us to [23]. We wish to thank him for his help on these points.

Lemma A.2. (i) The map I_{reg} : S(V (𝔸)^{3}) → A(G_{1}) factors through the space of coinvariants S(V (𝔸)^{3})_{H_{1}(𝔸)} = Π(V). (ii) For all β ∈ Sym_{3}(k),

I_{reg,β}(g_{1}, φ) = \frac{1}{2} · ∫_{H_{1}(𝔸)} ω(g_{1})φ(h^{−1}x) dh,

where x ∈ V (k)^{3} with Q(x) = β.

The second statement here is Corollary 6.11 of [15]; it asserts that the nonsingular Fourier coefficients behave as though no regularization were involved.

Next we have the analogue of Lemma 4.2, p.111 of [20]; the main point of the proof is the local uniqueness, and the ‘submersive set’ argument for the archimedean places carries over for a complex place.

Lemma A.3. For β ∈ Sym_{3}(k) with det(β) ≠ 0, let T_{β} be the space of distributions T ∈ S(V (𝔸)^{3})′ such that

(i) T is H(𝔸)–invariant. (ii) For all b ∈ Sym_{3}(𝔸_{f}),

T (ω(n(b)φ) = ψ_{β}(b) T (φ),

where ψ_{β}(b) = ψ(tr(βb)). (iii) For an archimedean place v of k and for all X ∈ 𝔫 = Lie(N),

T (ω(X)φ) = dψ_{β}(X) · T (φ).

Then T_{β} has dimension at most 1 and is spanned by the orbital integral

T (φ) = ∫_{H_{1}(𝔸)} φ(h^{−1}x) dh,

where x ∈ V (k)^{3} with Q(x) = β. In particular, T_{β} = 0 if an only if there is no such x.

sketch of the Proof of Theorem 4.1. First consider a global space V associated to a quaternion algebra B. We have two intertwining maps

(A.2)  E_{1}(0) : Π(V) ⟶ A(G_{1})  and  I_{reg} : Π(V) ⟶ A(G_{1})

from the irreducible representation Π(V) ≃ S(V (𝔸)^{3})_{H(𝔸)} of G_{1}(𝔸) to the space of automorphic forms. For a nonsingular β ∈ Sym_{3}(k), the distributions obtained by taking the βth Fourier coefficient of the composition of the projection S(V (𝔸)^{3}) → Π(V) with each of the embeddings in (A.2) satisfy the conditions of Lemma A.3 and hence are proportional. In particular, the β-th Fourier of the Eisenstein series vanishes unless β is represented by V. By the argument of pp. 111–115 of [20], the constant of proportionality is independent of β and so there is a constant c such that E_{1}(g, 0, [φ]) − c · I_{reg}(g, φ) has vanishing nonsingular Fourier coefficients. But then the argument at the top of p.28 of [15], cf. also, [20], implies that this difference is identically zero.

In the case of a component Π(V) ⊂ I_{1}(0), the nonsingular Fourier coefficients of E(g, 0, Φ) vanish by the argument on p. 28 of [15], so, again by ‘nonsingularity’ the map E_{1}(0) must vanish on Π(V). □

References

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

[2] B. Gross and S. Kudla, Heights and the central critical values of triple product L-functions, Compositio Math. 81 (1982), 143–209.

[3] M. Harris and S. Kudla, Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2), Duke Math. J. 66 (1992), 59–121.

[4] , The central critical value of a triple product L-function, Annals of Math. 133 (1991), 605–672.

[5] R. Howe and I.I. Piatetski–Shapiro, Some examples of automorphic forms on Sp_{4}, Duke Math. J. 50 (1983), 55–106.

[6] T. Ikeda, On the functional equations of triple L-functions, J. Math. Kyoto Univ. 29 (1989), 175–219.

[7] , On the location of poles of the triple L-functions, Compositio Math. 83 (1992), 187–237.

[8] , On the gamma factor of the triple L-function I, Duke Math. J. 97 (1999), 301–318.

[9] Dihua Jiang, Nonvanishing of the central critical value of the triple product L-functions, Internat. Math. Res. Notices 2 (1998), 73–84.

[10] H. H. Kim and F. Shahidi, Holomorphy of Rankin triple L-functions; special values and root numbers for symmetric cube L-functions, Israel J. Math. 120 (2000), 449-466.

[11] , Functorial products for GL_{2} × GL_{3} and functorial symmetric cube for GL_{2}, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 599–604.

[12] S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. of Math. 146 (1997), 545-646.

[13] S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), 25–45.

[14] , Ramified degenerate principal series, Israel J. Math. 78 (1992), 209–256.

[15] , A regularized Siegel–Weil formula: The first term identity, Annals of Math. 140 (1994), 1–80.

[16] H. Y. Loke, Trilinear forms of gl_{2}., Pacific J. Math. 197 (2001), 119–144.

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

[18] D. Prasad, Trilinear forms for representations of GL(2) and local (epsilon)-factors, Compositio Math. 75 (1990), 1-46.

[19] S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), 333–399.

[20] , L-functions and the oscillator representation, Lecture Notes in Math. 1245, Springer– Verlag, New York, 1987.

[21] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Annals of Math. 152 (2000), 45–111..

[22] B. Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317.

[23] S. Sahi, Jordan algebras and degenerate principal series, Crelle’s Jour 462 (1995), 1-18.

[24] H. Shimizu, Theta series and automorphic forms on GL_{2}, Jour. Math. Soc. Japan 24 (1972), 638–683.

[25] V. Tan, A regularized Siegel–Weil formula on U(2,2) and U(3), Duke Math. J. 94 (1998), 341–378.

[26] Chen-bo Zhu, Invariant distributions of classical groups, Duke Math. Jour. 65 (1992).

[27] , private communication.