ON CERTAIN AUTOMORPHIC DESCENTS to GL 1. Introduction

ON CERTAIN AUTOMORPHIC DESCENTS TO GL2

DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Abstract. We consider a new construction of automorphic representations of GL2(A), using the general idea of automorphic descent methods. In particular, four families of examples are considered. The representations τ of GL2(A) are obtained from automorphic representations π on reductive groups H, which contain a reductive subgroup G of GSp2n. We give criteria for nonvanishing and for cuspidality of the constructed representations τ, in terms of periods or co-periods and by the holomorphy at s = 1 of certain l-functions of π. We also calculate the relation of the unramiﬁed parameters in certain cases, which indicates that τ and π ﬁt partially to the Langlands functorial principle. We prove some low rank cases to support our conjectures.

1. Introduction

Let F be a number ﬁeld and A be its ring of adeles, and let H and G be two reductive groups deﬁned over F . Langlands functoriality conjecture predicts that if there is an L-homomorphism from the L-group of H to the L-group of G, then this map determines a corresponding functorial lift from automorphic representations of H(A) to automorphic representations of G(A). In [G-R-S], a constructive method called automorphic descent was introduced. The general idea of this method is to consider a reductive group M over F , and take a small, automorphic representation Θ of

Date: July, 2010 submitted. November, 2010 revised. 1991 Mathematics Subject Classiﬁcation. Primary 11F70, 11F67 ; Secondary 22E50, 22E55, 20G05. Key words and phrases. Automorphic forms, Poles of L-functions, Periods or co-periods of automorphic forms, Langlands functoriality. The work of the second named author is supported in part by NSF grant DMS- 1001672. The ﬁrst and the third named authors are partially supported by Grant No. 210/10 of The Israel Science Foundation Founded by the Israel Academy of Sciences and Humanities. 1 2 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

M(A), typical examples of which are given by residues of certain residual Eisenstein series. Then we consider certain integral operators with the automorphic functions in the space of Θ as kernel functions. The representation Θ may depend on an automorphic representation π of G(A) that we want to lift to, or may be ”universal” . The integral operators are obtained by integrating the automorphic functions in the space of Θ along an F -subgroup N of M against automorphic forms in the space of a certain automorphic representation σ of N(A). The general setting of the construction datum (M, Θ, G, N, σ) will be speciﬁed when we discuss families of examples in the following sections. The group N and the representation σ may be ”universal”. For example, we may take N to be a unipotent radical of a parabolic subgroup and σ to be a ﬁxed character. In some cases, N and σ may depend on G(A) and π, which is the representation that we want to lift to. When Θ depends on π, the smallness of Θ forces some conditions on π, such as nonvanishing of certain co-periods, or certain L-functions of π need to have a pole at a given point.

For example, in the case studied in [G-R-S], M = Sp4n, G = GL2n,Θ is the residual representation (at s = 1, in a suitable parametrization) corresponding to the induced representation from the Siegel parabolic subgroup, and on the Levi part G(A), we take the automorphic repre- S 2 S 1 sentation π. In this case we must assume that L (π, ∧ , s)L (π, s − 2 ) has a pole at s = 1. The kernel of the integral operator is simply a certain Fourier-Jacobi coeﬃcient of automorphic forms in the space of

Θ. It then defines a cuspidal representation τ of Spf2n(A), which lifts to π. One may also replace the conditions on L-functions by the nonvanishing condition of linear periods on π. This method may also be applied to study the local descents for supercuspidal representations of classical groups over p-adic local fields (as indicated in [JS03], [JNQ10], and [JS10]). In this paper we consider new examples which may potentially pro- duce new cases of Langlands functoriality. The formulation is general, ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 3 but for the moment, we can only offer preliminary results for some low rank cases. As an example, we consider H = GL2 ×GLn and G = GL2n L and the tensor product L-homomorphism from H = GL2(C)×GLn(C) L to G = GL2n(C). Then the Langlands functoriality conjecture asserts that for a given irreducible cuspidal automorphic representation τ ⊗ σ of H(A), there should exist an irreducible automorphic representation π(τ ⊗ σ) of G(A), which is the corresponding functorial lift. When n = 2, this was proved by D. Ramkrishnan ([R]) and when n = 3, this was proved by Kim and Shahidi ([KS]). For n ≥ 4, the problem is widely open. We formulate the following conjecture, which can be viewed as an approach towards a special case of the problem discussed above. ∗ ∗ Let ω denote a nontrivial quadratic character of F \A . Let O2,ω be the anisotropic orthogonal group in two variables, attached to ω. Let τ(ω) denote the an irreducible cuspidal automorphic representation of

GL2(A), which is a functorial lift from an irreducible automorphic representation of O2,ω(A), i.e. the automorphic induction from ω. Then we view the tensor product functorial lift of τ(ω) ⊗ σ from GL2(A) ×

GLn(A) to GL2n(A) as the functorial lift from O2,ω(A) × GLn(A) to

GL2n(A). We state a conjectural criterion for an irreducible cuspidal automorphic representation π of GL2n(A) to be a tensor product functorial lift from an irreducible cuspidal automorphic representation of

O2,ω(A) × GLn(A).

Conjecture 1.1. With the above notations, an irreducible cuspidal automorphic representation π of GL2n(A) is a functorial lift from an irreducible cuspidal automorphic representation of O2,ω(A) × GLn(A) if and only if there exists an irreducible cuspidal automorphic representation τ of GL2(A), such that the co-period integral Z ϕτ (g1)ϕπ(g2)θω,n+1((g1, g2))dg1dg2

0 0 Z(A)(GL2×GSp2n) (F )\(GL2×GSp2n) (A) 0 is nonzero for some choice of data. Here, (GL2 × GSp2n) denotes the subgroup of GL2 × GSp2n, which consists of all pairs (g1, g2) whose 4 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY similitude factor is the same, and Z denotes the center of that group.

Also, θω,n+1 is a residue of an Eisenstein series on GSp2(n+1)(A) (de- ﬁned in Section 2). Moreover, if the co-period integral is nonzero, for some choice of data, then the representation τ is a functorial lift from O2,ω(A).

Motivated by Conjecture 1.1, we construct the following descent from

GL2n(A) to GL2(A) deﬁned by Z (1.1) ϕ(g1) := ϕπ(g2s(g1))θω,n+1((g1, g2))dg2

Sp2n(F )\Sp2n(A) where g1 ∈ GL2(A) and all other notations are explained in Section

3. The functions ϕ(g1) generate an automorphic representation τ of

GL2(A). We want to know when is this automorphic representation nonzero and when is it cuspidal. Furthermore, we want to know the relation between π and τ, at least at the unramiﬁed local places. We formulate the criterion for cuspidality of τ as follows, while the criterion for nonvanishing of τ is closely related to Conjecture 1.1.

Conjecture 1.2. Let π be an irreducible cuspidal automorphic representation of GL2n(A), with central character µπ. Let ω be a nontrivial quadratic character of F ∗\A∗. Assume that n ≥ 2. The automorphic representation τ of GL2(A), deﬁned by integral (1.1) or (3.1) in Sec- tion 3, is cuspidal if and only if for all characters χ with the property n 2 that µπω χ = 1, the partial twisted exterior n-th power L-functions LS(π ⊗ χ, ∧n, s) are holomorphic at s = 1.

In Conjecture 1.2, the partial twisted exterior n-th power L-functions LS(π ⊗ χ, ∧n, s) are well defined for the real part of s large. However, their meromorphic continuation to the whole complex plane and the location of the possible poles are not known unless n ≤ 3 ([J-S] and [G-R]). By the generalized Ramanujan conjecture, one expects that the partial twisted exterior n-th power L-functions LS(π ⊗χ, ∧n, s) are holomorphic for Re(s) > 1 and may have a possible pole at s = 1. It is an easy exercise to understand Conjecture 1.2 for the case n = 2 in ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 5 terms of the Langlands functorial classification for π of GL4(A) to be of symplectic type or of orthogonal type. Conjecture 1.2 will be discussed in more detail in Section 5. In Sec- tions 5 and 8, we prove that Conjecture 1.2 is true for n = 3 (Theorem 5.3). One direction of Conjecture 1.2 for n = 2 is proved in Proposition 5.1. The other direction for n = 2 needs more work on the local theory of the global integral (5.2) and we will not discuss it here. The question concerning the nonvanishing of the integral (1.1) is a harder question, since we currently do not have a constructive way to define representations π on GL2n(A), which are a tensor product functorial lift from GL2(A)×GLn(A). As it often happens, in some low rank cases, the co-period we consider is a residue of a certain eulerian integral. This is the case here when n = 2, for instance. In such a circumstance, we are able to state a criterion for the nonvanishing of the integral (1.1) in terms of poles of certain L-functions. This will be discussed in Subsection 6.2 with more details.

It is clear that the automorphic representation τ of GL2(A), constructed via integral (1.1), from an irreducible cuspidal automorphic representation π of GL2n(A), contains only partial information about

π. Certainly, it can not determine π uniquely, since the GLn piece is missing. We may call such a construction a partial descent from

GL2n(A) to GL2(A). The notion of partial descent will be justified in Section 7. It remains open to find an explicit construction of the automorphic representation σ of GLn(A), such that τ ⊗ σ lifts to π. The paper is organized as follows. In Section 2, we fix some notation and we recall from [K-R1] the Siegel-Weil formula, which is needed in this paper. In particular, we define the kernel function θω,n+1 there. In Section 3, we define the partial descent by the integrals (3.1), and prove a simple criterion for its cuspidality. In Section 4, we list a set of four examples, which fit the general construction of Section 3. In Section 5, we study the question of cuspidality (of the descent) in these examples. We state a conjecture on this cuspidality, in terms of the holomorphy of 6 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY certain L-functions at s = 1. These conditions are satisfied in some low rank cases. Section 6 is devoted to the question of nonvanishing of the descent in the examples above. In general, we reduce the nonvanishing of the descent to a condition on automorphic forms on O2,ω(A), obtained by some sort of a theta correspondence. In some low rank cases, we reduce the nonvanishing to conditions on certain L-functions. In Section 7, we discuss the relation, at the unramified places, between the irreducible cuspidal representation π of GL2n(A) and the partial descent τ on GL2(A). Finally, in the last section (Section 8) we use some co-period integrals, mainly in exceptional groups, to complete Conjecture 1.2 for the case n = 3. Finally, we thank the referee for very helpful remarks and sugges- tions.

2. Preliminaries

2.1. Basic notations. Let J denote the n×n matrix with ones on the n − Jn second diagonal and zero elsewhere. Set J2n = . Consider −Jn the group of symplectic similitudes

t − − GSp2n = {g ∈ GL2n : g J2ng = λJ2n}.

We call λ = λ(g) the similitude character. We regard GSp2n as an algebraic group over the number field F , and over any one of its com- pletions. Let A denote the ring of adeles of the number field F . We fix a nontrivial character ψ of F \A.

We shall denote by ei,j a matrix whose (i, j) entry is one and and all its other coordinates are zero (its size will be understood).

2.2. The Siegel Eisenstein series and its residues. Let Qn denote the standard maximal parabolic subgroup of GSp2n whose Levi part is isomorphic to GL1 × GLn. Let δQn denote its modulus character, AX − n(n+1) n+1 δ = |λ| 2 | det(A)| . Qn λA∗ ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 7

Let ω be a multiplicative character of F ∗\A∗. It deﬁnes the following character of Qn, which we still denote by ω, AX ω = ω(λ)ω(det(A)). λA∗

Consider an Eisenstein series on GSp2n(A), associated with the induced s− 1 representation IndGSp2n(A)(ωδ 2 ) (normalized induction). Denote it Qn(A) Qn by Eω(g, s), or Eω,n(g, s). Its poles are studied in [K-R2], Sec. 1, for

Sp2n, and, since

Qn(F )\GSp2n(F ) = Qn(F ) ∩ Sp2n(F )\Sp2n(F ), their results apply to GSp2n, as well. It follows that the normalized ∗ 2 Eisenstein series Eω(g, s) is holomorphic if ω 6= 1. Henceforth, we shall assume that ω is a quadratic character which may be trivial. In ∗ this case, Eω(g, s) (as well as Eω(g, s)) has a simple pole at the point n s0 = n+1 . In the parametrization of [K-R2], this corresponds to the n−1 pole at 2 . Let Θω,n denote the corresponding residue representation of GSp2n(A), at s0. We list several properties of Θω,n.

Let V1,n denote the unipotent radical of the standard maximal parabolic subgroup of GSp2n, whose Levi part is isomorphic to GL1 ×

GSp2n−2. By the inductive formula (1.19) in [K-R2], for the constant term of Eω,n(g, s) along V1,n, we obtain, by taking the residue at s0, the decomposition of the residual representation

V1,n (2.1) Θω,n = Cω ⊕ Θω,n−1, where Cω is the one-dimensional representation of GL1(A) depending on ω, and Θω,n−1 is the residual representation of GSp2(n−1)(A) generated by the residue at the corresponding point s0 of the corresponding

Siegel Eisenstein series on GSp2(n−1)(A). Next, we recall the well-known Siegel-Weil identity for the group

Sp2n, corresponding to s0. In our context, the formula can be described as follows.

Assume ﬁrst that ω = 1. Let En(g, s) denote an Eisenstein series on s− 1 Sp (A), associated with the induced representation IndSp2n(A)(δ 2 ) 2n Pn(A) Pn 8 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

(normalized induction). Here, Pn denotes the standard maximal parabolic subgroup of Sp2n, whose Levi part is isomorphic to GL1 ×Sp2(n−1). By Cor. 6.3 in [K-R2] we have an identity of the form 1 (2.2) θn(g) = Ress= n E1,n(g, s) = c ·En(g, ), g ∈ Sp (A). n+1 2 2n Here c is a nonzero constant. Note that the identity holds for some choice of data. See also [J].

Assume that ω is a (nontrivial) quadratic character. Let O2,ω denote ψ the orthogonal group associated to ω by class ﬁeld theory. Let Θe 2n denote the automorphic representation of Spf4n(A) (the metaplectic ψ double cover of Sp4n(A)), generated by the theta functions θeφ (g), as φ varies in the space of Bruhat-Schwartz functions S(A2n). When there is no confusion, we shall omit the reference to ψ. In this case, the Siegel-Weil formula has the form (see [K-R1]) Z ψ (2.3) θω,n(g) = c · θeφ,2n((h, g))dh g ∈ Sp2n(A)

O2,ω(F )\O2,ω(A)

ψ ψ Here c is a nonzero constant, θeφ,2n is a vector in the space of Θe 2n, and we restrict it to O2,ω(A)×Spf2n(A), via the dual pair map into Spf4n(A). The formula is valid for appropriate corresponding data.

3. Partial Descent and Cuspidality

In this section, we shall construct explicitly the partial descent to

GL2(A), and discuss the issue of its cuspidality. Let G be a reductive F -subgroup of GSp . We assume that G con- 2n aIn tains the center of GSp2n and all similitude factors s(a) := . In 0 0 0 Denote G = G ∩ Sp2n. Then we also assume that G (F )\G (A) has a ﬁnite volume. Let σ be an automorphic representation of G(A), which may be reducible. Assume that the automorphic forms ϕσ in the space of σ a b are rapidly decreasing on G0(F )\G0(A). For h = ∈ GL (A), c d 2 ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 9 deﬁne (3.1)   Z a b f(h) := ϕσ(gs(det(h)))θω,n+1( gs(det(h)) )dg G0(F )\G0(A) c d

Since θω,n+1 is of moderate growth, the integral converges absolutely.

It is not hard to check that f is left GL2(F )-invariant. If σ has a central n+3 character µσ, then the function f(h) has the central character µσ ·ω , i.e. a f( h) = µ (a)ωn+3(a)f(h). a σ holds for all a ∈ A∗.

Let τ denote the automorphic representation of GL2(A) generated by all functions f(h) as defined by (3.1). We call τ the partial descent of σ to GL2(A). As discussed in Section 7, the Satake parameters of τ at unramified, finite local places are determined by those of π. Hence when τ is cuspidal, for example, the representation τ is irreducible due to the strong multiplicity one theorem for GL2(A). This may justify the notion of the partial descent of π.

Proposition 3.1. Let σ be an automorphic representation of G(A). Assume that

1. automorphic forms in the space of σ are rapidly decreasing on G0(F )\G0(A); 2. the integral Z (3.2) ϕσ(g)θω,n(g)dg

G0(F )\G0(A)

is zero for all choice of data; and 3. ω is a nontrivial quadratic character.

Then the partial descent τ of σ to GL2(A) is a cuspidal automorphic representation of GL2(A). 10 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Proof. We need to show that the integral

Z 1 x (3.3) f( h)dx 1 F \A is zero for all choice of data. In other words, we need to prove that the integral   Z Z 1 x ϕσ(g)θω,n+1  g  dgdx F \A G0(F )\G0(A) 1 is identically zero. Using the Siegel-Weil formula (2.3), it is enough to prove the vanishing of the integral     Z Z Z 1 x ψ ϕσ(g)θeφ,2(n+1) (m,  g ) dmdgdx

0 1 F \A [G ] [O2,ω]

0 0 0 where [G ] := G (F )\G (A) and [O2,ω] := O2,ω(F )\O2,ω(A), to sim- ψ plify the notation. We recall that θeφ,2(n+1) is a usual theta series on the metaplectic group Spf4(n+1)(A). 2n 2 Let φ = φ1 ⊗ φ2, where φ1 ∈ S(A ) and φ2 ∈ S(A ). Then, the usual factorization of theta series implies that the above integral is equal to Z Z Z ψ ψ 1 x (3.4) ϕσ(g)θe ((m, g))θe ((m, ))dmdgdx. φ1,2n φ2,2 1 0 F \A [G ] [O2,ω]

Here θψ is a theta series on Sp (A), and θψ is a theta series on eφ1,2n f4n eφ2,2 Sp (A). Since ϕ is rapidly decreasing on G0(F )\G0(A), θψ is of f4 σ eφ1,2n moderate growth, and F \A and O2,ω(F )\O2,ω(A) are compact, we can switch the order of integration in (3.4) and get

Z Z Z ψ ψ 1 x (3.5) ϕσ(g)θe ((m, g))θe ((m, ))dmdgdx. φ1,2n φ2,2 1 0 [G ] [O2,ω] F \A ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 11

Writing θψ as a series over F 2, and using the well known formulas eφ2,2 for the action of the Weil representation, we obtain Z Z ψ 1 x X 1 x θe ((m, ))dx = ωψ,2((m, ))φ2(ξ)dx φ2,2 1 1 2 F \A ξ∈F F \A X = ωψ,2((m, 1))φ2(ξ) = φ2(0). (ξ,ξ)=0

Here ωψ,2 is the Weil representation of Spf4(A) , and (ξ, ξ) is the quadratic form preserved by O2,ω. Since this quadratic form is anisotropic, only ξ = 0 contributes to the above sum. Therefore, the integral (3.5) is equal to Z Z (3.6) φ (0) ϕ (g)θψ ((m, g))dmdg. 2 σ eφ1,2n 0 0 G (F )\G (A) O2,ω(F )\O2,ω(A)

However, it follows from the Siegel-Weil formula (2.3) and from the vanishing assumption (3.2), that the integral (3.6) is identically zero. This proves the cuspidality of τ.

We may consider the space of automorphic functions on O2,ω(A), deﬁned by the integrals Z (3.7) ϕ (g)θψ ((m, g))dg, σ eφ1,2n G0(F )\G0(A) as the theta lifting of the representation σ to an automorphic representation of O2,ω(A). Since O2,ω is F -anisotropic, any automorphic representation of O2,ω(A) is a direct sum of irreducible ones. With this deﬁnition, (3.6) shows

Corollary 3.2. With assumptions as in Proposition 3.1, the partial descent τ of σ is a cuspidal automorphic representation GL2(A), if and only if the identity representation of O2,ω(A) is not a summand of the theta lift of σ. 12 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

4. Some Examples of Partial Descent

In this section, we consider four examples, which ﬁt the general setup of the previous section. In all the examples below, the rapid decay of the automorphic forms in the space of σ can be veriﬁed. In Sections 5 and 6, we will discuss the cuspidality and non-vanishing conditions for these examples.

A2n−1-Type: Let π be an irreducible cuspidal automorphic representation of GL2n(A). Take G = GSp2n. We restrict the cusp forms of π to G(A), and let σ be the resulting representation of G(A), on the 0 space of these restrictions. Here, G = Sp2n.

D2n-Type: Let GSO4n be the F -split group of orthogonal similitudes, acting in the 4n-dimensional column space, and preserving the quadratic form deﬁned by J4n. Let Rn denote the unipotent radical of the standard maximal parabolic subgroup of GSO4n, whose Levi part is isomorphic to GL1 × GL2n. In term of matrices, I2n X t Rn = { : X ∈ Mat2n×2n,X J2n + J2nX = 0}. I2n

Let ψRn denote the character of Rn(F )\Rn(A) deﬁned as follows. If

X = (xi,j) is as above, with coordinates in A, then

ψRn (r) = ψ(x1,1 + ... + xn,n).

It is not hard to check that the stabilizer of the form x1,1 + ... + xn,n, inside GL1 × GL2n, is the group GSp2n. Let G = GSp2n. Let π denote an irreducible cuspidal automorphic representation of

GSO4n(A). For g ∈ GSp2n(A) and a cusp form ϕπ in the space of π, we deﬁne the following Fourier coeﬃcient Z

(4.1) ϕ(g) = ϕπ(rg)ψRn (r)dr

Rn(F )\Rn(A) This is an automorphic function on G(A). The space generated by the

ϕ, as ϕπ varies, deﬁnes our representation σ of G(A).

E7-Type: Let GE7 denote the F -split similitude exceptional group of type E7. In [G2], Section 2.2, p. 110-112, a unipotent subgroup U ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 13 of GE7, and a character ψU of U(A), which is trivial on U(F ), were deﬁned. It was shown there that the group GSp6 embedded in GE7 in a suitable way, is the stabilizer of this character. Let π denote an irreducible, cuspidal, automorphic representation of the group GE7(A). Then, the integral Z ϕ(g) = ϕπ(ug)ψU (u)du

U(F )\U(A) deﬁnes an automorphic function on GSp6(A). In this example, G =

GSp6 and σ is the representation of G(A) in the space generated by the functions ϕ. n A1 -Type: In this example, we let G = GL2 ×· · ·×GL2, the product of n copies of GL2, embedded in GSp2n by the direct sum embedding. 0 Thus, G = SL2 × ... × SL2, the product of n copies of SL2, embedded inside Sp2n by the direct sum embedding. We take σ = π to be an irreducible, cuspidal, automorphic representation of G(A). In all the examples above, we will re-name the partial descent τ of σ, as constructed Sec. 3, and call it the partial descent of π.

5. Some conjectures on the cuspidality of partial descents

In this section we study the cuspidality of the partial descents introduced in the previous section. In order to get this cuspidality, we want to verify the condition of Proposition 3.1. We shall state a few conjectures, in which we claim that the cuspidality of the partial descent in each case is equivalent to the holomorphy at s = 1 of a certain L-function of π twisted by characters. In some cases, for low rank groups, we will show that these conjectures hold. To motivate our conjectures, we start with the A2n−1-Type examples, deﬁned in the previous section.

5.1. Cuspidality of the partial descent of A2n−1-Type. Let π be an irreducible cuspidal automorphic representation of GL2n(A). By 14 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Proposition 3.1, we need to show that the following integral Z (5.1) ϕπ(g)θω,n(g)dg

Sp2n(F )\Sp2n(A) is identically zero, in order for the partial descent to be cuspidal. When n = 1, this integral is clearly zero, for any choice of data. Suppose that n = 2. To study this case, we consider the following function on A∗, Z ξ(a) = ϕπ(gs(a))θω,2(gs(a))dg.

Sp4(F )\Sp4(A) ∗ Clearly, this function is F -invariant, and if µπ is the central character 2 of π, then ξ(t a) = µπ(t)ξ(a). Therefore, the integral (5.1), with n = 2, is identically zero, if and only if for all characters χ of F ∗\A∗, such that 2 µπχ = 1, the integral Z Z ϕπ(gs(a))θω,2(gs(a))χ(a)dgda

∗ 2 ∗ ∗ (A ) F \A Sp4(F )\Sp4(A) is zero for any choice of data. The above integral can also be written as Z ϕπ(g)θω,2(g)χ(λ(g))dg.

Z(A)GSp4(F )\GSp4(A)

Here, Z is the center of GSp4. Recall that λ(g) denotes the similitude factor of g. Replacing the residue by the Eisenstein series, we obtain an integral Z ϕπ(g)Eω,2(g, s)χ(λ(g))dg

Z(A)GSp4(F )\GSp4(A) with Eω,χ,2(g, s) being the Eisenstein series on GSp4(A), associated to the induced representation (normalized)

s− 1 IndGSp4(A)(ω δ 2 ). Q2(A) χ Q2

Here for (a, h) ∈ GL1 × GL2, h x ω = ωχ(a)ω(det(h)). χ 0 ah∗ ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 15

With these notations, the last integral is equal to Z (5.2) ϕπ(g)Eω,χ,2(g, s)dg.

Z(A)GSp4(F )\GSp4(A)

If we relate GL4 with GSO6, and GSp4 with SO5, then this integral, with χ = 1, is a special case of the integrals considered in [B-F-G], and hence is eulerian. Unfolding the above integral (5.2), for Re(s) large enough, we obtain, as inner integration, the integral

Z Z IX g (5.3) ϕ ( )ψ(trX)dXχ(detg)dg. π I g [GL2] [Mat2×2]

Here the integration dg is over [GL2] := Z2(A)GL2(F )\GL2(A), and the integration dX is over [Mat2×2] := Mat2×2(F )\Mat2×2(A). It is clear that integral (5.3) is the Shalika period integral. Hence, by [J-S], this integral is zero for ϕπ if and only if the partial exterior square L-function LS(π ⊗ χ, ∧2, s) is holomorphic at s = 1. Therefore we may conclude the following.

Proposition 5.1. Let π be an irreducible cuspidal automorphic representation of GL4(A). Assume that the partial L-functions

LS(π ⊗ χ, ∧2, s)

2 2 are holomorphic at s = 1 for all characters χ, which satisfy µπω χ = 1. Then the integral (5.1) (with n = 2) is identically zero, and hence the partial descent of π to GL2(A) is cuspidal.

This proves one direction of Conjecture 1.2 for the case n = 2. The other direction needs more work on the local theory of the global integral (5.2) and we will not discuss it here. However, we are able to complete the proof of Conjecture 1.2 for n = 3 in this paper. The analogue for n = 3 will be discussed in Section 8 by calculating certain co-period of residues of Eisenstein series on F4(A). To prove the other direction of Conjecture 1.2, we give details for the case n = 3 below. 16 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

3 Assume that n = 3. We study the residue at s = 4 of the integrals Z (5.4) ϕπ(g)Eω,χ,3(g, s)dg

Z(A)GSp6(F )\GSp6(A) 2 where χ is a character which satisﬁes µπωχ = 1, and Eω,χ,3(g, s) is an Eisenstein series, associated to the induced representation

s− 1 IndGSp6(A)(ω δ 2 ). Q3(A) χ Q3

Here, for (a, h) ∈ GL1 × GL3, h ∗ ω = ωχ(a)ω(det(h)). χ 0 ah∗

2 2 Since µπωχ = 1 = ω , we can write the character above as

3 2 (µπχ )(a)(µπχ )(det(h)). The integrals (5.4) were studied in [G-R] and were shown to represent the partial L-function LS(π ⊗ χ, ∧3, s) (after a simple linear change in 2 s). Since ω is a nontrivial quadratic character, it follows that µπχ 6= 1, 2 4 but µπχ = 1. Thus, it follows from Theorem 3.3 in [G-R] that under these assumptions, if the partial L-function LS(π⊗χ, ∧3, s) has a simple pole at s = 1 for some χ as above, then the residue at the corresponding 3 point (which is s = 4 ) of the integrals (5.4) is nonzero for some choice of data. A closer look at the structure of the integrals (5.4) reveals, as in case n = 2, that the integral (5.1) is an inner integral of the residue 3 at s = 4 of the integral (5.4), and hence is nonzero for some choice of data. By Prop 3.1, this proves

Proposition 5.2. Let π be an irreducible cuspidal automorphic representation of GL6(A). Assume that the partial descent of π to GL2(A) is cuspidal. Then the partial L-functions LS(π ⊗ χ, ∧3, s) are holomor- 3 2 phic at s = 1 for all characters χ, which satisfy µπω χ = 1.

This proves the other direction of Conjecture 1.2 for n = 3. Com- bining with Proposition 8.2, we have

Theorem 5.3. Conjecture 1.2 holds for n = 3. ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 17

5.2. Cuspidality of the partial descents of D2n-Type, E7-Type, n and A1 -Type. The partial descent to GL2 of D2n-Type can be treated in a similar way in the low rank cases n = 2, 3. We sketch some of the details. We start with n = 2. This case is similar to the GL2n case when n = 2. Let π denote an irreducible generic cuspidal representation of

GSO8(A). Consider the Fourier coeﬃcient of π given by (4.1) with n = 2. The stabilizer of this Fourier coeﬃcient is the group GSp4, and as in the GL2n case, the integral that we consider is Z Z

(5.5) ϕπ(rg)θω,2(g)ψR2 (r)drdg.

Sp4(F )\Sp4(A) R2(F )\R2(A)

In other words, the lift to GL2 is cuspidal if and only if integral (5.5) is zero for all choice of data. Arguing as in the GL4 example in Sec. 5.1, we deduce that if integral (5.5) is not identically zero, then, for Re(s) large, the integral Z Z

ϕπ(rg)Eω,χ,2(g, s)ψR2 (r)drdg.

Z(A)GSp4(F )\GSp4(A) R2(F )\R2(A) is not identically zero. Unfolding this integral, we obtain the integral Z Z

(5.6) ϕπ(vg)ψV2 (v)χ(detg)dvdg.

Z2(A)GL2(F )\GL2(A) V2(F )\V2(A) as an inner integration. Here, the group V2 is the unipotent radical of the standard parabolic subgroup of GSO8, whose Levi part is isomorphic to GL2 × GL2. Thus, V2 consists of upper unipotent matrices.

The character ψV2 is deﬁned as follows. For v = (vi,j) ∈ V2, deﬁne

ψV2 (v) = ψ(v1,3 + v2,4 + v3,5). The stabilizer of this character is the ∗ ∗ group GL2 embedded in GSO8 as g → diag(g, g, g , g ). To relate this integral with the Spin L-function of π, we consider the integral Z Z

(5.7) ϕπ(vg)ψV2 (v)E(g, χ, s)dvdg.

Z2(A)GL2(F )\GL2(A) V2(F )\V2(A)

Here E(g, χ, s) is the standard GL2 Eisenstein series attached to the character χ. This integral unfolds to an integral with the Whittaker 18 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY coeﬃcient of π, and can be shown to represent the partial Spin L- function LS(π ⊗ χ, Spin, 2s − 1/2). Thus, we can deduce that if the partial Spin L-function LS(π ⊗ χ, Spin, 2s − 1/2) has a simple pole at s = 1, then integral (5.7) is not zero for some choice of data. Un- fortunately, the converse result which applies to integral (5.3) is not proved yet. Namely, it is not proved that if the partial Spin L-function is holomorphic at s = 1, then integral (5.7) is zero for all choice of data. However, the above discussion clearly motives the conjecture given below. When n = 3, the results in [G3], Section 5 also suggests that there is a relation between the cuspidality of the partial descent to GL2 and the holomorphy of the twisted Spin L-function attached to π. As in the previous case, we need to assume that π is generic. These two cases lead to the following conjecture, where we assume that π is globally generic.

Conjecture 5.4. For n ≥ 2, let π be an irreducible cuspidal automorphic representation of GSO4n(A) with central character µπ. Assume that π is globally generic. The partial descent τ, deﬁned by (3.1), is a cuspidal automorphic representation of GL2(A) if and only if for all n 2 characters χ with the property that µπω χ = 1, the partial L-functions LS(π ⊗ χ, Spin, s) are holomorphic at s = 1.

Similarly, the work ([G2]), which is analogous to the above two cases when n = 3, motivates the following

Conjecture 5.5. Let π be an irreducible cuspidal automorphic representation of GE7(A), with central character µπ. Assume that π is globally generic. The partial descent τ, deﬁned by (3.1), is a cuspidal automorphic representation of GL2(A) if and only if for all characters 2 S χ, such that µπωχ = 1, the partial 27- degree L-functions L (π ⊗ χ, s) are holomorphic at s = 1.

The results of [PS-R] for n = 3 motivate the conjecture given below. Indeed, in [PS-R], a global Rankin-Selberg construction is given, and ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 19 represents the triple L-function π1 ⊗ π2 ⊗ π3, where πi are cuspidal representations of GL2(A). In that reference, they show a relation between the poles of the this triple L-function and the integral Z ϕπ(g)θω,3(h)dh

Z2(F )H(F )\H(A)

Here, H is the group of all triples (g1, g2, g3) of matrices in GL2, which have equal determinants. It is embedded in GSp6 in the standard way. Thus we have

Conjecture 5.6. For n ≥ 2, let π = π1 ⊗· · ·⊗πn be an irreducible cuspidal automorphic representation of the product of n copies of GL2(A).

Let µπ be its central character. The partial descent τ deﬁned by (3.1), is a cuspidal automorphic representation of GL2(A) if and only if for n 2 all characters χ, such that µπω χ = 1, the partial (tensor) L-functions LS(π ⊗ χ, s) are holomorphic at s = 1.

As the referee pointed out, the case n = 2 of Conjecture 5.6 may be veriﬁed using the doubling method of Piatetski-Shapiro and Rallis.

The right set-up is to use the doubling integral for SL2 × SL2, in stead for GL2 × GL2. In this case, Conjecture 5.6 describes the irreducible components when the irreducible cuspidal automorphic representations

πi of GL2(A) restricted to SL2(A). We omit the details.

6. On the nonvanishing of the partial descents

In Subsection 6.1, we give some equivalent conditions for the nonvanishing of the partial descents deﬁned in Section 4, and in Subsection 6.2 we discuss some lower rank examples.

6.1. Conditions for the nonvanishing of the partial descents. Let ρ ∈ F ∗ be non-square, such that ω corresponds to the quadratic √ extension F ( ρ). We realize O2,ω as the group of isometries of the norm form x2 − ρy2. 20 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

It follows from (3.1) that the partial descent to GL2(A) is nonzero if and only if there is a ∈ F , such that the integral   Z Z 1 x (6.1) ϕσ(g)θω,n+1  g  ψ(ax)dgdx F \A G0(F )\G0(A) 1 is not identically zero. We use the notation introduced in the proof of Proposition 3.1. Plugging the Siegel-Weil formula (2.3), the above integral is a nonzero multiple of   Z Z Z 1 x ψ ϕσ(g)θeφ,2(n+1)((m,  g ))ψ(ax)dmdgdx,

0 1 F \A [G ] [O2,ω] 0 0 0 where [G ] := G (F )\G (A) and [O2,ω] := O2,ω(F )\O2,ω(A). Choose 2n 2 φ = φ1 ⊗ φ2, where φ1 ∈ S(A ) and φ2 ∈ S(A ). Unfolding the theta function, and switching the order of integration, as in the proof of Proposition 3.1, we obtain Z Z ϕ (g)θψ ((m, g))φ (ξ m)dgdm. σ eφ1,2n 2 a a 0 0 O2,ω(F )\O2,ω(A) G (F )\G (A) 2 a Here, ξa ∈ F is a ﬁxed vector, such that (ξa, ξa) = a, and O2,ω is a the stabilizer of ξa inside O2,ω. Thus, if a = 0, then O2,ω = O2,ω, and √ if a 6= 0 (necessarily a norm from a nonzero element in F ( ρ)) then a ∼ 2 O2,ω = Z2. Since φ2 is an arbitrary Schwartz-Bruhat function in S(A ), it follows that the last integral is not identically zero, if and only if the integral Z Z (6.2) ϕ (g)θψ ((m, g))dgdm σ eφ1,2n a a 0 0 O2,ω(F )\O2,ω(A) G (F )\G (A) is not identically zero. As in Section 3, we can view this last integral as a the co-period along O2,ω of the theta lift of the representation σ from 0 G (A) to the group O2,ω(A). We summarize

Theorem 6.1. Notations are as in Sec.3. Part (a): The following are equivalent.

(1) The partial descent τ to GL2(A) as deﬁned by (3.1) is nonzero. ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 21

(2) The theta lift of σ to O2,ω(A) has a nontrivial co-period along a √ a subgroup of the form O2,ω, for some a ∈ NF ( ρ)/F . (3) At least one of the following co-period integrals is not identically zero. Z Z ψ (6.3) ϕσ(g)θeφ,2n((m, g))dgdm;

0 0 O2,ω(F )\O2,ω(A) G (F )\G (A) or Z Z −1 ψa ψ−a ρ (6.4) ϕσ(g)θeφ0,n(m, g)θeφ00,n (g)dgdm,

0 0 Z2(F )\Z2(A) G (F )\G (A)

a −a−1ρ √ ψ ψ for some 0 6= a ∈ NF ( ρ)/F , and where θeφ0,n and θeφ00,n are a theta series on Spf2n(A), with respect to the characters ψ , and −a−1ρ ψ , respectively; Z2 is viewed as the orthogonal group O1 . Part (b): The partial decent τ from σ is cuspidal if and only if the identity representation is not a summand of the theta lift of σ to

O2,ω(A). Part (c): If the partial descent τ is nonzero and cuspidal, then the representation obtained by restricting the cusp forms of τ to SL2(A) has a nontrivial theta lift to O2,ω(A).

Proof. We already proved the equivalence of (1) and (2) in the ﬁrst part. The integral (6.3) is the integral (6.2), when a = 0, and when a 6= 0, the integral (6.4) is equal to the integral (6.2), when φ1 is of the form φ0 ⊗ φ00. This follows from the usual factorization of theta functions. This factorization corresponds to an orthogonal factorization of the quadratic space F 2, equipped with the norm form x2 − ρy2, as F 2 =

F ξa ⊕ F ξ−ρa−1 . The second part is exactly Corollary 3.2. Let us prove the third part. Assume that τ is cuspidal and nonzero. Then from (3.1), we conclude that the following integral is not identically zero, Z Z ϕτ (h)ϕσ(g)θω,n+1((h, g))dhdg

0 0 SL2(F )\SL2(A) G (F )\G (A) 22 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

By the Siegel-Weil formula (2.3), and the usual factorization of theta functions, we deduce that the integral Z

(6.5) ϕτ,φ2,ψ(m)ϕσ,φ1,ψ(m)dm

O2,ω(F )\O2,ω(A) is nonzero for some choice of data, where Z ϕ (m) := ϕ (h)θψ ((m, h))dh, τ,φ2,ψ τ eφ2,2

SL2(F )\SL2(A) Z ϕ (m) := ϕ (g)θψ ((m, g))dg. σ,φ1,ψ σ eφ1,2n G0(F )\G0(A) This implies that the integral Z ϕ (h)θψ ((m, h))dh τ eφ2,2

SL2(F )\SL2(A) is nonzero for some choice of data. This last integral is the theta lift −1 (with respect to ψ ) of the cuspidal representation τ from SL2(A) to O2,ω(A). n Let us consider the examples of A1 -Type in Section 4. In this case 0 G = SL2 × · · · × SL2, the n-copy product of SL2, and σ = π =

π1 ⊗ · · · ⊗ πn, where each πi is an irreducible cuspidal automorphic representation of GL2(A). We have

Theorem 6.2. With notation as in the last paragraph, the partial descent τ is nonzero if and only if for each 1 ≤ i ≤ n, the (restriction to

SL2(A) of the) representation πi has a nontrivial theta lift to an auto- √ morphic representation i of O2,ω(A), and there exists a ∈ NF ( ρ)/F , such that Z

(6.6) ϕ1 (m) ··· ϕn (m)dm

a a O2,ω(F )\O2,ω(A) is not identically zero. Moreover, the representation τ is cuspidal if and only if the integral Z

(6.7) ϕ1 (m) ··· ϕn (m)dm

O2,ω(F )\O2,ω(A) ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 23

is identically zero . Here, ϕi is a vector in the space of the representation of i. Finally, if the partial descent τ is nonzero and cuspidal, then it has a nontrivial theta lift (from SL2(A)) to O2,ω(A).

Proof. The last assertion is a special case of the third part of the last theorem. In the ﬁrst part of the last theorem, we proved that τ is nonzero if and only if the integral (6.2) is nonzero for some choice of √ data, for some a, which is a norm from F ( ρ). Write g = (g1, . . . , gn) 0 0 0 where gi ∈ SL2(A), and φ1 = φ1 ⊗ · · · ⊗ φn, where φi is a Schwartz- Bruhat function on A2. By the usual factorization of theta functions, it follows that, for a given a, as above, the integral (6.2) is nonzero for some choice of data if and only if

Z n Z Y ψ ϕπi (gi)θe 0 (m, gi)dgidm φi,2 a a i=1 O2,ω(F )\O2,ω(A) SL2(F )\SL2(A) is nonzero for some choice of data. Note that each dgi-integral in the last product is the ψ-theta lift of ϕπ, from SL2(A) to an automorphic form ϕi on O2,ω(A), and now the last integrals becomes the integral (6.6). This proves the ﬁrst part of the theorem. The special case a = 0 gives, by Proposition 3.1, the constant term of τ, and then the last integral becomes the integral (6.7). This completes the proof of the theorem.

6.2. Some low rank cases. In this subsection, we will consider two Rankin-Selberg integrals, which will provide us with examples for the non-vanishing of the partial descents. Let τ and π be two irreducible cuspidal automorphic representations of GL2(A) and GL4(A), respectively. Given a quadratic character ω, recall the Siegel Eisenstein series Eω,3(g, s) on GSp6(A) (Sec. 2.2). Deﬁne

G := {(g1, g2) ∈ GL2 × GSp4 : det(g1) = λ(g2)}. 24 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Assume that µτ µπω = 1. Let ϕτ and ϕπ be cusp forms in the spaces of τ and π, respectively. We consider the global integral Z (6.8) ϕτ (g1)ϕπ(g2)Eω,3((g1, g2), s)dg1dg2,

Z(A)G(F )\G(A) 3 whose residue at s = 4 is the integral (3.1). Here Z is the center of G, and the integral is well deﬁned since µτ µπω = 1. A simple unfolding process implies that the integral is eulerian. In fact, one can easily show that, for Re(s) large, it is equal to Z (6.9) Wτ (g1)Wπ(g2)fω(w3w2w1z(1)(g1, g2), s)dg1dg2.

Z(A)U(A)\G(A) Here,  1 x y z   1 r  1 r y  U = { ,   ∈ G},  1  1 −x 1 and Wτ and Wπ are the Whittaker coefficient of ϕτ and ϕπ, with re- −1 spect to ψ and ψ , respectively. Also, fω is a vector in the induced representation which was defined in Subsection 2.2. Finally, we have z(1) = I6 + e1,3 − e4,6, and for 1 ≤ i ≤ 3, wi are the three simple reflection of the Weyl group of GSp6, where w3 corresponds to the long simple root. It follows from the above that the integral (6.8) is factorizable. A routine computation shows that, at a finite place v, where τv, πv, ωv,

ψv are unramiﬁed, and we substitute in the local version at v of the integral (6.9) normalized, unramiﬁed vectors, then this local integral is equal to L(π × τ , ∧2 × St, 2s − 1/2) (6.10) v v . L(ωv, 4s)ζv(8s − 2) Here, the numerator is the local L-function which corresponds to the tensor product representation of the second fundamental representation of GL4(C) and the standard representation of GL2(C). Also, L(ωv, 4s) is the one dimensional L-function attached to the character ωv. It is also a routine matter to prove that, given a point s ∈ C, the local ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 25 integral above is holomorphic and nonzero, for some choice of data. From this we conclude

Proposition 6.3. With the above notation, let π and τ be such that the partial L-function LS(π × τ, ∧2 × St, 2s − 1/2) has a simple pole at 3 s = 4 . Then the integral (3.1) is nonzero for some choice of data, and the partial descent of π to GL2(A) is nontrivial.

Proof. By (6.10), the global integral (6.8) is equal to, for some choice of data, LS(π × τ, ∧2 × St, 2s − 1/2) α(s) , LS(ω, 4s)LS(1, 8s − 2) where α is a meromorphic function, which is holomorphic and nonzero 3 at some neighborhood of s = 4 . The assumption of the proposition 3 implies that the residue at s = 4 of the global integral (6.8) is not identically zero. This residue is equal to the integral (6.8), with the

Eisenstein series replaced by its residue, i.e. by θω,3. This integral contains as an inner integral, the integral (3.1), and hence it is not identically zero. Thus, the partial descent of π to GL2(A) is nontrivial.

A similar example is available in case of D2n-Type, n = 2, from Section 4. Let τ and π denote two irreducible cuspidal automorphic representations of GL2(A) and GSO8(A), respectively. Assume that π is globally generic, and that µτ µπω = 1. Recall that the representation

σ of G(A) = GSp4(A) is as constructed from π in Sec. 4. The automorphic functions of σ are obtained by applying the ψR2 -coeﬃcient to R2,ψ the cusp forms ϕπ of π. Denote this coeﬃcient by ϕπ . Let ϕτ and

ϕπ be cusp forms in the spaces of τ and π, respectively. Consider the global integral Z V,ψ (6.11) ϕτ (g1)ϕπ (g2)Eω,3((g1, g2), s)dg1dg2, Z(A)G(F )\G(A) where G is the same group as in (6.8). Again, if the integral (6.11) has 3 a pole at s = 4 for some data, then, by taking the residue, we see that 26 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY in particular, the integral (3.1) is not identically zero, and hence the partial descent of π to GL2(A) is nontrivial. As in the previous case, one can show that integral (6.11) is eulerian and we can prove

Proposition 6.4. With the above notation, let π and τ be such that the partial L-function LS(π × τ, Spin × St, 2s − 1/2) has a simple pole 3 at s = 4 . Then the integral (3.1) is nonzero for some choice of data, and hence the partial descent of π to GL2(A) is nontrivial. Here, Spin is the eight dimensional Spin representation of GSO8(C).

7. On the unramified correspondence

In this section, we will exhibit the relation between the unramiﬁed parameters of π and τ in the case of A2n−1-Type (Section 4), at a

finite place, where O2,ω is anisotropic (and hence compact). Thus, let F be a non-archimedean local field of characteristic zero. Let ω be the nontrivial quadratic unramified character of F ∗. Let π be an irreducible unitary generic unramified representation of GL2n(F ). Realize π as the representation induced from the standard Borel subgroup B2n from the unramified character, whose value on an element of B2n with diagonal

diag(t1, ..., t2n) is

µ1(t1) ··· µ2n(t2n), ∗ where µ1, ··· , µ2n are unramified characters of F . Let τ be an irreducible unitary generic unramified representation of GL2(F ), and realize it as IndGL2(F )(η ⊗ η ), B2(F ) 1 2 ∗ where η1, η2 are unramified characters of F . Let ψ be a nontrivial character of F . Consider the Weil representation of Spf4n+4(F ), corresponding to ψ, restricted to the dual pair O2(F ) × Sp2n+2(F ). Here,

O2(F ) = O2,ω(F ). Denote this restriction by ωψ,2n+2,2. By analogy to (3.1), we assume that

(7.1) HomO2(F )×Sp2n(F )×SL2(F )(ResSp2n(F )π⊗ωψ,2n+2,2, ResSL2(F )τ) 6= 0, ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 27 and also

n+1 (7.2) η1η2 = µπω .

Here µπ = µ1 ··· µ2n is the central character of π.

In (7.1), SL2(F ) × Sp2n(F ) is embedded in Sp2n+2(F ) by the direct sum embedding. Its action in the space of π is through the projection to Sp2n(F ), and its action in the space of τ is through the projection to SL2(F ), and O2(F ) acts trivially in both the spaces of π and τ.

Proposition 7.1. Under the assumptions above, up to a permutation of the unramiﬁed characters µ1, ..., µ2n, π has the form

(7.3) π = IndGL2n(F )(µ ⊗ · · · ⊗ µ ) ⊗ ω(µ ⊗ · · · ⊗ µ ), B2n(F ) 1 n n 1 and we have, up to switching the order of η1 and η2,

(7.4) η1 = ωµ1 ··· µn, η2 = µ1 ··· µn.

Proof. By Frobenius reciprocity, (7.1) implies that

−1 (7.5) HomO2(F )×T ×Sp2n(F )(ResSp2n(F )π ⊗ JZ (ωψ,2n+2,2), αη1η2 ) 6= 0. Here, α(x) = |x| is the absolute value character of F ∗; Z is the image of the standard unipotent subgroup in SL2(F ) inside Sp2n+2(F ), under the direct sum embedding above, i.e. the subgroup of matrices 1 0 z u(z) =  I2n  ; 1 and JZ denotes the application of the Jacquet functor along Z. Finally,

T is the diagonal subgroup of SL2(F ), embedded in Sp2n+2(F ) as above, and identiﬁed with F ∗, as well. We claim that, as representations of

O2(F ) × T × Sp2n(F ) ∼ (7.6) JZ (ωψ,2n+2,2) = ωα ⊗ ωψ,2n,2.

In order to see this, let us realize ωψ,2n+2,2 in the Schwartz-Bruhat space S(En+1), where E is the two dimensional quadratic space corre- 2 sponding to O2(F ). In Sec. 6, we took E to be F , equipped with the 28 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY norm form x2 − ρy2. Then, from the formulas of the Weil representation, for u(z) ∈ Z as above, and φ ∈ S(En+1), we have 1 ω (u(z))φ(x , ..., x ) = ψ( z(x , x ))φ(x , ..., x ). ψ,2n+2,2 1 n+1 2 1 1 1 n+1 Here, (·, ·) denotes the quadratic form on E. See, for example, [Ku], p.37. From this formula, it is clear that the subspace of S(En+1), which deﬁnes the kernel of the Jacquet module JZ (ωψ,2n+2,2) is the subspace of all φ ∈ S(En+1), such that

φ(0, x2, ..., xn+1) = 0. This follows from the fact that the quadratic form on E is anisotropic. Thus, the map from S(En+1) to S(En), which sends φ to φ0 ∈ S(En), where 0 φ (y1, ..., yn) = φ(0, y1, ..., yn), factors through JZ (ωψ,2n+2,2) and induces an isomorphism i of this Jacquet module with S(En). The action of

O2(F ) × T × Sp2n(F )

n on S(E ) is obtained as follows. Let h ∈ O2(F ), g ∈ Sp2n(F ) and t = diag(a, a−1) ∈ T . Then

i(ωψ,2n+2,2(htg)φ)(y1, ..., yn)

= ωψ,2n+2,2(htg)φ(0, y1, ..., yn) −1 −1 = ω(t)|t|ωψ,2n+2,2(g)φ(0, h y1, ..., h yn)(7.7)

= ωα(t)ωψ,2n,2(hg)i(φ)(y1, ..., yn). By (7.5) and (7.6), we conclude that

−1 ∗ HomO2(F )×F ×Sp2n(F )(ResSp2n(F )π ⊗ ωψ,2n,2, ωη1η2 ) 6= 0. This implies that

−1 (7.8) η1η2 = ω, and

(7.9) HomO2(F )×Sp2n(F )(ResSp2n(F )π ⊗ ωψ,2n,2, 1) 6= 0. ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 29

Let us re-denote (2n) (2n) π = π = πµ1,...,µ2n . Then we write

π(2n) ∼= IndGL2n(F ) (π(2n−1) ⊗ µ ), µ1,...,µ2n P2n−1,1(F ) µ1,...,µ2n−1 2n where P2n−1,1 is the standard parabolic subgroup of GL2n(F ), corresponding to the partition (2n − 1, 1), and the rest of notation is self evident. Since GL2n(F ) = P2n−1,1(F )Sp2n(F ), we have

(7.10) Res π ∼= IndSp2n(F )(π(2n−1)), Sp2n(F ) Q1(F ) µ,Q1 where Q1 is the standard parabolic subgroup of Sp2n(F ), whose Levi ∗ part is isomorphic to F ×Sp2n−2(F ). The induction is normalized and the inducing representation is deﬁned as follows x u z  x u (7.11) π(2n−1) g u0 = µ−1(x)π(2n−1) . µ,Q1   2n µ1,...,µ2n−1 g x−1

Note that the unipotent radical of Q1 does not act trivially in (7.10). By Frobenius reciprocity, we conclude from (7.9) that

− 1 (7.12) Hom (δ 2 π(2n−1) ⊗ Res (ω ), 1) 6= 0. O2(F )×Q1 Q1 µ,Q1 O2(F )×Q1 ψ,2n,2

Note that the center C of the unipotent radical of Q1 acts trivially − 1 through the representation δ 2 π(2n−1). Thus, in (7.12), we may replace Q1 µ,Q1 the Weil representation by its Jacquet module along C, which, in turn, is isomorphic to

ωα ⊗ ωψ,2n−2,2,

as a module over MQ1 , the Levi part of Q1. This is the content of (7.6) (with 2n + 2 replaced by 2n). When we transfer, through the analog in this case of the isomorphism i above, the action of the unipotent radical UQ1 of Q1, we see that

UQ1 acts trivially on the image of i. This follows immediately from the formulas of the Weil representation. Thus, in (7.12), we may also replace π(2n−1) by its Jacquet module along U . From the definition µ,Q1 Q1 30 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY of π(2n−1) (7.11), we now get that the following Hom-space µ,Q1 (7.13) 1−n −1 (2n−1) Hom ∗ (ωα µ J (π ) ⊗ ω , 1) O2(F )×F ×Sp2n−2(F ) 2n UP1,2n−2 µ1,...,µ2n−1 ψ,2n−2,2 is nonzero. Here, J denotes the Jacquet functor, along U , UP1,2n−2 P1,2n−2 the unipotent radical of P1,2n−2. Write now π(2n−1) ∼= IndGL2n−1(F )(µ ⊗ π(2n−2) ). µ1,...,µ2n−1 P1,2n−2(F ) 1 µ2,...,µ2n−1 ∗ As modules over F × GL2n−2(F ), we have J (IndGL2n−1(F )(µ ⊗ π(2n−2) ))(7.14) UP1,2n−2 P1,2n−2(F ) 1 µ2,...,µ2n−1 2n−1 n−1 − 1 (2n−2) ≡ ⊕ α µ ⊗ (α ◦ det) 2 π . i=1 i µ1,...,µî,...,µ2n−1 Here, the decomposition is only in the sense of semi-simplification. This is obtained by standard Mackey theory. See [B-Z], Section 2.12. From this and from (7.13), we conclude that there is 1 ≤ i ≤ 2n − 1, such that (7.15) −1 (2n−2) Hom ∗ (ωµ µ π ⊗ ω , 1) 6= 0. O2(F )×F ×Sp2n−2(F ) i 2n µ1,...,µî,...,µ2n−1 ψ,2n−2,2

This implies that µ2n = ωµi. Up to a permutation, we may assume that i = 1, and then we must also have

(2n−2) (7.16) HomO2(F )×Sp2n−2(F )(πµ2,,...,µ2n−1 ⊗ ωψ,2n−2,2, 1) 6= 0. This is (7.9), with n replaced by n − 1. Now, we continue by induction. We conclude that after a suitable reordering of the characters µi, we must have

µ2n−1 = ωµ2, µ2n−2 = ωµ3, ..., µn+1 = ωµn, and so π ∼= IndGL2n(F )(µ ⊗ · · · ⊗ µ ) ⊗ ω(µ ⊗ · · · ⊗ µ ). B2n(F ) 1 n n 1 In particular, (7.2) now implies that

n+1 2 2n+1 2 η1η2 = χπω = (µ1 ··· µn) ω = (µ1 ··· µn) ω. By (7.8) we conclude that

2 2 η2 = (µ1 ··· µn) . ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 31

Since all characters µi, ηj are unramiﬁed, we conclude from (7.8) that, up to switching the order of η1, η2,

η1 = µ1 ··· µn

η2 = ωµ1 ··· µn.

This proves the proposition.

Consider now the case where O2,ω is split over F . The setup is the same. Then the component of ω at F is the trivial character. We know that at F , the component of θω,n+1 is the unramiﬁed quotient of

GSp2n+2(F ) Ind 00 (1), Q1 (F ) 00 where Q1 is the standard parabolic subgroup of GSp2n+2(F ), which preserves a line, and the induction is normalized. Thus, by analogy to (3.1), we assume that

GSp2n+2(F ) 0 (7.17) Hom(GSp (F )×GL2(F )) (ResGSp (F )π ⊗ Ind 00 (1), τ) 6= 0. 2n 2n Q1 (F ) Here, π and τ are exactly as before. Similarly, as in (7.2), we assume that

(7.18) η1η2 = µπ. In this case, ﬁnding a similar description of the unramiﬁed parameters is more complicated. We will make a certain reduction of the computation of this Hom-space, and carry it out for n = 2. The reduction is as follows.

Lemma 7.2. The space (7.17) is nonzero, if and only if

GSp2n(F ) (7.19) HomGSp (F )(ResGSp (F )π, Ind 0 (χη)) 6= 0. 2n 2n Q1(F ) 0 Here, Q1 is the standard parabolic subgroup of GSp2n(F ), which preserves a line, and, up to switching the order of η1, η2, χη is the following 0 character of Q1, ax u z  0 −1 χη  g u  = η1(a)η1η2 (x). x−1 32 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Proof. As in the previous case, we apply Frobenius reciprocity to (7.17) and obtain, with the same notation, that

1 GSp2n+2(F ) 2 (7.20) Hom (π ⊗ J (Ind 00 (1)), δ (η ⊗ η )) 6= 0. M 00 Z Q (F ) B2 1 2 Q1 1

00 Here, M 00 is the Levi part of Q . We compute the Jacquet module Q1 1 GSp2n+2(F ) JZ (Ind 00 (1)) by Bruhat theory. First note that Q1 (F )

00 00 Q1\GSp2n+2(F )/Q1 has three elements, whose representatives can be chosen as follows.

I2n+2, w1, w, where 0 1 0 1 w = diag( , I , ) 1 1 0 2n−2 1 0

 1 and w =  I2n . The contribution of I2n+2 to (7.20) is zero. −1 Indeed, this contribution is equal to

1 1 2 2 (7.21) Hom (π ⊗ δ 00 , δ (η ⊗ η )). M 00 Q B2 1 2 Q1 1

∗ For g ∈ GSp2n(F ), with similitude factor a, and x ∈ F , we have ax  1 n+1 2 2 n+1 π ⊗ δQ00  g  = |a| |x| π(g), 1 x−1

1 1 −1 while δ 2 (η ⊗η ) assigns to this element the value |a| 2 |x|η (ax)η (x). B2 1 2 1 2 −1 n Thus, if the space (7.21) is nonzero, we must have η1η2 = |x| . This is impossible, since by our assumption τ is unitary and generic. Then

zi 1 we know that ηi has the form |x| , where |Re(zi)| < 2 . Let us examine the contribution of w1. This is done in several steps. 00 0Q1 First, the following subquotient indR (ξn) contributes to

GSp2n+2(F ) ResQ00 (Ind 00 (1)), 1 Q1

−1 00 00 0 where R = w1 Q1w1 ∩ Q1, ind denotes compact, unnormalized induction, and ξn is the following character of R. R consists of the following ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 33 elements in GSp2n+2(F ): ax 0 v r z   ay u s r0   0 0   g u v  ,  −1   y 0  x−1

n+1 n+1 and ξn assigns to such an element the value |a| 2 |y| . 00 0Q1 Next, consider ResM 00 (indR (ξn)). Again, by Bruhat theory, we Q1 00 get two subquotients parametrized by R\Q /M 00 . This set has two 1 Q1 0 elements, whose representatives can be chosen as I2n+2, w1, where 1 1 1 −1 w0 = diag( , I , ). 1 0 1 2n−2 0 1

The contribution of I2n+2 to (7.20) is zero, since the corresponding 0M 00 Q1 −1 subquotient is indR∩M 00 (ξn), and now note thatx ˆ := diag(x, I2n, x ) Q1 lies in R ∩ M 00 , is central in M 00 , and ξ (ˆx) = 1, while Q1 Q1 n 1 δ 2 η ⊗ η (ˆx) = |x|η η−1(x) B2 1 2 1 2 cannot be the trivial character by the same reason as before. The sub- 00 0M 00 0Q1 0 Q1 0 quotient of ResM 00 (indR (ξn)), which corresponds to w1 is indL (ξn). Q1 Here, 0 −1 0 L = (w ) Rw ∩ M 00 , 1 1 Q1 and it consists of all elements of GSp2n(F ) of the form ax   ax u s   0   g u  ,  −1   x  x−1 0 and ξn is the character of L, which assigns to such an element the value n+1 n+1 |a| 2 |x| . Thus, the contribution of this subquotient to (7.20) is

0M 00 1 Hom (Res π ⊗ ind Q1 (ξ0 ), δ 2 η ⊗ η ). M 00 GSp2n(F ) L n B2 1 2 Q1 1 2 Recall that δ η ⊗ η acts through the projection of M 00 to the diago- B2 1 2 Q1 nal subgroup of GL2(F ). By Frobenius reciprocity, the last Hom-space is isomorphic to the one in (7.19). 34 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

Finally, consider the contribution of w to (7.20). First, the following 00 0Q1 GSp2n+2(F ) subquotient ind (β ) contributes to Res 00 (Ind 00 (1)), where M 00 n Q1 Q Q1 1

−1 − n+1 −(n+1) βn(diag(ax, g, x ) = |a| 2 |x| , λ(g) = a.

Next, it is easy to see that

0Q00 0Q00 1 ∼ 1 0 JZ (indM 00 (βn)) = indM 00 Z (βn), Q1 Q1 where ax 0 z  1−n 0 2 1−n βn  g 0  = |a| |x| . x−1

0Q00 Now, consider, as before, the restriction Res (ind 1 (β0 )). The MQ00 M 00 Z n 1 Q1 00 set of double cosets M 00 Z\Q /M 00 has two elements, whose represen- Q1 1 Q1 tatives may be chosen to be I2n+2 and   1 u0 0 0 v0 =  I2n u0 , 1 where u0 = (0, ..., 0, 1). As in the previous case, the contribution of

I2n+2 to (7.20) is zero, and the contribution of v0 is isomorphic to the space in (7.19), with η1, η2 switched. This proves the lemma.

Let us carry out the computation of the Hom-space in (7.19), for n = 2.

Lemma 7.3. With the notation of Lemma 7.2, assume that n = 2. Assume that the Hom-space in (7.19) is nonzero. Then, up to a permutation of the unramiﬁed characters µ1, µ2, µ3 and µ4, we have η1 = µ1µ2 and η2 = µ3µ4.

Proof. The proof is by Bruhat theory. We sketch the details. They are similar to the previous cases. As in (7.10), we may write

∼ GSp4(F ) (3) ResGSp (F )π = Ind 0 (π 0 ), 4 Q1 µ,Q1 ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 35 where   ax u z (3) 0 −1 (3) ax u π 0  g u  = µ4 (x)πµ ,µ ,µ . µ,Q1 1 2 3 g x−1 Thus, by Frobenius reciprocity,

1 GSp4(F ) (3) 2 (7.22) HomQ0 (Ind 0 (π 0 ), δ 0 χη) 6= 0. 1 Q1 µ,Q1 Q1

Consider the three representatives I4, w1, w of

0 0 Q1\GSp4(F )/Q1, as we did right after (7.20) (with 2n + 2 = 4). The contribution of I4 to (7.22) is

(3) (3) 0 ∼ ∗ (7.23) HomQ (π 0 , χη) = HomF ×GL2(F )(JUP (πµ ,µ ,µ ), χη,µ4 ), 1 µ,Q1 1,2 1 2 3 ∗ (3) (3) ax 0 where F × GL2(F ) acts in JU (πµ ,µ ,µ ) via π , with P1,2 1 2 3 0 g g ∈ GL2(F ), such that det(g) = a. The character χη,µ4 assigns to −1 the last matrix the value η1(a)µ4η1η2 (x). As in (7.14), up to semi- simpliﬁcation,

(3) − 1 (2) J (π ) ≡ | · |µ ⊗ | det | 2 π ⊕ UP1,2 µ1,µ2,µ3 1 µ2,µ3 1 1 − 2 (2) − 2 (2) (7.24) | · |µ2 ⊗ | det | πµ1,µ3 ⊕ | · |µ3 ⊗ | det | πµ1,µ2 . Consider, for example the contribution of the ﬁrst piece to (7.23). If this is nontrivial, then we must have that

1 (2) − 2 −1 HomGL2(F )(πµ2,µ3 , | det | η1µ1 ◦ det) 6= 0. (2) This is impossible, since πµ2,µ3 is irreducible and generic. Similarly, the second and third terms in (7.24) contribute zero. Thus, the contribution of I4, i.e. the Hom-space (7.23) is zero.

The contribution of w1 to (7.22) is

(7.25) Hom (J (π(3) ), χ0 ), D UP2,1 µ1,µ2,µ3 η,µ4 where D is the diagonal subgroup of GL3(F ), and

0 −1 −1 −1 χη,µ4 (diag(a, b, c)) = |ac |µ4(a)µ4 η1(b)µ4 η2(c). 36 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

As in (7.24), we have

(3) 1 (2) −1 J (π ) ≡ | det | 2 π ⊗ | · | µ ⊕ UP2,1 µ1,µ2,µ3 µ1,µ2 3 1 1 2 (2) −1 2 (2) −1 (7.26) | det | πµ1,µ3 ⊗ | · | µ2 ⊕ | det | πµ2,µ3 ⊗ | · | µ1. For example, if the ﬁrst term in (7.26) contributes to the Hom-space in (7.25), then η2 = µ3µ4, and from (7.18), it follows that η1 = µ1µ2. The two other terms in (7.26) will give similar relations, with suitable permutations of µ1, ..., µ4. The contribution of w to (7.22) is

(7.27) Hom (π(3) , χ00 ), MP1,2 µ1,µ2,µ3 η,µ4 where MP1,2 is the Levi part of the standard parabolic subgroup P1,2 of GL3(F ), and

00 −1 −1 χη,µ4 (diag(x, g)) = µ4 η1(det(g))µ4η1 η2(x). Writing π(3) ∼ IndGL3(F )(µ ⊗ π(2) ), µ1,µ2,µ3 = P1,2 1 µ2,µ3 we use Bruhat theory once again and conclude, with exactly the same reasoning as above, that if the Hom-space in (7.27) is nonzero, then there is 1 ≤ i ≤ 3, such that η1 = µiµ4 (and this determines η2 by (7.18)). This ﬁnishes the proof for n = 2.

8. A proof of Conjecture 1.2 in case n = 3

In this section we complete the proof of Conjecture 1.2 for the case of n = 3. Conjectures 5.4 and 5.5, for n = 3, may be treated in a similar way, but we omit the details here. Let π denote an irreducible cuspidal automorphic representation of

GL6(A). In Proposition 5.2, we proved one direction of Conjecture 1.2, when n = 3. In the second direction, we assume that the partial L-functions LS(π⊗χ, ∧3, s) are holomorphic at s = 1, for all characters 2 χ, such that µπωχ = 1. We want to show that the partial descent of

π to GL2(A) is cuspidal. By Proposition 3.1, it is enough to show that ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 37 the integral (3.2) is identically zero. This is equivalent to showing the vanishing of the following co-period integrals, for all χ, as above. Z (8.1) ϕπ(g)θω,3(g)χ(s(g))dg.

Z(A)GSp6(F )\GSp6(A) We will prove that if the above co-period integral is nonzero, for some χ, then the partial L function LS(π ⊗ χ, ∧3, s) has a simple pole at s = 1. To show this, we will construct a certain Eisenstein series on the (split) exceptional group G = E6, and show that the non-vanishing of (8.1) implies that this Eisenstein series has a simple pole, and using the result of [K], Theorem 4.11, this will imply that LS(π ⊗ χ, ∧3, s) has a simple pole at s = 1.

For basic notation related to the group E6 we refer the reader to [G2]. Let R denote the maximal parabolic subgroup of the exceptional group

G = E6, whose Levi part contains the group of type A5. Let Eπ,χ(m, s) denote the Eisenstein series associated to the induced representation (normalized) 1 E6(A) 0 s− 2 IndR(A) (πχ δR ).

If we parametrize the maximal torus of E6 by t = h(t1, t2, . . . , t6), then 0 0 the character χ is deﬁned by χ (t) = χ(t2). From a simple calculation of the constant term of the Eisenstein series, it follows that the poles of the Eisenstein series Eπ,χ(m, s) are determined by the poles of the product 11 LS(π ⊗ χ, ∧3, 11s − )LS(µ χ, 22s − 11). 2 π Here, the second partial L-function is the one dimensional L-function attached to the character µπχ. It follows from [K] that if the Eisenstein 13 series Eπ,χ(m, s) has a simple pole at s = 22 , then the partial L-function LS(π⊗χ, ∧3, s) has a pole at s = 1. If the Eisenstein series has a pole at 13 s = 22 , we shall denote by Eπ,χ(m) its residue. More precisely we will show that the non-vanishing assumption of the integral (8.1) implies 13 that Eπ,χ(m, s) has a simple pole at s = 22 . In fact we are going to show that the residue Eπ,χ(m) of the Eisen- 13 stein series Eπ,χ(m, s) at s = 22 has a certain nontrivial co-period over 38 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY the split exceptional group H = F4 of type F4 if the integral (8.1) is not identically zero. To show that, we have introduce a special residual representation deﬁned on the exceptional group H = F4. For basic notation related to the group F4, we refer the reader to [G-J]. Let L denote the maximal parabolic subgroup of H = F4 whose Levi part is GSp6. Take Θω,3 to be the residue of the Eisenstein series on GSp6 considered in §2.2. Let E(h, ω, s) denote the Eisenstein series associated with the induced representation 1 F4(A) s− 2 IndL(A) (Θω,3δL ), A constant term calculation of this Eisenstein series implies that the poles of this Eisenstein series are determined by the poles of the product of L-functions 11 11 ζS(8s − )ζS(16s − 8)ζS(16s − 10)LS(ω, 8s − )LS(ω, 16s − 8). 2 2 13 Thus the Eisenstein series has a simple pole at s = 16 , and we shall denote by Θω the residual representation of H(A) generated by these residues, which we typically denote by θω(h). We will prove that if the integral (8.1) is nonzero for some choice of data, then the integral Z (8.2) Eπ,χ(h)θω(h)dh

H(F )\H(A) is nonzero for some choice of data. In particular Eπ,χ(m, s) has a sim- 13 ple pole at s = 22 . Note that integral (8.2) is deﬁned through the regularization in terms of the Arthur truncation method, which will be discussed below. More precisely, we apply the Arthur truncation to the

Eisenstein series Eπ,χ(m, s). This method was used in various cases, see for example [G-J], and therefore, our proof here will be sketchy.

The embedding of H inside E6 is as follows. For 1 ≤ i ≤ 6, we label the simple roots of E6 by αi, as in [G2]. We label the simple roots of H by βj, for 1 ≤ j ≤ 4, where the simple roots in GSp6 are

β2, β3 and β4. For a root δ, we denote by xδ(q) the one dimensional ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 39 unipotent subgroup generated by this root. With these notations, the four unipotent subgroups of H corresponding to the four simple roots, are embedded in E6 as follows

xβ1 (q) = xα2 (q);

xβ2 (q) = xα4 (q);

xβ3 (q) = xα3 (q)xα5 (q);

xβ4 (q) = xα1 (q)xα6 (q).

We recall Arthur’s truncation formula, in our special case. Since R is maximal parabolic subgroup of G = E6, aR is one dimensional. We c identity aR with R, as usual. Let c be a real number c > 1. Let τ be the function on R>0, which is the characteristic function of the subset c R≥c. Put τc = 1R>0 − τ . By Sec. 2.13 of [M-W], the truncation of the Eisenstein series is deﬁned as follows:

c X c (8.3) Λ Eπ,χ(m, s) = Eπ,χ(m, s) − Eπ,χ;R(γm, s)τ (H(γm)) γ∈R\G where Eπ,χ;R(m, s) is the constant term of Eπ,χ(m, s) along the maximal parabolic subgroup R. We may take c as large as we may need. We write, for Re(s) large, X Eπ,χ(m, s)(g) = fπ,χ(γg, s), γ∈R\G where fπ,χ(γg, s) is a standard section. As in Sec. 4.3, [G-J], or Sec. 6 in [G-J-S] we have an identity of the form, for c suﬃciently large, and 13 s0 = 22 . Z c c c (8.4) Eπ,χ(h)θω(h)dh = ress=s0 [I1 − I2] + I3 H(F )\H(A)

In proving this identity, we also show that |Eπ,χ(h)θω| is integrable on c H(F )\H(A). The Ij have the following form Z c c Ij = ξj (h)θω(h)dh, H(F )\H(A) 40 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY where c c X ξ1(g) = ξ1(g, s) = fπ,χ(γg, s)τc(H(γg)), γ∈R\G

c c X c ξ2(g) = ξ2(g, s) = M(s)fπ,χ(γg, s)τ (H(γg)), γ∈R\G

c X c ξ3(g) = ress=s0 M(s)fπ,χ(γg, s)τ (H(γg)). γ∈R\G Here, M(s) is the corresponding intertwining operator. The integral c I2 converges absolutely at all points s, where the intertwining operator c is holomorphic (c is suﬃciently large). The integral I1 converges absolutely, for Re(s) suﬃciently large and has a meromorphic continuation c to the whole plane. Of course, I2 is meromorphic in C. Similarly, we c show that I3 converges absolutely, and hence, we conclude that the integral (8.2) converges absolutely. Since

c c Λ Eπ,χ = Eπ,χ − ξ3, we have Z c c c (8.5) Λ Eπ,χ(h, s)θω(h)dh = I1 − I2. H(F )\H(A)

Taking residues at s0, we get (8.4). We also have

c c (8.6) ress=s0 I2 = I3. Hence, from (8.4), Z c (8.7) Eπ,χ(h)θω(h)dh = ress=s0 I1. H(F )\H(A) In what follows, we give some details of the proof of the last identity c and the meromorphic continuation of I1; will omit the similar details c for I2. We have Z c X (8.8) I1 = fπ,χ(γh, s)τc(H(γh))θω(h)dh. H(F )\H(A) γ∈R\G ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 41

When Re(s) is suﬃciently large, we have Z c X (8.9) I1 = fπ,χ(γh, s)τc(H(γh))θω(h)dh, γ γ∈R\G/H H \H(A)

γ −1 where H := γ R(F )γ ∩ H(F ). Since H = F4 is a spherical subgroup in G = E6, it is easy to check that H(F ) has two F -rational orbits acting in the projective variety R(F )\G(F ). The representatives can be taken to be the identity e and the Weyl group element w2w4w5.

For the double coset Rw2w4w5H, one has

w2w4w5 H = w5w4w2Rw2w4w5 ∩ H = GSpin6 · U, where U is the unipotent radical of the maximal parabolic of H whose

Levi part is GSpin7. Using the smallness of Θω, and the cuspidality of π, one can show (by taking further Fourier expansions) that the integral Z fπ,χ(γh, s)τc(H(γh))θω(h)dh w2w4w5 HF \H(A) is identically zero. The convergence of this integral should be proved as in the other cases treated in ([G-J], [G-J-S]); we omit the details. Hence we obtain Z c (8.10) I1 = fπ,χ(h, s)τc(H(h))θω(h)dh. e HF \H(A) It is clear that He = R ∩ H = L, which is the maximal parabolic subgroup of H with Levi decomposition

L = GSp6 · V. Q Using the Iwasawa decomposition H(A) = L(A)·K, where K = v Kv is the standard maximal compact subgroup of H(A), we have (8.11) Z Z Z c −1 I1 = fπ,χ(gk, s)τc(H(g)) θω(vgk)χ(λ(g))δL (g)dvdgdk K g V (F )\V (A) R where g is the integration over GSp6(F )\GSp6(A). 42 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY

First we consider the inner integral Z V (8.12) θω (gk) := θω(vgk)dv. V (F )\V (A)

13 It is the constant term of the residue at s = 16 of an Eisenstein series E(h, ω, s).

Denote by eh(t1, t2, t3, t4) the maximal torus of H. Here eh(t1, 1, 1, 1) is the maximal torus of the copy of SL2 which corresponds to the root β1 etc. Via the embedding of H inside E6, this maximal torus is embedded in the maximal torus of E6 as h(t4, t1, t3, t2, t3, t4). The center of GSp6, inside H is given by

de(z) = eh(z2, z3, z2, z), and its embedding in E6 is as

d(z) = h(z, z2, z2, z3, z2, z).

Then we have 22 δR(d(z)) = |z| and 16 δL(de(z)) = |z| . We have 13 V 1− 16 V θω (de(z)gk) = ω(z)δL (de(z))θω (gk). Similarly, we have

−1 2 −1 22s −1 fπ,χ(d(z)gk, s)χ◦λδL (de(z)g) = µπχ (z)δL (de(z))|z| fπ,χ(gk, s)χ◦λδL (g).

By factoring the dg-integration through the center of GSp6, we ob- 2 tain, using that ωµπχ = 1, (8.13) Z Z Z c 1 −1 V 22s−13 × I1 = fπ,χ(gk, )χ◦λδL (g)θω (gk)ddgdk |z|A dz . × × K g 2 F \A ,|z|A≤c R where g is the integration over ZAGSp6(F )\GSp6(A). This integra- 0 0 tion can be carried out in GSp6(F )\GSp6(A), where GSp6(A) is the ON CERTAIN AUTOMORPHIC DESCENTS TO GL2 43 subgroup of GSp6(A) of all elements g, such that |λ(g)| = 1. It is easy to check that Z 22s−13 22s−13 × × 1 c (8.14) |z|A dz = µ(F \A ) · , × × F \A ,|z|A≤c 22s − 13 where µ(F ×\A1) is the volume of the quotient F ×\A1 with the canon- c ical Haar measure. Now it is clear that I1 converges absolutely, when 13 Re(s) > 22 , and it has a meromorphic continuation to the whole plane, 13 with at most a simple pole at s = 22 . We have (8.15) Z Z c × 1 1 −1 V res 13 I = µ(F \A ) · fπ,χ(gk, )χ ◦ λδ (g)θ (gk)dgdk. s= 22 1 L ω K g 2 We summarize the above calculation as

13 Theorem 8.1. The co-period of the residue at s = 22 , Eπ,χ(g), of the Eisenstein series Eπ,χ(g, s) satisﬁes the following identity: Z Z Z 1 E (h)θ (h)dh = c f (gk, )χ ◦ λδ−1(g)θV (gk)dgdk π,χ ω π,χ 2 L ω [H] K [GSp6] where [H] = H(F )\H(A), [GSp6] := ZAGSp6(F )\GSp6(A) and c = µ(F ×\A1).

In order to ﬁnish the proof of Conjecture 1.2 for n = 3, we ﬁrst deduce that the inner integration (i.e. the dg-integration) is the same V as the co-period given in (8.1). We consider the constant term θω (h) 13 of the residue θω(h). Recall that θω(h) is the residue at s = 16 of the V E(h, ω, s). It is a routine to check that the constant term θω (h) is equal 13 to the residue at s = 16 of the standard intertwining operator from the 1 F4(A) s− 2 (normalized) induced representation IndL(A) (Θω,3δL ) to the induced 1 F4(A) 2 −s V representation IndL(A) (Θω,3δL ). Hence the constant term θω (h) of the residue θω(h) represents a section φ 1 13 (h) belonging to the in- ω, 2 − 16 1 13 F4(A) 2 − 16 duced representation IndL(A) (Θω,3δL ). Now the dg-integration is taken place over the Levi subgroup of L, which is GSp6. It follows that if co-period (8.1) is nonzero for some choice of data, then the dg- integration as the inner integration in Theorem 8.1 is nonzero for some choice of data. Then we show that the dk-integration does not destroy 44 DAVID GINZBURG, DIHUA JIANG, AND DAVID SOUDRY this nonvanishing property. See the argument in the similar cases of [G-J] and [G-J-S], for example. Hence, by the identity in Theorem 8.1, the co-period of the residue Eπ,χ is nonzero. In particular, the residue itself is nonzero. As we explained in the beginning of this section, this implies that the partial L-function LS(π ⊗ χ, ∧3, s) has a simple pole at s = 1. This completes the proof of the converse of Proposition 5.2, which is analogue of Proposition 5.1 for n = 3.

Proposition 8.2. Let π be an irreducible cuspidal automorphic representation of GL6(A). Assume that the partial L-functions LS(π ⊗ χ, ∧3, s)

3 2 are holomorphic at s = 1 for all characters χ, which satisfy µπω χ = 1. Then the integral (5.1) (with n = 3) is identically zero, and hence the partial descent of π to GL2(A) is cuspidal.

Hence we complete the proof of Theorem 5.3, that is, Conjecture 1.2 holds for n = 3.

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School of Mathematical Sciences, Sackler Faculty of Exact Sci- ences, Tel-Aviv University, Israel 69978 E-mail address: [email protected]

School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA E-mail address: [email protected]

School of Mathematical Sciences, Sackler Faculty of Exact Sci- ences, Tel-Aviv University, Israel 69978 E-mail address: [email protected]