ENTIRENESS OF THE SPINOR

ENTIRENESS OF THE SPINOR L-FUNCTIONS FOR CERTAIN GENERICCUSP FORMS ON GSp(2)

By TOMONORI MORIYAMA

Abstract. Let G = GSp(2) be the symplectic group with similitude of degree two, which is definedover Q. For a generic cusp form F on the adelized group GA whose archimedean type is either a(limit of) large discrete series representation or a certain principal series representation, we showthat its spinor L-function is continued to an entire function and satisfies the functional equation.

0. Introduction. Let G = GSp(2) be the symplectic group with similitudeof degree two, defined over a global field k. Suppose that Π = ⊗′vΠv is a genericcuspidal automorphic representation of the adelized group GAk , that is, Π hasa nonvanishing global Whittaker model. In the 1970s, Novodvorsky announcedthat the spinor L-function L(s, Π) of Π, which is originally defined for Re(s)� 0,is continued to an entire function and satisfies the functional equation when thebase field k is a function field. Later the missing detailed proofs of the resultsin Novodvorsky’s paper [N] are supplied by Bump [B1, §3] and Takloo-Bighash[TB].

The purpose of this paper is to establish similar results to Novodvorsky’sfor the completed spinor L-function Λ(s, Π) when the base field k is a totallyreal number field, provided that the local factor Πv at each infinite place v of kis either a (limit of) square integrable representation modulo center or a certainprincipal series representation. In the text, we actually take the field Q of rationalnumbers as the base field k, for the sake of simplicity.

In order to show the result mentioned above, Novodvorsky [N, §1] introduceda zeta integral ZN(s, F) (F ∈ Π) that represents the spinor L-function L(s, Π) ofΠ. It is known that the zeta integral ZN(s, F) equals an integral transform of theglobal Whittaker function WF of F. If a cusp form F ∈ Π is decomposable inthe restricted tensor product Π = ⊗′vΠv , then the global Whittaker function WF

of F can be written as a product of local Whittaker functions:

WF(g) =∏v

W (v)F (gv), g = (gv) ∈ GAk .

Hence the zeta integral ZN(s, F) is decomposed into a product of local zeta in-tegrals Z(v)

N (s,W (v)F ), which are integral transforms of local Whittaker functions

Manuscript received September 27, 2002.American Journal of Mathematics 126 (2004), 899–920.

899

900 TOMONORI MORIYAMA

W (v)F . For every unramified place v, it is known that Z(v)

N (s,W (v)F ) coincides with

the local factor of the spinor L-function.Our method in this paper is also based on Novodvorsky’s theory of zeta

integrals. The difference from the function field case is that we have to computethe local zeta integrals Z(v)

N (s,W (v)F ) at the real places v for some local Whittaker

functions W (v)F . To this end, we use an explicit formula of the local Whittaker

function on Gkv∼= GSp(2, R). The formula is given by an integral of Mellin-

Barnes type, which is essentially obtained in our previous paper [Mo].We stress that the genericity of Π implies that each local component Πv

of Π must have a Whittaker model. In particular, if v is archimedean, then atheorem of Kostant [Ko, Theorem 6.8.1] tells us that the representation Πv ofGkv must be large in the sense of Vogan. Since (limits of) holomorphic discreteseries representations of Sp(2, R) are not large, Novodvorsky’s method is notapplicable for holomorphic modular forms. By [V, Proposition 6.2 (f)], we knowthat the irreducible large representations of Gkv are divided into four classes,each of which corresponds to the conjugacy classes of the parabolic subgroupsof G. The representations of Gkv that we treat in this paper occupy two ofthem corresponding to G itself or the non-Siegel parabolic subgroup of G. Onthe other hand, we do not succeed in handling the other two cases at present,although there are some explicit formula of Whittaker functions ([I],[Ni]) for suchcases.

Here we remark that there is a different method to obtain the same spinorL-function due to Andrianov [An]. Hori [H] extended it, which was originallydiscussed for holomorphic modular forms, to the automorphic forms whose archi-medean types are spherical principal series representations, and Miyazaki [Mi]to the automorphic forms with large discrete series at the infinite place.

The organization of this paper is as follows. In the first section we formulateour main results. In the second section we collect known facts on Novodvorsky’szeta integral. The main result is proved in the final section.

Acknowledgments. I would like to thank Professor Takayuki Oda for hisencouragement and valuable advice. I am also grateful to Professor Hisayosi Ma-tumoto for useful conversations on Whittaker models. I should add gratefully thatProfessor Ramin Takloo-Bighash suggested an improvement to the first versionof this paper.

Notation and conventions. For each place v of the field Q of rationalnumbers, we denote by Qv the completion of Q at v. The module of an elementx ∈ Qv is denoted by |x|v . For a finite place p of Q, Zp stands for the ring ofintegers in Qp. The adele ring (resp. the idele group) is denoted by A (resp. A×).The module of an element x ∈ A× is denoted by |x|A. Unless otherwise stated, weunderstand that all the measures on locally compact unimodular groups are Haarmeasures. For a finite place p of Q, we normalize the Haar measure on Qp (resp.

SPINOR L-FUNCTIONS 901

Q×p ) so that vol(Zp) = 1 (resp. vol(Z×p ) = 1). As usual we set ΓC(s) = 2(2π)−sΓ(s)(s ∈ C), where Γ(s) is the Gamma function.

1. Preliminaries and the main theorem. In this section we introduce basicingredients of this paper and state our main result.

1.1. Global Whittaker functions on GSp(2). Let G = GSp(2) be the sym-plectic group with similitude of degree two, which is defined by

G := {g ∈ GL(4) | tgJ4g = ν(g)J4 for some ν(g) ∈ Gm}, J4 :=

(12

−12

).

We regard G as an algebraic group defined over Q. For any Q-algebra R, thegroup of R-valued points of G is denoted by GR. We adopt the same conventionfor algebraic subgroups of G. The space of cusp forms on GA is denoted byAcusp(GQ\GA). Let Π be a cuspidal automorphic representation of GA. We regardΠ as a subspace of Acusp(GQ\GA).

We fix a maximal unipotent subgroup of G defined over Q as follows:

N =

n(x0, x1, x2, x3) :=

1 x0

11−x0 1

·

1 x1 x2

1 x2 x3

11

|xi ∈ A

.

Define a character ψ : NA → C(1) of NA by

ψ(n(x0, x1, x2, x3)) = eA(− x0 − x3).

Here eA : A/Q → C(1) is the additive character of A characterized by eA(t∞) =exp (2π

√−1t∞) (t∞ ∈ R). For an automorphic form F on GA, the global Whit-

taker function WF attached to F is defined by

WF(g) :=∫

NQ\NA

F(ng)ψ(n−1)dn, g ∈ GA.

Here dn stands for the usual NA-invariant measure on NQ\NA.

1.2. The (limits of) discrete series representations of Sp(2, R). In thissubsection, we recall some basic facts on the (limits of) discrete series repre-sentations of connected semisimple Lie groups in the case of G0 := Sp(2, R) ={g ∈ GR|ν(g) = 1} ([Kn, Theorems 9.20 and 12.26]). We first fix some nota-tion. A maximal compact subgroup K (resp. K0) of GR (resp. G0) is given byK := GR ∩ O(4) (resp. K0 := Sp(2, R) ∩ O(4)). The group K0 is isomorphic to


the unitary group U(2) := {g ∈ GL(2, C)| tgg = I2} of degree two. Define anisomorphism κ : U(2) ∼= K0 by

κ : U(2) � A +√−1B �→

(A B−B A

)∈ K0, (A, B ∈ M(2, R)).

We write the Lie algebra of GR, G0 and K0 by g, g0 and k, respectively. We use thesame symbol for a continuous representation of GR (resp. G0) and its underlying(g, K)-module (resp. (g0, K0)-module). For an arbitrary Lie subalgebra l of g, wedenote its complexification l ⊗ C by lC. We write the dual space HomC (lC, C)of lC by l∗C. The differential κ∗ of κ defines an isomorphism of Lie algebras:κ∗ : gl(2, C) ∼= kC. The simple Lie algebra g0 has a compact Cartan subalgebrah := RT1 ⊕ RT2, where

T1 := κ∗

((√−1 00 0

)); T2 := κ∗

((0 00√−1

)).

Define a C-basis {β1,β2} of h∗C by βi(Tj) =√−1δij (1 � i, j � 2) and fix an

inner product 〈 , 〉 on h∗C as 〈βi,βj〉 = δi,j (1 � i, j � 2). Then the root system ∆ =∆(g0,C, hC) for the pair (g0,C, hC) is given by ∆(g0,C, hC) = {±2β1,±2β2,±(β1±β2)}. Denote by ∆c the set of compact roots in ∆: ∆c = {±(β1− β2)}. We take apositive system ∆+

c of ∆c as ∆+c := {β1 − β2}.

The irreducible finite dimensional representations of K0 are parameterized bythe set of their highest weights relative to ∆+

c :

{q = q1β1 + q2β2 = (q1, q2) ∈ h∗C|qi ∈ Z, q1 � q2}.

For each dominant integral weight q = (q1, q2), we set d = dq = q1 − q2( � 0).Then the degree of the representation (τ(q1,q2), V(q1,q2)) with highest weight (q1, q2)is dq + 1. We can take a basis {vk|0 � k � d} of V(q1,q2) so that the representationof kC associated with τ = τ(q1,q2) is given by

τ

(κ∗

((1 00 0

)))vk = (q2 + k)vk;

τ

(κ∗

((0 00 1

)))vk = (q1 − k)vk;

τ

(κ∗

((0 10 0

)))vk = (k + 1)vk+1;

τ

(κ∗

((0 01 0

)))vk = (d + 1− k)vk−1.

We call the basis {vk|0 � k � d} the standard basis of V(q1,q2).


There are four positive systems of ∆ containing ∆+c :

∆+I = {(1,−1), (2, 0), (1, 1), (0, 2)};

∆+II = {(1,−1), (0,−2), (2, 0), (1, 1)};

∆+III = {(1,−1), (− 1,−1), (0,−2), (2, 0)};

∆+IV = {(1,−1), (− 2, 0), (− 1,−1), (0,−2)}.

Let J be a variable running over the set of indices {I, II, III, IV}. We set ∆+J,n :=

∆+J \ ∆+

c . For each index J, we denote by ΞJ the set of integral weights Λ =(Λ1, Λ2) ∈ h∗C (Λi ∈ Z) satisfying (i) 〈Λ,β〉 � 0 for all β ∈ ∆+

J,n and (ii)〈Λ,β〉 > 0 for each β ∈ ∆+

c that is a simple root in ∆+J . For example,

ΞI = {(Λ1, Λ2) ∈ h∗C|Λi ∈ Z, Λ1 > Λ2, Λ2 � 0};

ΞII = {(Λ1, Λ2) ∈ h∗C|Λi ∈ Z, Λ1 + Λ2 � 0, Λ2 � 0}.

Then the set {(J, Λ)|I � J � IV , Λ ∈ ΞJ} gives the Harish-Chandra parame-terization of the (limits of) discrete series representations for G0. We denote byπ(Λ, ∆+

J ) the (limit of) discrete series representation of G0 associated to (J, Λ). If〈Λ,β〉 > 0 for all β ∈ ∆+

J , then π = π(Λ, ∆+J ) is a discrete series representation of

G0. Otherwise, it is a limit of discrete series representation of G0. The Blattnerparameter λmin ∈ h∗C of π = π(Λ, ∆+

J ) is given by

λmin := Λ +12

∑α∈∆+

J

α−∑β∈∆+

c

β.

The highest weights of the K0-types of π are of the form λmin +∑α∈∆+

Jmαα

with mα ∈ Z�0. Further, τλmin occurs in π(Λ, ∆+J ) with multiplicity one, and

we call it the minimal K0-type of π(Λ, ∆+J ). We denote the (limit of) discrete

series representation of G0 with minimal K0-type τ(q1,q2) by D(q1,q2). A (limit of)discrete series representation π(Λ, ∆+

J ) of G0 is called large if J = II or III (see [V,Theorem 6.2, (f)]). We note that π(Λ, ∆+

II) = DΛ+(1,0) and π(Λ, ∆+III) = DΛ+(0,−1).

1.3. The P1-principal series representations of Sp(2, R). We fix a maximalparabolic subgroup P1 of Sp(2, R) with Langlands decomposition P1 = M1A1N1

by

M1 :=

ε

a bε

c d

∣∣∣∣∣ε ∈ {±1},

(a bc d

)∈ SL(2, R)

;

A1 := {diag(t, 1, t−1, 1)|t > 0}; N1 := {n(x0, x1, x2, 0) ∈ NR|xi ∈ R}.


We shall introduce principal series representations of Sp(2, R) induced from P1.A (limit of) discrete series representation (σ, Vσ) of the semisimple part M1 ofP1 is of the form σ = ε� Dn(|n| � 1), where ε : {±1} → C× is a character, Dn

is the (limit of) discrete series representation of SL(2, R) with Blattner parametern, that is, the extremal weight vector v of Dn satisfies

Dn

((cos x sin x− sin x cos x

))v = e

√−1nxv, x ∈ R.

For each ν1 ∈ C, we define a quasi-character exp (ν1) of A1 by

exp (ν1)(a1) = tν1 , for a1 = diag(t, 1, t−1, 1) ∈ A1.

We call an induced representation

I(P1;σ, ν1) := C∞- IndG0P1

(σ ⊗ exp (ν1 + 2)⊗ 1N1 )

the P1-principal series representation of G0. The representation space of I(P1;σ,ν1) is given by

{F : G0 −→ Vσ| C∞-class, F(m1a1n1g) = σ(m1) exp (ν1 + 2)(a1)F(g),

∀(m1, a1, n1, g) ∈ M1 × A1 × N1 × G0},

on which G0 acts by right translation. For ν1 ∈ C in a general position, therepresentation I(P1;σ, ν1) of G0 is irreducible.

1.4. The main result. In order to state our main theorem, we will make thefollowing fundamental assumptions A.1 and A.2 on Π:

A.1. For some cusp form F in Π, the global Whittaker functionWF attachedto F does not vanish.

This is equivalent to saying that the global Whittaker function WF does notvanish for all nonzero F ∈ Π. The cuspidal automorphic representation Π canbe written as a restricted tensor product Π = ⊗′vΠv , where v runs through allthe places of Q and Πv is an irreducible admissible representation of GQv or anirreducible (g, K)-module according as v is finite or infinite.

A.2. The restriction of Π∞ to Sp(2, R) is either (α) a direct sum of two(limits of) discrete series representations D(λ1,λ2) and D(−λ2,−λ1) or (β) a directsum of two irreducible P1-principal series representations I(P1; ε ⊗ Dn, ν1) andI(P1; ε⊗ D−n, ν1).


We denote the central character of Π by ωΠ, that is ω(z14) = ωΠ(z)idΠ(z ∈ A×). Define a complex number ω∞ ∈ C by ωΠ(t) = tω∞ (t ∈ R, t > 0).Note that from the assumption A.1, the irreducible (g, K)-module Π∞ has alocal Whittaker model (in the sense of (2.1) below). Hence, by a theorem ofKostant [Ko, Theorem 6.8.1] on the existence of local Whittaker models, therepresentations of G0 occurring in Π∞ must be large. In the case of (α), we musthave 1− λ1 � λ2 � 0 or 1 + λ2 � −λ1 � 0. Without any loss of generality wemay and do assume that 1− λ1 � λ2 � 0 (resp. n � 1) in the case of (α) (resp.(β)).

The assumption A.1 tells us that, for each finite place p <∞, the irreducibleadmissible representation Πp of GQp has a local Whittaker model. Thus Propo-sition 4 in the next section enables us to attach the local L-factor (resp. ε-factor)L(s, Πp) and ε(s, Πp,ψp) to Πp. Define the spinor L-function L(s, Π) of Π by

L(s, Π) :=∏

p<∞L(s, Πp),(1)

where p runs through all the finite places of Q. As we shall see in the next section,the product converges absolutely for Re(s) > (5 − Re(ω∞))/2. Multiplying thefollowing gamma factor

L(s, Π∞) :=

{ΓC(s + ω∞+λ1+λ2−1

2 )ΓC(s + ω∞+λ1−λ2−12 ) for the case (α);

ΓC(s + ω∞+ν1+n−12 )ΓC(s + ω∞−ν1+n−1

2 ) for the case (β),

we have the completed spinor L-function Λ(s, Π):

Λ(s, Π) := L(s, Π∞)× L(s, Π).

The ε-factor ε(s, Π∞,ψ∞) for Π∞ is defined by ε(s, Π∞,ψ∞) := ( − 1)λ1 (resp.( − 1)n) in the case of (α) (resp. (β)). Then the global ε-factor ε(s, Π) is givenby ε(s, Π) :=

∏v ε(s, Πv ,ψv), where the product is taken over all the places v of

Q. For a cusp form F ∈ Π, we set

F(g) = ωΠ(ν(g))−1F(gη), η :=

1

−1−1

1

∈ GQ.

Then Π∨ := {F|F ∈ Π} is a cuspidal automorphic representation of GA. Asan admissible representation of GA, Π∨ is isomorphic to the restricted tensorproduct ⊗′vΠ∨v , where Π∨v stands for the contragredient representation of Πv .This follows from [TB, Proposition 2.3] and the assumption A.2. Since Π∨also satisfies the assumptions A.1 and A.2, we can define the spinor L-function


L(s, Π∨) =∏

p<∞ L(s, Π∨p ) and the completed L-function Λ(s, Π∨). Now we statethe main result of this paper:

THEOREM 1. Suppose that Π is a cuspidal automorphic representation of GA

satisfying the assumptions A.1 and A.2. Then:(i) the completed L-function Λ(s, Π) of Π is continued to an entire function of

s ∈ C ;(ii) we have the functional equation

Λ(s, Π) = ε(s, Π)Λ(1− s, Π∨).

2. A review on Novodvorsky’s zeta integral. As we said in the Introduc-tion, our proof of the main theorem has its basis on Novodvorsky’s zeta integralfor GSp(2). In this section we shall recall Novodvorsky’s theory. Basic referencesare [N, §1], [B1] and [TB].

2.1. Local Whittaker functions and local zeta integrals. Let us begin withthe definition of local Whittaker functions. For each place v of Q, the restrictionof ψ to NQv is denoted by ψv . Consider the representation of GQv induced fromthe character ψv of NQv :

C∞ψv(NQv\GQv ) := {W : GQv → C| smooth,

W(ng) = ψv(n)W(g), (∀(n, g) ∈ NQv ×GQv )},

on which GQv acts by right translation.Suppose that v = p is a finite place of Q. Then for an arbitrary irreducible ad-

missible representation πp of GQp , the intertwining space HomGQp(πp, C∞ψp

(NQp\GQp)) is at most one-dimensional ([R, Theorem 3]). If there is a nonzero inter-twining operator

Ψ ∈ HomGQp(πp, C∞ψp(NQp\GQp)),

then we say πp is generic and call the image Wu := Ψ(u) of u ∈ πp the localWhittaker function corresponding to u ∈ πp. The space of all Wu (u ∈ πp) iscalled the Whittaker model of πp with respect to ψp and denoted by Wh(πp,ψp).

Next we suppose that v = ∞ is the infinite place. We say that a C-valuedfunction W on GR is of moderate growth if there exist C > 0 and M > 0 suchthat |W(g)| � C‖g‖M for all g ∈ GR. Here the norm ‖g‖ of g = (gi,j) ∈ GR

is defined by ‖g‖ := max{|gi,j|, |(g−1)i,j| |1 � i, j � 4}. The space of functionsW ∈ C∞ψ∞(NR\GR) which is of moderate growth is denoted by Aψ∞(NR\GR).Improving Shalika’s local multiplicity one theorem [S, Theorem 3.1], Wallach[Wa, Theorem 8.8 (1)] showed that for an arbitrary irreducible (g, K)-module π∞the intertwining space Hom(g,K) (π∞,Aψ∞(NR\GR)) is at most one dimensional.


Again, if there is a nonzero intertwining operator

Ψ ∈ Hom(g,K)(π∞,Aψ∞(NR\GR)),

then we say π∞ is generic and call the image Wu := Ψ(u) of u ∈ π∞ the localWhittaker function corresponding to u ∈ π∞. The space of all Wu (u ∈ π∞) isdenoted by Wh(π∞,ψ∞). We call Wh(π∞,ψ∞) the Whittaker model of π∞ withrespect to ψ∞.

Let v be a finite or infinite place of Q. For each local Whittaker functionW (v) ∈ Wh(πv ,ψv), we define the local zeta integral Z(v)

N (s, W (v)) associated toW (v) by

Z(v)N (s, W (v)) :=

∫Q×v

d×y∫

Qv

dxW (v)

yy

1x 1

|y|s−3/2

v .

2.2. Global zeta integrals and the basic identity. For a cusp form F ∈Acusp(GQ\GA), Novodvorsky introduced the following global zeta integral

ZN(s, F) :=∫

A×/Q×

∫(A/Q)⊕3

F

1 x0

1x1

z1−x0 1

·

y

y1

1

× eA(x0)|y|s−1/2A dx0dx1dzd×y.

By the cuspidality of F, this integral converges for all s ∈ C and defines an entirefunction of s ∈ C.

Let Π be a cuspidal automorphic representation of GA satisfying the assump-tion A.1. We say that a cusp form F ∈ Π is decomposable if F can be written inthe form ⊗′vξF

v in the restricted tensor product Π = ⊗′vΠv . If F is decomposable,then the uniqueness of local Whittaker models implies that the global Whittakerfunction WF is decomposed into a product of local Whittaker functions:

WF(g) =∏v

W (v)F (gv), g = (gv) ∈ GA.(2)

Here W (v)F ∈ Wh(Πv ,ψv) is the local Whittaker function corresponding to the

vector ξFv ∈ Πv . Here we note the following proposition:


PROPOSITION 2. For each place v of Q there exists a constant Mv > 0 such that

|W (v)F (gv)| � Mv |ν(gv)|ω∞/2

v , ∀gv ∈ GQv .

Proof. Since we assume that F is a cusp form, the restriction of F to G1A :=

{g ∈ GA||ν(g)|A = 1} is bounded ([M-W, I.2.12, I.2.18]). If we write g ∈ GA inthe form g = z∞g1 with z∞ ∈ R>0 and g1 ∈ G1

A, then we have

|F(g)| = |F(z∞g1)| = |zω∞∞ F(g1)| = |ν(g)|Re(ω∞)/2A |F(g1)|.

Hence there exists a constant M > 0 such that

|WF(g)| � M|ν(g)|Re(ω∞)/2A , g ∈ GA.

Take an element g0 = (g0,v) ∈ GA such that WF(g0) �= 0. For each place v of Q,the estimate in the proposition holds for

Mv := M ×∏

w(=v)

|ν(g0,w)|Re(ω∞)/2w

|W (w)F (g0,w)|

,

where almost all the factors in the product are one.

We know from [C-S, Proposition 6.1] that for each finite place p there existsa constant C > 0 such that the support of the integrand of the local zeta integralZ(p)

N (s,W (p)F ) is contained in the set {(x, y) ∈ Q×p × Qp||x|p � C and |y|p � C}.

Hence, by Proposition 2, the local zeta integral Z(p)N (s,W (p)

F ) at each finite placep <∞ converges absolutely for Re(s) > (3− Re(ω∞))/2. As explained in [B1,§3], we have the following basic identity:

PROPOSITION 3. (Basic identity, see [B1, §3]) Let F ∈ Π be a decomposablecusp form. Suppose that there exists a real number σ0 ∈ R such that the local zetaintegral Z(∞)

N (s,W (∞)F ) at the infinite place converges absolutely for Re(s) > σ0.

Then we have(i) the integral

∫A×

∫AWF

yy

x1

1

|y|s−3/2

A dx d×y

converges absolutely for Re(s) > max{(5−Re(ω∞))/2,σ0}and is equal to ZN(s, F);


(ii) the global zeta integral ZN(s, F) associated to F is decomposed into aproduct of local zeta integrals:

ZN(s, F) =∏v

Z(v)N (s,W (v)

F ),

where the product converges absolutely for Re(s) > max{(5− Re(ω∞))/2,σ0}.

2.3. Local L and ε factors at finite places. Fix a finite place p of Q. Letπp be an irreducible admissible representation of GQp having a local Whittaker

model Wh(πp,ψp). For each W ∈ Wh(πp,ψp), the local zeta integral Z(p)N (s, W)

converges absolutely for Re(s) � 0 and equals a rational function in p−s. Foreach vector u ∈ πp, we put u′ := πp(diag(p, p, 1, 1))u. Then it is a simple matterto check that

Z(p)N (s, Wu′) = ps−1/2Z(p)

N (s, Wu), ∀u ∈ πp.

Hence the totality

{Z(p)N (s, W) ∈ C(p−s)| W ∈Wh(πp,ψp)}

of local zeta integrals forms a C[p−s, ps]-module in C(p−s). For each place v ofQ, we denote by ηv the image of η ∈ GQ by the natural inclusion GQ ↪→ GQv . Fora local Whittaker function W ∈Wh(πp,ψp), we set W(g) := ωπp(ν(g))−1W(gηp)(g ∈ GQp), where ωπp stands for the central character of πp. Then W belongs tothe local Whittaker model Wh(π∨p ,ψp) of the contragredient representation π∨p ofπp. Then the local L (resp. ε) factor L(s,πp) (resp. ε(s,πp,ψp)) of πp is definedin the following proposition:

PROPOSITION 4. ([B1, §3] and [TB, Theorems 2.1, 3.6 and 4.1]) Let πp be anirreducible admissible representation of GQp which has a local Whittaker modelWh(πp,ψp).

(i) There exists a unique polynomial Qπp(X) ∈ C[X] of one variable satisfyingQπp(0) = 1 and

{Z(p)N (s, W (p)) ∈ C(p−s)| W (p) ∈Wh(πp,ψp)} = Qπp(p−s)−1C[p−s, ps].

In particular, there exists a local Whittaker function W ∈ Wh(πp,ψp) such thatZ(p)

N (s, W) = Qπp(p−s)−1. The L-factor L(s,πp) of the representation πp is definedby

L(s,πp) := Qπp(p−s)−1.


(ii) There exists a unique monomial ε(s,πp,ψp) = ap−fs (a ∈ C×, f ∈ Z) inp−ssuch that

Z(p)N (1− s, W)L(1− s,π∨p )

= ε(s,πp,ψp)× Z(p)N (s, W)L(s,πp)

, for all W ∈Wh(πp,ψp).

(iii) Suppose thatπp is an irreducible unramified principal series representationwith Satake parameter Ap ∈ GSp(2, C). Let W0 be the local Whittaker functioncorresponding to the unramified vector in πp normalized so that W0(14) = 1. Then

Z(p)N (s, W0) = [ det (14 − Ap · p−s)]−1.

Moreover we have L(s,πp) = [ det (14 − Ap · p−s)]−1 and ε(s,πp,ψp) = 1.

From Proposition 4 and what we remarked before Proposition 3, the Eulerproduct (1) arising from a generic cuspidal automorphic representation Π of GA

converges absolutely for Re(s) > (5− Re(ω∞))/2.

Remark. (i) The first assertion of Proposition 4 (iii) is proved by using the explicitformula of W0 ([Ka], [C-S]).

(ii) Takloo-Bighash [TB] determines the L-factors in the above sense for allgeneric representations πp of GQp .

(iii) Let πp be an irreducible admissible representation of GQp which has anonzero (GQp ∩ GL(4, Zp))-fixed vector. Then it is known that πp has a nonzeroWhittaker model if and only if πp is equivalent to some irreducible unramifiedprincipal series representation of GQp ([B-M, Theorem 5.1],[Li, Theorem 2.7]).

3. Proofs of the main results. We are ready to prove our global andarchimedean results. Throughout this section, Π is a cuspidal automorphic rep-resentation of GA satisfying the assumptions A.1 and A.2.

3.1. Reduction of the problem. We make a reduction of the proof ofTheorem 1 to the archimedean local problems, which is the usual procedure ofthe theory of zeta integrals (see [Ge-Sha, Ch I] for example). Firstly it is easy toverify that

ZN(s, F) = ZN(1− s, F).

Suppose that F ∈ Π is a decomposable cusp form. Then the global Whittakerfunction attached to F can be written as (2). Moreover, by Proposition 4 (i) and(iii), we may and do assume that the local zeta integral Z(p)

N (s,W (p)F ) at each finite

place p < ∞ equals the local L-factor L(s, Πp) of Πp. For any local Whittakerfunction W ∈Wh(Π∞,ψ∞), we set W(g) := ωΠ(ν(g))−1W(gη∞), which belongsto the local Whittaker model Wh(Π∨∞,ψ∞) of the contragredient representation


Π∨∞ of Π∞. Then, by Propositions 2 and 3, we have

ZN(s, F) = Z(∞)N (s,W (∞)

F )× L(s, Π),

and

ZN(s, F) = Z(∞)N (s, W (∞)

F )×∏

p<∞ε(1− s, Πp,ψp)× L(s, Π∨).

Hence Theorem 1 follows from the following two propositions:

PROPOSITION 5. There exists a local Whittaker function W ∈ Wh(Π∞,ψ∞)satisfying the following two conditions:

(i) the local zeta integrals Z(∞)N (s, W) and Z(∞)

N (s, W) converge absolutely forRe(s) � 0 and are continued to nonzero meromorphic functions on the wholes-plane;

(ii) the following equality holds:

Z(∞)N (1− s, W)L(1− s, Π∨∞)

= ε(s, Π∞,ψ∞)× Z(∞)N (s, W)L(s, Π∞)

.

PROPOSITION 6. For each complex number s0 ∈ C, there exists a local Whittakerfunction W ∈Wh(Π∞,ψ∞) such that the associated local zeta integral Z(∞)

N (s, W)converges absolutely for Re(s) � 0 and that the ratio Z(∞)

N (s, W)/L(s, Π∞) iscontinued to an entire function of s ∈ C which does not have zeros at s = s0.

Until the end of this paper, we devote ourselves to showing Propositions 5and 6. We omit proofs for the case of (β), because they are quite similar to thosefor the case of (α) (see the proof of Proposition 7 below). Moreover it is a simplematter to reduce our problems to the case where ω∞ = 0. Hence, from now on,we assume that ω∞ = 0.

3.2. Explicit formulae of local Whittaker functions on GR. In order toprove Propositions 5 and 6, we use explicit formulae for some Whittaker functionson GR. Recall that the representation Π∞ is a direct sum of two (limits of) discreteseries representations D(λ1,λ2) and D(−λ2,−λ1) of Sp(2, R) (1− λ1 � λ2 � 0). Let{vk|0 � k � d = λ1−λ2} be the standard basis of the minimal K0-type τ(−λ2,−λ1)

of D(−λ2,−λ1) (see subsection (1.2)). We denote by Wvk ∈Wh(Π∞,ψ∞) the localWhittaker functions corresponding to vk ∈ Π∞ (0 � k � d). Define a vectorsubgroup A of GR by A := {diag(a1, a2, a−1

1 , a−12 )|ai > 0 (i = 1, 2)}. Then we

have the following explicit formula for the values of the functions Wvk on A:

PROPOSITION 7. (cf. [Mo]) (i) For each 0 � k � d, the support of Wvk iscontained in the identity component of GR.


(ii) Take a pair (σ1,σ2) of two real numbers satisfying

σ1 + σ2 + 1 > 0 and σ1 > 0 > σ2.(3)

Then there exists a nonzero constant C ∈ C× independent of 0 � k � d such that

Wvk |A(diag(a1, a2, a−11 , a−1

2 )) = C ×(

dk

)× (2√−1)−k × exp (− 2πa2

2)

×∫

L(σ1)

ds1

2π√−1

(s1)k × (4π3a21)−s1+λ1+1−k

2

×∫

L(σ2)

ds2

2π√−1

(4πa22)−s2+λ2+k

2

× Γ(

s1 + s2 − 2λ2 + 12

)Γ(

s1 + s2 + 12

)× Γ

(s1

2

)Γ(−s2

2

).

Here (s1)k = Γ(s1 + k)/Γ(s1) and the path of integration L(σj) ( j = 1, 2) is thevertical line from σj −

√−1∞ to σj +

√−1∞.

Proof. Firstly we remark that the assertion (i) follows from (ii). Indeed, thesecond assertion implies that there is a nonzero (g0, K0)-homomorphism fromD(−λ2,−λ1) to

Aψ∞(NR\G0) := Aψ∞(NR\GR) ∩ C∞(G0).

We put δ1 := diag(1, 1,−1,−1) ∈ GR and suppose that the function G0 � g0 �→Wv0 (g0δ1) ∈ C does not vanish. Then it defines a nonzero (g0, K0)-homomorphismfrom D(λ1,λ2) to Aψ∞(NR\G0). On the other hand, by Wallach’s multiplicity onetheorem of Whittaker models [Wa, Theorem 8.8] combined with Kostant [Ko,Theorem 6.2.2], we have

Hom(g0,K0)(D(−λ2,−λ1),Aψ∞(NR\G0)) + Hom(g0,K0)(D(λ1,λ2),Aψ∞(NR\G0)) � 1.

This is a contradiction. Hence we have shown (i) by assuming (ii).We shall prove (ii). We first assume that 1 − λ1 < λ2 < 0. Then Oda [O]

constructed a system of differential equations satisfied by the Whittaker functionsWvk , which we now recall. Let {v ′k} be the standard basis of the contragredientrepresentation τ(λ1,λ2) of τ(−λ2,−λ1). Then it is easy to see that

〈vk, v ′l 〉 = (− 1)k ×(

dk

)× δk,d−l × 〈v0, v ′d〉 0 � k, l � d,


where 〈 , 〉 is the canonical pairing between τ(−λ2,−λ1) and τ(λ1,λ2). Hence thefunctions Wvk (a) are related to the functions ck(a) in [O] by

Wvk (a) = C × (− 1)k ×(

dk

)× cd−k(a), a ∈ A,

where C is a nonzero constant independent of 0 � k � d. We introduce a newcoordinate x = (x1, x2) on A by

x1 =√

4π3a1, x2 :=√

4πa2.

Define a new set of functions {hk(x)|0 � k � d} by

ck(x) = xλ2+1+k1 xλ1−k

2 exp (− x22/2)hk(x), (0 � k � d).

By [O,§8, (H − 1), (H − 2) and (E)k], we have the following system of differentialequations:

[∂1∂2 + 4(x1/x2)2]hd(x) = 0;[(∂1 + ∂2 + 2λ2 − 1)(∂1 + ∂2 − 1)− 2x2

2]hd(x) = 0;(∂1 + k + 1− d)hk+1(x)− 2

√−1hk(x) = 0, (0 � k � d − 1).

(4)

Next we consider the case of 1 − λ1 = λ2 or λ2 = 0, that is, D(−λ2,−λ1) is alimit of discrete series representation of G0. The construction of this system ofdifferential equations works as long as the dimension of the minimal K0-type ofD(−λ2,−λ1) is greater than one. Therefore the above system (4) of differential equa-tions remains valid for (λ1,λ2) �= (1, 0). Finally we suppose that (λ1,λ2) = (1, 0).Then, by constructing a system of differential equations for hk(x) in the samemanner as [M-O1, M-O2] by using the fact that the Harish-Chandra parameter ofD(0,−1) is (0, 0) ∈ h∗C, we can confirm that the system (4) is still valid. A solutionof the system (4) is given by

hk(x1, x2) =∫

L(σ1)

ds1

2π√−1

x−s11

∫L(σ2)

ds2

2π√−1

x−s22

× (s1)d−k

(− 2π√−1)d−k

× Γ(

s1 + s2 − 2λ2 + 12

)Γ(

s1 + s2 + 12

)× Γ

(s1

2

)Γ(−s2

2

).

By Stirling’s formula, the functions on G0 corresponding to this solution are ofmoderate growth. Hence (ii) follows.


3.3. Computation of the archimedean local zeta integrals. By usingthe explicit formulae in Proposition 7, we can evaluate the local zeta integralsZ(∞)

N (s, Wvk ). For later purpose it is better to compute the following slightly gen-eral integral Z(∞)

N (s, y1; W) (y1 > 0, W ∈Wh(Π∞,ψ∞)):

Z(∞)N (s, y1; W) :=

∫R×

d×y∫

Rdx W

yy1

y

xy−1

11

|y|s−3/2.

The introduction of this integral is inspired by a paper [H-M] of Hoffstein andMurty.

PROPOSITION 8. The integrals Z(∞)N (s, y1; Wvk ) (0 � k � d) converge absolutely

for Re(s) > (− λ1 − λ2 + 1)/2 and

Z(∞)N (s, y1; Wvk )

L(s, Π∞)= C1 ×

(dk

)× (√−1)−k × (4π)s(5)

×∫

L(σ1)

ds1

2π√−1

(4πy1)−s1−s+λ1+1

×Γ(s1 + −λ1−λ2+1

2 )Γ(s1 + −λ1+λ2+12 )

Γ( s1−s−d+k+22 )Γ( s1+s−k+1

2 )

with a nonzero constant C1 ∈ C× independent of 0 � k � d. Here σ1 ∈ R is takenso that σ1 > (λ1− λ2− 1)/2. Moreover, for each fixed y1( > 0), the integral in (5)is continued to an entire function of s.

Proof. Since

(y 0x 1

)=

(1 xy/(1 + x2)0 1

)(y/√

1 + x2 00

√1 + x2

)1√

1 + x2

(1 −xx 1

),

for each y > 0 and x ∈ R, we have

Wvk

yy1

y

xy−1

11

= exp

(−2π√−1xy

1 + x2

)(1 +√−1x√

1 + x2

)λ2+k

×Wvk

diag

√yy21,√

y1 + x2 ,

√1

yy21

,

√1 + x2

y

.


Hence, by Proposition 7, we get

Z(∞)N (s, y1; Wvk )

= C ×(

dk

)× (2√−1)−k

∫ ∞0

ys−3/2 dyy

×∫ ∞−∞

dx exp( −2πy

1−√−1x

)(1 +√−1x√

1 + x2

)λ2+k

×∫

L(σ1)

ds1

2π√−1

(4π3yy21)(−s1+λ1+1−k)/2

∫L(σ2)

ds2

2π√−1

(4πy

1 + x2

)(−s2+λ2+k)/2

× (s1)k × Γ(

s1 + s2 − 2λ2 + 12

)Γ(

s1 + s2 + 12

)Γ(

s1

2

)Γ(−s2

2

).

It is readily seen from Stirling’s formula ([Ah, Ch.5, §2.5]) that the above four-fold integral converges absolutely if we choose (σ1,σ2) ∈ R2 so that

12

(σ1 + σ2 − λ1 − λ2 + 2) < Re(s) <12

(σ1 − λ1 + k + 1).(6)

Note that for each s ∈ C with Re(s) > (−λ1−λ2 +1)/2 we can take (σ1,σ2) ∈ R2

satisfying (3) and (6). Hence we may interchange the order of integration freelyunder the conditions (3) and (6). By carrying out the integration with respect tody, we find that Z(∞)

N (s, y1; Wvk ) equals

C ×(

dk

)× (2√−1)−k

∫L(σ1)

ds1

2π√−1

∫L(σ2)

ds2

2π√−1

y−s1+λ1+1−k1

× 2(−2s−s1−s2+λ1+λ2+4)/2 × π−s−s1+λ1−k+5/2(s1)kΓ(

s1

2

)× Γ

(s1 + s2 − 2λ2 + 1

2

)× Γ

(s1 + s2 + 1

2

)Γ(−s2

2

)Γ(

2s− s1 − s2 + λ1 + λ2 − 22

)

×∫ ∞−∞

dx (1 + x2)(2s−s1+s2+d−2k−2)/4

(1 +√−1x

1−√−1x

)(−2s+s1+s2−d+2k+2)/4

.

Here we understand that | arg ( 1+√−1x

1−√−1x

)| < π for x ∈ R. In order to carry out theintegration with respect to dx, we need the following lemma:

LEMMA 9. For two complex numbers α and β with Re(α) < −1/2, we have

∫ ∞−∞

(1 + x2)α(

1 +√−1x

1−√−1x

)βdx = 22α+2 × π × Γ(− 1− 2α)

Γ(− α− β)Γ(− α + β).


For a proof, we refer to [B2, Proposition 2.6.3], for example. By using this,we know that the local zeta integral Z(∞)

N (s, y1; Wvk ) is equal to

C ×(

dk

)× (2√−1)−k ×

∫L(σ1)

ds1

2π√−1

y−s1+λ1+1−k1 × 2−s1+λ1−k+3

× π−s−s1+λ1−k+7/2

× (s1)kΓ(

s1

2

)Γ(−2s + s1 − d + 2k + 2

2

)−1 ∫L(σ2)

ds2

2π√−1

× Γ(

s1 + s2 − 2λ2 + 12

)Γ(

s1 + s2 + 12

)Γ(

2s− s1 − s2 + λ1 + λ2 − 22

)× Γ

(−2s + s1 − s2 − d + 2k2

).

The integral with respect to ds2 can be evaluated by the following formula(Barnes’ first lemma):

LEMMA 10. [W-W, p. 289] Take four complex numbers a, b, c, d ∈ C suchthat a + c, a + d, b + c, b + d /∈ Z�0. Then we have

1

2π√−1

∫ +√−1∞

−√−1∞

Γ(a + s)Γ(b + s)Γ(c− s)Γ(d − s)ds

=Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d).

Here the path of integration is curved so that the poles of Γ(c − s)Γ(d − s) lie onthe right of the path and the poles of Γ(a + s)Γ(b + s) lie on the left.

Finally, making a change of variables s1 �→ s1−s+k, we arrive at the desiredformula. To prove the second assertion, we note that, for an arbitrary ε > 0, thereexists a constant M > 0 such that

∣∣∣∣∣∣Γ(σ1 +√−1τ1 + −λ1−λ2+1

2 )Γ(σ1 +√−1τ1 + −λ1+λ2+1

2 )

Γ(σ1+√−1τ1−s−d+k+2

2 )Γ(σ1+√−1τ1+s−k+1

2 )

∣∣∣∣∣∣� M exp (− (π − ε)|τ1|/2), ∀τ1 ∈ R.

(7)

This is an easy consequence of Stirling’s formula. Thus the integral on the righthand side of (5) converges uniformly on every compact subset of the s-plane.Therefore it defines an entire function in s ∈ C.


Remark. If λ2 < 0, we can express the local integrals Z(∞)N (s, Wvk ) as follows:

Z(∞)N (s, Wvk )

L(s, Π∞)= C1 ×

(dk

)× (√−1 )−k ×

−λ2∑l=1

(− 4π)−l (l− 1)!(− λ2 − l)!

(8)

× Γ(

s− k + l + λ1+λ2+12

2

)−1

Γ(−s + k + l + −λ1+3λ2+3

2

2

)−1

.

Here C1 ∈ C× is a nonzero constant independent of 0 � k � d. This expressionis itself interesting and easily obtained from Proposition 8 by a simple residuecalculus. On the other hand, if λ2 = 0, then it follows from a residue calculus againthat all the integrals Z(∞)

N (s, Wvk ) vanish. This is the reason why we introduce theintegrals Z(∞)

N (s, y1; Wvk ).

3.4. End of the proof. Now we shall show Propositions 5 and 6 to completethe proof of Theorem 1. We first prove Proposition 6. Set X1 := diag(1, 0,−1, 0) ∈g0. Then we have, for each u ∈ Π∞,

WΠ∞(X1)u(diag(a1, a2, a−11 , a−1

2 )) = a1∂

∂a1(Wu(diag(a1, a2, a−1

1 , a−12 ))).

The same argument as in the proof of Proposition 8 shows that Z(∞)N (s, WΠ∞(Xm

1 )vk)

(m � 0, 0 � k � d) converges absolutely for Re(s) > ( − λ1 − λ2 + 1)/2 andthat the ratio Z(∞)

N (s, WΠ∞(Xm1 )vk

)/L(s, Π∞) is continued to an entire function ofs ∈ C. Therefore we have

(y1

∂

∂y1

)m

|y1=1

[Z(∞)

N (s, y1; Wvk )

L(s, Π∞)

]=

Z(∞)N (s, WΠ∞(Xm

1 )vk)

L(s, Π∞).(9)

From (7), we know that for an arbitrary ε > 0, there exists a constant M > 0such that∣∣∣∣∣∣Γ(σ1 +

√−1τ1 + −λ1−λ2+1

2 )Γ(σ1 +√−1τ1 + −λ1+λ2+1

2 )

Γ(σ1+√−1τ1−s−d+k+2

2 )Γ(σ1+√−1τ1+s−k+1

2 )

∣∣∣∣∣∣ exp (− t(σ1 +√−1τ1))

� M exp (− ε|τ1|/2), ∀τ1 ∈ R and ∀t ∈ C with |Im(t)| < π/2− ε.

Hence, by Proposition 8, Z(∞)N (s, y1; Wv0 )/L(s, Π∞) is a real analytic function of

y1 > 0 for each s ∈ C and the following equality holds:

Z(∞)N (s, y1; Wv0 )

L(s, Π∞)=∞∑

m=0


1 )v0 )

L(s, Π∞)( log y1)m

m!, if | log y1| < π/2.

(10)


For each fixed s0 ∈ C, it follows from Proposition 8 that the left hand side of(10) is a nonzero function of y1( > 0). Hence there exists an integer m � 0such that Z(∞)

N (s, WΠ∞(Xm1 )v0 )/L(s, Π∞) does not vanish at s = s0. This proves

Proposition 6.In order to prove Proposition 5, we consider the following integral attached

to a local Whittaker function W ∈Wh(Π∞,ψ∞):

Z(∞)N (s, y1; W) :=

∫R×

d×y∫

Rdx W

yyy−1

1

xy−11

1y1

|y|s−3/2,

y1 > 0.

By a change of variables (x, y) �→ (xy21, yy2

1), we find that

Z(∞)N (s, y1; W) = y2s−1

1 × Z(∞)N (s, y1; W).(11)

Since Wv0 = (√−1)d(− 1)λ1Wvd , it follows from (11) and Proposition 8 that

Z(∞)N (1− s, y1; Wv0 )

L(1− s, Π∨∞)= (− 1)λ1 × Z(∞)

N (s, y1; Wv0 )L(s, Π∞)

.(12)

Next we set X2 := Ad(η∞)(X1) = diag(0,−1, 0, 1). Noting

X2 = X1 − diag(2, 2, 0, 0) + diag(1, 1, 1, 1),

we have, for each u ∈ Π∞ and m � 0,

WΠ∞(Xm2 )u

yy

xy1

1

=(

y1∂

∂y1− 2y

∂

∂y

)m

Wu

yy1

y

xyy−1

11

=m∑

j=0

(mj

)(−2y

∂

∂y

)m−j

WΠ∞(Xm1 )u

yy

xy1

1

.


From this, we conclude that the integral Z(∞)N (s, WΠ∞(Xm

2 )vd) converges absolutely

for Re(s) > (− λ1 − λ2 + 1)/2. Further, integration by parts gives the following:


2 )vd)

L(s, Π∞)=

m∑j=0

(mj

)× (2s− 1)m−j ×

Z(∞)N (s, WΠ∞(Xj

1)vd)

L(s, Π∞)

=(

y1∂

∂y1

)m

|y1=1

[y2s−1

1 × Z(∞)N (s, y1; Wvd )

L(s, Π∞)

].

From this and (11), we have

(y1

∂

∂y1

)m

|y1=1

[Z(∞)

N (s, y1; Wvd )

L(s, Π∞)

]=


2 )vd)

L(s, Π∞), m � 0.(13)

By comparing (9), (12) and (13), we conclude that

Z(∞)N (1− s, WΠ∞(Xm

1 )v0 )

L(1− s, Π∨∞)= (− 1)λ1 ×


1 )v0 )

L(s, Π∞), m � 0.

As we noted above, the right hand side does not vanish for some m � 0. HenceProposition 5 follows.

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO,3-8-1 KOMABA MEGURO-KU, TOKYO 153-8914, JAPAN

E-mail: [email protected]

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