SOME RESULTS ON THE SPINOR L-FUNCTION FORTHE GROUP GSp(4)

RAMIN TAKLOO-BIGHASH

To Siavash Shahshahani on the occasion of his 60th birthday

Abstract. We collect some known and new results on the localEuler factors of the spinor L-function of the similitude symplecticgroup of order four using the integral of Novodvorsky.

Contents

Introduction 1Notation 41. The work of Jacquet and Langlands 52. The Spinor L-function for GSp(4) 112.1. The integral 112.2. Non-Archimedean Theory 132.3. Archimedean Theory 162.4. A possible application 192.5. Open problems 21References 21

Introduction

In his article “Problems in the theory of automorphic forms”, RobertLanglands set forward a series of conjectures that were to revolution-ize the theory and practice of modular forms. Langlands’ approach tothe theory was via his theory of L-functions. In this theory, the mostimportant L-series were the ones that had an Euler product, an infi-nite product of simple functions, one for each prime ideal of the basefield, including the primes “at infinity.” One also expects a functionalequation similar to the one satisfied by the Riemann Zeta function.In Langlands’ theory, one starts with a global field F , a reductive al-gebraic group G defined over F , an irreducible automorphic cuspidalrepresentation π of G(AF ), and a finite dimensional algebraic repre-sentation r of the dual of G, denoted by LG. Then π = ⊗vπv, where

1

for each place v, πv is a representation of G(Fv). Also, there is a finiteset of place S, including the places at infinity, such that for v /∈ S, therepresentation πv is unramified. Next, by a process, essentially due toSatake, one associates to πv, v /∈ S, a conjugacy class Cv(π) in LG. Wedefine

Lv(s, π, r) =1

det(I − r(Cv(π))Nv−s).

For example, if F = Q, G = Gm, π the trivial representation, and rthe trivial representation, then for v = p a prime number, we have

Lp(s, π, r) =1

1− p−s,

the p-Euler factor of the Riemann zeta function!Langlands’ first conjecture, and in some sense the most basic one,

asserts that it is possible to extend the definition of Lv to v ∈ S, insuch a way that if we set

L(s, π, r) =∏v

Lv(s, π, r),

then L is “nice.” Here “nice” means that L is meromorphic in theentire complex plane with only a finite number of poles and satisfiesthe functional equation

L(s, π, r) = ε(s, π, r, ψ)L(1− s, π, r).

Here r is the contragradient of the finite dimensional complex repre-sentation r.

In this generality, the conjecture is still open. There are, however,many instances where it has been verified. For example, if G = GL(n)and r the standard representation of LG = GLn(C), then the classicalresults of Godement, Tate, and Jacquet, which predate the paper ofLanglands, affirm the conjecture. In the works of Godement-Jacquet,and Tate, the L-function is presented as an integral on some adelicspace, which in turn is an infinite product, the factors of which areindexed by the places of the global field. For appropriate choices ofthe local data, each of these local integrals gives the correspondinglocal Euler factor. In modern terminology, this is the method of Inte-gral Representations or the Rankin-Selberg Method. We also note thepowerful method of Langlands and Shahidi which has yielded manyinteresting examples where the conjecture holds.

In this article, we concentrate on a special Rankin-Selberg Integralfor the symplectic similitude group of order four GSp(4) over a totallyreal field . The group GSp(4) is important, both from a historical point

2

of view and from the point of view of applications. This group pro-vides the natural group theoretic framework for Siegel modular formsof genus two, and for that reason has immediate connection to vari-ous fundamental problems in arithmetic geometry and number theory.Also in a systematic development of the theory, it is necessary to haveworked out various non-trivial lower dimensional examples in detailsto provide the required empirical data for the revelation of a universalvision.

This paper is organized as follows. Section One gives an overview ofthe zeta integral machine through an example from the classical workof Jacquet and Langlands [14]. Our overview is heavily based on theexposition of [10]. The reader who is familiar with classical modularforms is encouraged to consult Gelbart’s wonderful monograph [9] inorder to understand the relevance of the material presented here to theclassical theory. Section two contains the theory of Novodvorsky’s in-tegral and the L-function represented by it. In this section, we have in-cluded the functional equation, the sketch of unramified computationsfrom [5], a review of the computations at the bad primes from [20],and some thoughts on the archimedean calculations. The archimedeantheory described in this section is the main contribution, if any, of thiswork. Here, we have introduced a global method in order to performthe local computation at the real place for the generic (limit of) dis-crete series. Our idea is to utilize the global theta correspondence forthe dual reductive pair (GO(2, 2), GSp(4)). In this report, however, wehave chosen to ignore the effects of the disconnectedness of GO(2, 2),and we have worked only with the connected component of the iden-tity. Consequently, for the time being, all the “theorems” announcedin Section 2.3 will remain conjectures until the appearance of [21] infinal form. We also note the work in progress of Miller and Schmidton the real unramified principal series. In this section, we have alsoincluded a possible arithmetic application of our results to the workin progress of Furusawa and Shalika. Our exposition of their researchclosely follows an unpublished manuscript by Furusawa.

In the preparation of this work, we have largely benefited from con-versations with Jeff Adams, Masaaki Furusawa, Freydoon Shahidi, andmost importantly my graduate advisor Joseph A. Shalika under whomguidance the non-archimedean computations of section two were pre-pared.

It is our utmost pleasure to dedicate this article to Professor SiavashShahshani on the occasion of his 60th birthday. This author owesa great deal of the development of his mathematical and intellectualcharacter, including his passion for analysis, to what he heard and

3

learned in Professor Shahshahani’s classes and lectures. We believethat because of Professor Shahshahani’s teachings the mathematicalcommunity of Iran is one long step closer to the realization of thedream outlined in his article in the Memoires of the 25th MathematicalConference of Iran.

Notation. In this paper, the group GSp(4) over an arbitrary fieldK is the group of all matrices g ∈ GL4(K) that satisfy the followingequation for some scalar ν(g) ∈ K:

tgJg = ν(g)J,

where J =

1

1−1

−1

. It is a standard fact that G = GSp(4)

is a reductive group. The map (F×)3 −→ G, given by

(a, b, λ) 7→ diag(a, b, λb−1, λa−1)

gives a parameterization of a maximal torus T in G. Let χ1, χ2 and χ3

be quasi-characters of F×. We define the character χ1 ⊗ χ2 ⊗ χ3 of Tby

(χ1 ⊗ χ2 ⊗ χ3)(diag(a, b, λb−1, λa−1)) = χ1(a)χ2(b)χ3(λ).

We have three standard parabolic subgroups: The Borel subgroup B,The Siegel subgroup P , and the Klingen subgroup Q with the followingLevi decompositions:

B = {

a

bb−1λ

a−1λ

1 x1

1 −x1

1 r s1 t r

11

},

P = {(g

ατg−1

)1 r s

1 t r1

1

|g ∈ GL(2)}.

Here τg is the transposed matrix with respect to the second diagonal,and finally

Q = {

α gα−1 det g

1 x1

1 −x1

1 r s1 r

11

|g ∈ GSp(2)}.

4

Over a local field, we will use the notation χ1×χ2oχ3 for the parabol-ically induced representation from the minimal parabolic subgroup, bythe character χ1⊗χ2⊗χ3. If π is a smooth representation of GL(2), andχ a quasi-character of F×, then πoχ (respectively χoπ) is the paraboli-cally induced representation from the Levi subgroup of the Siegel (resp.Klingen) parabolic subgroup. We define a character of the unipotentradical N(B) of the Borel subgroup by the following:

θ(

1 x

11 −x

1

1 r s1 t r

11

) = ψ(x+ t).

We call an irreducible representation (Π, VΠ) of GSp(4) over a localfield generic, if there is a functional λΠ on VΠ such that

λΠ(Π(n)v) = θ(n)v,

for all v ∈ VΠ and n ∈ N(B). If such a functional exists, it is unique upto a constant [19]. Freydoon Shahidi has given canonical constructionsof these functionals in [17] for representations induced from genericrepresentations. We define Whittaker functions on G× VΠ by

W (Π, v, g) = λΠ(Π(g)v).

When there is no danger of confusion, after fixing v and suppressingΠ, we write W (g) instead of W (Π, v, g). For a character Ψ of theunipotent radical of the Borel subgroup, we denote by πN,Ψ the twistedby Ψ Jacquet module of the representation π. For any representationπ, we will denote by ωπ the central character of π. We will also useShahidi’s notation for intertwining operators and local coefficients from[17].

1. The work of Jacquet and Langlands

In this section, we examine the group GL(2). This section servesas motivation for Section Two which contains the main results of thepaper. Our exposition is heavily based on [10], to the point of copying,especially pages 5-19. For the sake of familiarity and simplicity, wework over Q.

Let G = GL(2). Suppose χ is a unitary character of A×. By a χ-cuspform ϕ on GL(2), we mean an L2(ZAG(Q)\G(A)) function satisfying

ϕ(

(a

a

)g) = χ(a)ϕ(g),

5

and ∫Q\A

ϕ(

(1 x

1

)g) dx = 0,

for almost all g ∈ G(A). It is clear that if ϕ is a χ-cusp form, andg ∈ G(A), then the function g.ϕ on G(A) defined by

g.ϕ(h) = ϕ(hg),

is again a χ-cusp form. This defines a representation of G(A) on thevector space of χ-cusp forms L2

0(χ). It is a fundamental fact that L20(χ)

is a discrete direct sum of irreducible subspaces, each of which appearswith multiplicity one. An irreducible representation π of GL2(A) whichis realized as an irreducible subspace Hπ of L2

0(χ) is called a cuspidalautomorphic representation.

Suppose π is an irreducible cuspidal automorphic representation ofGL2(A), and ϕ ∈ Hπ. We introduce a global zeta integral

Z(ϕ, s) =

∫Q×\A×

ϕ(

(a

1

))|a|s−

12

Ad×a.(1)

If ϕ is “nice enough”, Z(ϕ, s) defines an entire function in C. Also, thezeta function Z(ϕ, s) satisfies a functional equation:

Z(ϕ, s) = Z(ϕw, 1− s),

where w =

(1

−1

), and ϕw(g) = ϕ(gw). Also

Z(ϕ, s) =

∫Q×\A×

ϕ(

(a

1

))|a|s−

12χ−1(a) d×a,(2)

with χ the central character of π.The problem is to relate the function Z(ϕ, s) to an automorphic L-

function L(s, π, r) for some representation r of LG = GL2(C). For thispurpose, we start by the Fourier expansion of ϕ

ϕ(g) =∑ξ∈Q×

Wψϕ (

(ξ

1

)g).(3)

Here

Wψϕ (g) =

∫Q\A

ϕ(

(1 x

1

)g)ψ(x) dx,(4)

6

where ψ is a non-trivial character of Q\A. It follows from (1) and (3)that

Z(ϕ, s) =

∫A×Wψϕ (

(a

1

))|a|s−

12 d×a,

for <s large enough.Now we recall some of the properties of the Whittaker functions Wψ

ϕ .From now on we suppress ψ. We assume that ϕ is right K-finite. Inthis situation, Wϕ is rapidly decreasing at infinity and satisfies

Wϕ(

(1 x

1

)g) = ψ(x)Wϕ(g),(5)

for all x ∈ A. The space of all such Wϕ provides the ψ-Whittakermodel of π. It is known that such a model is unique ([14], or [19]), andit is equal to the restricted tensor product of local Whittaker modelsW(πp, ψp), where π = ⊗pπp and ψ =

∏p ψp. In particular, we can

assume that

Wϕ(g) =∏p

Wp(gp),(6)

where eachWp ∈ W(πp, ψp) and for almost all finite p, Wp is unramified,i.e. Wp(k) = 1 for k ∈ Kp = GL2(Zp).

Finally, we obtain for <s large

Z(ϕ, s) =∏p

Z(Wp, s),(7)

where

Z(Wp, s) =

∫Q×p

Wp(

(a

1

))|a|s−

12 d×a.(8)

First, we collect some of the properties of the local zeta functionsZ(W, s). The fundamental fact for p < ∞ is the following: There area finite number of finite functions c1, . . . , cN on Q×p , depending onlyon πp, such that for every W ∈ W(πp, ψp), there are Schwartz-Bruhatfunctions Φ1, . . . ,ΦN on Qp satisfying

W (

(a

1

)) =

N∑i=1

ci(a)Φi(a).(9)

Here, a finite function is a function whose space of right translates byQ×p is finite dimensional; finite functions on Q×p are thus characters,

integer powers of the valuation function, or products and linear com-binations thereof. Taking the asymptotic expansion just mentioned forgranted, we obtain from Tate’s thesis that the integral defining Z(W, s)

7

converges for <s large (independent of s), and in the domain of conver-gence is equal to a rational function of p−s. In particular, the integralhas a meromorphic continuation to all of C. Furthermore, the familyof rational functions {Z(W, s) |W ∈ W(πp, ψp)} admits a common de-nominator, i.e. a polynomial P such that P (p−s)Z(W, s) ∈ C[p−s, ps],for all W . Also, there exists a W ∗ in W(πp, ψp) with the property thatZ(W ∗, s) = 1. The analogous result in the archimedean situation isthat there is a W ∗ such that Z(W ∗, s) has no poles or zeroes.

As in Tate’s thesis [22], we also need to establish the functional equa-tion and perform the local unramified computations. The functionalequation asserts that there exists a meromorphic function γ(πp, ψp, s)(rational function in p−s when p <∞) such that

Z(Ww, 1− s) = γ(πp, ψp, s)Z(W, s).(10)

Here Ww(g) = W (gw) and

Z(W, s) =

∫Q×p

W (

(a

1

))|a|s−

12χ−1

p (a) d×a,

with χp the central character of πp. The (non-trivial) proof of this equa-

tion follows from the fact that the integrals Z and Z define functionals(depending on s) satisfying a certain invariance property. Next, oneproves that the space of such functionals is at most one dimensional,implying that Z and Z must be proportional. The factor γ is simplythe factor of proportionality.

We recall πp is called unramified when

πp = Ind(µ1 ⊗ µ2),

with µ1 and µ2 unramified characters of Q×p . We also suppose that W 0

is the unique Kp-invariant function inW(πp, ψp). We also suppose thatψp is unramified. Then a direct calculation, using the results of [6] forexample, shows that

Z(W 0, s) =1

(1− µ1($p)p−s)(1− µ2($p)p−s),

where $p is the local uniformizer at p. Next the conjugacy class in

LG = GL2(C) canonically associated with πp is tp =

(µ1($p)

µ2($p)

).

In particular, we have

Z(W 0, s) = Lp(s, π, r),(11)

with r the standard two dimensional representation of GL2(C).After this preparation, we can prove the conjecture of Langlands

for L(s, π, r) with r as above. For simplicity, we write L(s, π) instead8

of L(s, π, r). We start by extending the definition of Lv(s, π) to theramified and archimedean places. We observe that in equation (11),the right hand side is indeed the greatest common denominator of thefamily of rational functions {Z(W, s)}. Since we have already notedthat such a g.c.d. exists, even when the given representation is notunramified, we set

Lv(s, π) = g.c.d. {Z(W, s)},(12)

when v <∞. Also, when v =∞, we can choose an appropriate productof Tate’s archimedean L-functions, denoted by L∞(s, π, r), such thatthe ratio

Z(W, s)

L∞(s, π)(13)

is an entire function for all W ∈ W(πv, ψv), and it is a nowhere van-ishing function for some choice of W .

With this extension, we now proceed to outline the proof. Let S bea set of places, including the place at infinity, such that for v /∈ S, allthe data is unramified. We set

LS(s, π) =∏v/∈S

Lv(s, π).(14)

For each “ramified” non-archimedean place p, we choose Wp such thatZ(Wp, s) = 1. Also for the archimedean v, we choose Wv such thatZ(Wv, s) is a non-vanishing entire function eg(s). If we set W =

∏vWv,

with Wp = W 0p for p ∈ S, we have

Z(ϕ, s) = eg(s)LS(s, π),(15)

implying the holomorphicity of LS. This immediately implies the con-tinuation of L to a meromorphic function with only a finite number ofpoles.

9

We finally turn to the functional equation of the completed L-function.Choosing Wp so that Z(Wp, s) = Lp(s, π), we have

L(s, π) = Z(ϕ, s)

= Z(ϕw, 1− s)

=(∏p∈S

Z(Wwp , 1− s)

)LS(1− s, π)

=(∏p∈S

Z(Wwp , 1− s)

Lp(1− s, π)

)L(1− s, π)

=(∏p∈S

γ(πp, ψp, s)Z(Wp, s)

Lp(1− s, π)

)L(1− s, π)

=(∏p∈S

γ(πp, ψp, s)Lp(s, π)

Lp(1− s, π)

)L(1− s, π)

=(∏p∈S

ε(s, πp, ψp))L(1− s, π),

where

ε(s, πp, ψp) =γ(πp, ψp, s)Lp(s, π)

Lp(1− s, π).

Hence, if we set

ε(s, π) =∏p∈S

ε(s, πp, ψp),

we have the functional equation

L(s, π) = ε(s, π)L(1− s, π),(16)

as anticipated by Langlands. One last note is that the function ε(s, π)is a monomial function of s. In particular, it has no poles or zeroes.

Remark 1.1. In equation (9), if ci = µi.vri , with µi a quasi-character,

we have

Lp(s, π) =N∏i=1

L(s, µi)ri .

The L-functions appearing on the right hand side are Tate’s local L-factors for the quasi-characters µi. This implies that in order to givean explicit calculations of the local L-factors, we need to determinethe finite functions ci. In [14], this is established by a case by caseanalysis of representation types for πp, i.e. principal series vs. specialrepresentations vs. supercuspidals.

10

2. The Spinor L-function for GSp(4)

In this section, we examine the integral representation given byNovodvorsky [15] for G = GSp(4). The details of the material inthe following paragraphs appear in [5], [20], and [21].

2.1. The integral. Let ϕ be a cusp form on GSp(4,A), belongingto the space of an irreducible cuspidal automorphic representation π.Consider the integral

Z(ϕ, s) =

∫F×\A×

∫(F\A)3

ϕ(

1 x2 x4

1z 1 −x2

1

y

y1

1

)ψ(x2)|y|s−

12 dz dx2 dx4 d

×y.

(17)

If w =

1

1−1

−1

, then

Z(ϕ, s) = Z(ϕw, 1− s),(18)

where as before ϕw(g) = ϕ(gw). Also a usual unfolding process, asdescribed in [5], shows that

Z(ϕ, s) =

∫A×

∫A

Wϕ

(y

yx 1

1

)|y|s− 32 dx d×y.(19)

Here the Whittaker function Wϕ is given by

Wϕ(g) =

∫(F\A)4

ϕ(

1 x2

11 −x2

1

1 x3 x4

1 x1 x3

11

g)

ψ(x1 + x2) dx1 dx2 dx3 dx4.

(20)

Equation (19) implies that, in order for Z(ϕ, s) to be non-zero, weneed to assume that Wϕ is not identically equal to zero. A represen-tation satisfying this condition is called “generic.” Every representa-tion of GL(2) is generic. On other groups, however, there may ex-ist non-generic cuspidal representations. In fact, those representationsof GSp(4) which correspond to holomorphic cuspidal Siegel modularforms are not generic.

11

From this point on, we assume that all the representations of GSp(4),local or global, which appear in the text are generic.

If ϕ is chosen correctly, the Whittaker function may be assumed todecompose locally as W (g) =

∏vWv(gv), a product of local Whittaker

functions. Hence, for <s large, we obtain

Z(ϕ, s) =∏v

Z(Wv, s),(21)

where

Z(Wv, s) =

∫F×v

∫Fv

Wv

(y

yx 1

1

)|y|s− 32 dx d×y.(22)

As usual, we have a functional equation: There exists a meromorphicfunction γ(πv, ψv, s) (rational function in Nv−s when v <∞) such that

Z(Wv, s) = γ(πv, ψv, s)Z(Wwv , 1− s),(23)

with w as above,

Z(Wv, s) =

∫F×v

∫Fv

Wv

(y

yx 1

1

)χ−1v (y)|y|s−

32 dx d×y,

and χv the central character of πv.We also consider the unramified calculations. Suppose v is any

nonarchimedean place of F such thatWv is right invariant byGSp(4,Ov)and such that the largest fractional ideal on which ψv is trivial is O.Then the Casselman-Shalika formula [6] allows us to calculate the lastintegral (cf. [5]). The result is the following:

Z(Wv, s) = L(s, πv, Spin).(24)

Let us explain the notation. The connected L-group LG0 is GSp(C)(cf. [4]). Let LT be the maximal torus of elements of the form

t(α1, α2, α3, α4) =

α1

α2

α3

α4

,

where α1α4 = α2α3. The fundamental dominant weights of the torusare λ1 and λ2, where

λ1t(α1, α2, α3, α4) = α1,

12

and

λ2t(α1, α2, α3, α4) = α1α−13 .

The dimensions of the representation spaces associated with these dom-inant weights are four and five, respectively. In our notation, Spin is therepresentation of GSp(4,C) associated with the dominant weight λ1,i.e. the standard representation of GSp(4,C) on C4. The L-functionL(s, π, Spin) is called the Spinor, or simply the Spin, L-function ofGSp(4).

Next step is to use the integral introduced above to extend thedefinition of the Spinor L-function to ramified non-archimedean andarchimedean places. Our definition at these places would be analogousto equations (12) and (13) above.

2.2. Non-Archimedean Theory. In this paragraph, we sketch thecomputation of the local non-archimedean Euler factors of the Spin L-function given by the integral representation of the previous paragraph.In order for this to make sense, we need the following lemma:

Lemma 2.1 (Theorem 2.1 of [20]). Suppose Π is a generic represen-tation of GSp(4) over a non-archimedean local field K, q order of theresidue field . For each W ∈ W(Π, ψ), the function Z(W, s) is a ratio-nal function of q−s, and the ideal {Z(W, s)} is principal.

Sketch of proof. For W ∈ W(Π, ψ), we set

Z(W, s) =

∫K

W(

yy

11

)|y|s− 32 d×y.(25)

The first step of the proof is to show that the vector space {Z(W, s)}is the same as {Z(W, s)} (cf. Proposition 3.2 of [20]). Next, we usethe asymptotic expansions of the Whittaker functions along the torusto prove the existence of the g.c.d. for the ideal {Z(W, s)}. Indeed,Proposition 3.5 of [20] (originally a theorem in [6]) states that thereis a finite set of finite functions SΠ, depending only on Π, with thefollowing property: for any W ∈ W(Π, ψ), and c ∈ SΠ, there is aSchwartz-Bruhat function Φc,W on K such that

W(

yy

11

) =∑c∈SΠ

Φc,W (y)c(y)|y|32 .

The lemma is now immediate.13

We have the following theorem:

Theorem 2.2. Suppose Π is a generic representation of the groupGSp(4) over a non-archimedean local field K. Then

1. If Π is supercuspidal, or is a sub-quotient of a representation in-duced from a supercuspidal representation of the Klingen parabolicsubgroup, then L(s, π, Spin) = 1.

2. If π is a supercuspidal representation of GL(2) and χ a quasi-character of K×, and Π = πoχ is irreducible, we have

L(s,Π, Spin) = L(s, χ).L(s, χ.ωπ).

3. If χ1, χ2, and χ3 are quasi-characters of K×, and Π = χ1×χ2oχ3

is irreducible, we have

L(s,Π, Spin) = L(s, χ3).L(s, χ1χ3).L(s, χ2χ3).L(s, χ1χ2χ3).

4. When Π is not irreducible, one can prove similar statements forthe generic subquotients of Π = πoχ (resp. Π = χ1 × χ2oχ3)according to the classification theorems of Sally-Tadic [16] andShahidi [18] (cf. theorems 4.1 and 5.1 of [20]).

Remark 2.3. Sally and Tadic [16] and Shahidi [18] have completedthe classification of representations supported in the Borel and Siegelparabolic subgroups. In particular, they have determined for whichrepresentations the parabolic induction is reducible. From their re-sult, one can immediately establish a classification for all the genericrepresentations supported in the Borel or Siegel parabolic subgroups.

Sketch of proof. By the proof of the lemma, we need to determine theasymptotic expansion of the Whittaker functions in each case. Theargument consists of several steps:

Step 1. Bound the size of SΠ. Fix c ∈ SΠ, and define a functional Λc

on W(Π, ψ) by

Λc(W ) = Φc,W (0).(26)

If c, c′ ∈ SΠ, and c 6= c′, the two functionals Λc and Λc′ are linearlyindependent. Furthermore, the functionals Λc belong to the dual of acertain twisted Jacquet module ΠN,θ (notation from [20], page 1095).Hence #SΠ = dim ΠN,θ. Then one uses an argument similar to those of[19], distribution theory on p-adic manifolds, to bound the dimensionof the Jacquet module. The result (proposition 3.9 of [20]) is that if Πis supercuspidal or supported in the Klingen parabolic subgroup (resp.Siegel parabolic, resp. Borel parabolic), then #SΠ = 0 (resp. ≤ 2, resp.≤ 4). Note that this already implies the first part of the theorem.

14

¿From this point on, we concentrate on the Siegel parabolic sub-group, the Borel subgroup case being similar. We fix some notation.Suppose Π = πoχ, with π supercuspidal of GL(2). Let λΠ (resp. λπ)be the Whittaker functional of Π (resp. π) from [17]. It follows fromthe proof of the lemma 2.1 that, for f ∈ Π, there is a positive numberδ(f), such that

λΠ(Π(

yy

11

)f) =∑c∈SΠ

Λc(f)c(y)|y|32 ,

for |y| < δ(f). Here, Λc is the obvious functional on the space of Π.

Step 2. Uniformity. For f ∈ Ind(π × χ|P ∩K,K), and τ ∈ C, definefτ on G by

fτ (pk) = δP (p)τ+ 12π ⊗ χ(p)f(k).

It is clear that fτ is a well-defined function on G, and that it belongsto the space of a certain induced representation Πτ . The UniformityTheorem (Proposition 3.9 of [20]) asserts that one can take δ(fτ ) =δ(f).

Step 3. Regular representations. This is the case where ωπ 6= 1. In thissituation, we have

λΠ(Π(

yy

11

)f) =

λπ(A(w,Π)(f)(e))χ(y)|y|32 + C(wΠ, w−1)−1λπ(f(e))χ(y)ωπ(y)|y|

32 ,

(27)

for |y| < δ(f). Here w =

1

1−1

−1

, A(w,Π) is the intertwin-

ing integral of [17], and C(wΠ, w−1) is the local coefficient of [17]. Theproof of this identity follows from the the above lemma 2.1, and theMultiplicity One Theorem [19]. The idea is to find one term of theasymptotic expansion using the open cell; then apply the long inter-twining operator to find the other term.

Note that the identity of Step 3 also applies to reducible cases. Forexample, if f ∈ Π is in the kernel of the intertwining operator A(w,Π),the first term of the right hand side vanishes.

15

Step 4. Irregular Representations. The idea is the following: we twisteverything in Step 3 by the complex number τ , so that the resulting rep-resentation Πτ is regular. By Step 2, the identity still holds uniformlyfor all τ . By a theorem of Shahidi [17] (essentially due to Casselmanand Shalika [6]), we know that the left hand side of the identity isan entire function of τ . This implies that the poles of the right handside, coming from the intertwining operator and the local coefficient,must cancel out. Next, we let τ → 0. An easy argument (l’Hopital’s

rule!) shows the appearance of χ(y)|y| 32 and χ(y)|y| 32 logq |y| in theasymptotic expansion.

This finishes the sketch of proof of the theorem.

2.3. Archimedean Theory. In this section, we collect some resultson the computation of the L-factor at the real place. For reasons thatwill soon be clear, it will be convenient to work with a different real-ization of the group GSp(4), and also a different formulation of Novod-vorsky’s integral. Here, in the definition of the similitude group from

the Notation, we take J =

1

1−1

−1

. Also, if ϕ is a cusp

form on GSp(4,A), belonging to the space of an irreducible automor-phic cuspidal representation π, we set

Z(ϕ, s) =

∫F×\A×

∫(F\A)3

ϕ(

1 u v1 v w

11

y

11

y

)ψ(v)|y|s−

12 du dv dw d×y.

(28)

This is obtained from the integral of the previous section after a simplechange of coordinates. We also note that in this new form, the integralis nothing but a split Bessel model ([7]).

We now recall the definition of the group under consideration [11].Let V be the vector space M2, of the two by two matrices, equippedwith the quadratic form det. Let (, ) be the associated non-degenerateinner product, and H = GO(V, (, )) be the group of orthogonal simili-tudes of V , (, ). The group GL(2) × GL(2) has a natural involution tdefined by t(g1, g2) = (tb−1

2 ,t b−11 ), where the superscript t stands for the

transposition. Let H = (GL(2) × GL(2))o < t > be the semi-directproduct of GL(2)×GL(2) with the group of order two generated by t.

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There is an exact sequence

1 −→ Gm −→ H −→ H −→ 1,(29)

where the homomorphism ρ : H → H is defined by ρ(g1, g2)(v) =g1vg

−12 , and ρ(t)v = tv, for all g1, g2 ∈ GL(2) and v ∈ V . Also,

Gm → H is the natural map z 7→ (z, z)×1. It follows that the image ofthe subgroup GL(2)×GL(2) ⊂ H under ρ is the connected componentof the identity of H.

Suppose π1 and π2 are two irreducible cuspidal automorphic repre-sentations of GL2(A) satisfying

ωπ1 .ωπ2 = 1.

Then for ϕ1 and ϕ2 cusp forms in the spaces of π1 and π2, respectively,one can think of

ϕ(h1, h2) = ϕ1(h1)ϕ2(h2),

as a cusp form on the algebraic group ρ(H). We extend the definitionof ϕ to H by defining it to be right invariant under the compact totallydisconnected group < t > (A) =

∏v < t >.

Warning! In all of our computations, we have ignored the role of thegroup < t > (A).

Next, let f be a Schwartz-Bruhat function on the space M2×2(A)×M2×2(A). By a process described in [11], one associates to the couple(ϕ, f) a cusp form on the group GSp(4), denoted by θ(f ;ϕ). Here, inorder to stress the dependence on ϕ1 and ϕ2, we will use the notationθf (ϕ1, ϕ2). We then set

Z(ϕ1, ϕ2, f ; s) = Z(θf (ϕ1, ϕ2), s).(30)

We have our first result as follows.

Theorem 2.4. We have

Z(ϕ1, ϕ2, f, s) =

∫ρ(D)(A)\ρ(G1)(A)

f(h−11

(1 00 0

)h2, h

−11

(0 00 1

)h2)(∫

F\A×ϕ1(

(a

1

)h1)|a|s−

12 d×a

)(∫

F\A×ϕ2(

(b

1

)h2)|b|−s+

12 d×b

)dh1 dh2.

17

with subgroups ρ(G1) and ρ(D) of ρ(H) defined by

G1 = {(h1, h2) | deth1 = deth2},and

D = {((α

β

),

(α

β

)) |α, β ∈ Gm}.

The proof of this theorem is a standard exercise in the theory of theoscillator representation (cf. for example [12], proof of lemma 5.)

It follows from the functional equation of the Jacquet-Langlands Zetafunction and the Fourier-Whittaker expansion of the cusp form ϕ2 that

Corollary 2.5. We have

Z(ϕ1, ϕ2, f, s) =∫ρ(D′)(A)\ρ(G1)(A)

f(h−11

(1 00 0

)w−1h2w, h

−11

(0 00 1

)w−1h2w)(∫

A×Wϕ1(

(a

1

)h1)|a|s−

12 d×a

)(∫

A×Wϕ2(

(b

1

)h2w)|b|s−

12ω−1

π2(b) d×b

)dh1 dh2,

where w =

(1

−1

), and

D′ = {((α

β

),

(β

α

)) |α, β ∈ Gm}.

Next, if f = ⊗vfv, Wϕi = ⊗vW iv, then we obtain

Z(ϕ1, ϕ2, f, s) =∏v

Z(W 1v ,W

2v , fv, s),

with the appropriate local zeta functions. The local zeta function in-troduced here satisfy local functional equations similar to the onesconsidered before. Also, if all the data is unramified, one can easilyperform the local computation.

We now consider the local archimedean computations. Suppose vis a real archimedean place, and W1, W2 in the spaces of π1 and π2,respectively, then we can prove the following result:

Theorem 2.6. For every Schwartz-Bruhat function fv as above, thecomplex function Z(W 1

v ,W2v , fv, s) has an analytic continuation to a

meromorphic function on the complex plane. Furthermore, the ratio

Z(W 1v ,W

2v , fv, s)

L(s, πv1)L(1− s, πv2)(31)

18

has an analytic continuation to an entire function. Also, there is achoice of the data in such a way that the resulting entire function isnowhere vanishing.

The idea of the proof is the same as that of the proof of Theorem5.15 of [14].

The most interesting case in the above analysis is when the repre-sentations πv1 and πv2 are in the discrete series, and the data W v

1 , W v2 ,

and f are K-finite. In this situation, one can explicitly calculate theratio (31) in the terms of hypergeometric functions and gamma func-tions. Here, the main idea is to use a two-variable version of the zetafunction introduced above. There is a description of the discrete se-ries in terms of the Weil representation (paragraph 1, [14]). The finalresults are indeed given in terms of Meijer’s G-function, in which thevariable s appears in the parameters, not the argument. The particularG-function which appears in this work is a finite linear combination offunctions of the form

Γ(s+ A)2F3(s+ A,B;C,D,E;F )

for various values of the parameters A, . . . , F .

2.4. A possible application. In [2], Bocherer has proclaimed thefollowing conjecture:

Conjecture 2.7. Let Φ be a holomorphic cuspidal Siegel eigenform ofdegree two and weight k with respect to Sp4(Z). Let

Φ(Z) =∑T>0

a(T,Φ) exp(2π√−1tr(TZ))

be its Fourier expansion. For a fundamental discriminant −D, i.e. adiscriminant of an imaginary quadratic field Q(

√−D), let

BD(Φ) =∑

{T | det(T )=D4}/∼

a(T,Φ)

ε(T ),

where ∼ denotes the equivalence relation defined by T1 ∼ T2 when T1 =t

γT2γ for some γ ∈ SL2(Z) and ε(T ) = #{γ ∈ SL2(Z) | tγTγ = T}.Then there exists a constant CΦ which depends only on Φ such that

L(1

2, πΦ ⊗ χD) = CΦ.D

−k+1.|BD(Φ)|2,

where πΦ is the automorphic representation of GSp4(AQ) associatedwith Φ, χD is the quadratic character of A×

Qassociated with Q(

√−D)

and the left hand side denotes the central critical value of the quadratictwist by χD of the degree four Spinor L-function for πΦ.

19

Bocherer proved this assertion in the cases of the Klingen Eisensteinseries and the Saito-Kurokawa lifting in [2]. Later he and Schulze-Pillotproved this in the case of the Yoshida lifting in [3]. More recentlyMasaaki Furusawa and Joseph Shalika have started investigating thisconjecture from a different angle. Their approach to the problem is togeneralize Herve Jacquet’s relative trace formula for GL(2) to GSp(4).Jacquet has used his GL(2) relative trace formula in [13] to give an-other proof for an important theorem of Waldspurger [23]. Bocherer’sconjecture provides a natural generalization of this theorem for Siegelmodular forms.

¿From Furusawa-Shalika’s conjectural relative trace formula, oneshould be able to draw a conclusion similar to the one in [13], namely:

Assertion 2.8. For a globally generic cuspidal automorphic represen-tation of GSp4(AF ) with the trivial central character, E a quadraticextension of the base number field F , and χE, the quadratic characterof A×F corresponding to E, we have:

L(1

2, π).L(

1

2, π ⊗ χE) 6= 0(32)

if and only if there exists a triple (D, πD,ΨD), wherre D is a centralsimple quaternion algebra over F containing E, πD is a cuspidal auto-morphic representation of GD, the quaternion similitude unitary groupof degree two over D, which corresponds to π in the functorial sense,and ΨD is a cusp form in the space of πD such that∫

A×FRD(F )\RD(AF )

ΨD(r)τ(r) dr 6= 0.(33)

Here RD denotes the Bessel subgroup of GD and τ is a certain characterof RD(AF ) (cf. [8] for precise definitions).

Moreover, the detailed analysis should yield an identity that ex-presses the special value (32) as the square norm of one of the periodintegrals (33) multiplied by a positive constant Cπ which itself dependsonly on π and not on the quadratic extension E. Also a simple explicitformula for Cπ, given essentially as a ratio of Petersson inner products,is expected. It is believed that since the expected formula for Cπ in-volves the Petersson inner product of certain vectors in the Whittakermodel of the representation π , the proof of Bocherer’s conjecture ingeneral is out of the reach of classical methods.

The special L values considered above appear in Furusawa-Shalika’srelative trace formula as Novodvorsky’s integral. For this reason, inorder to prove the conjecture of Bocherer, one needs precise informationabout the local behavior of Novodvorsky’s integrals at all the places of

20

the number field F , including the ramified places and the places atinfinity.

2.5. Open problems. Below we list two problems directly related tothe subject matter of this work.

2.5.1. Comparison with the Langlands-Shahidi Method. It is a natu-ral question to determine whether the local L-factors given by Novod-vorsky’s integral are the same as the ones given by the Langlands-Shahidi method (see page 116 of [10]). The ambient group to consideris the group GSpin7 which has a parabolic subgroup with Levi factorisomorphic to GSpin5 ×GL1. It is well-known that GSpin5 is isomor-phic to GSp(4). Then we can use the results of [1] at least for the caseswhere the given representations are tempered. We hope to address thisproblem in a subsequent joint work with Mahdi Asgari.

2.5.2. Test Vectors. In [20], we have determined the local L-functiongiven by Novodvorsky’s integral. It is easy to see that this local L-factoris actually the value of the integral at a distinguished vector in the spaceof the representation. For trace formula applications, it is necessary tohave some information about this distinguished vector. This is still anopen problem which we would like to address in a subsequent paper.

References

[1] M. Asgari, Local L-functions for split spinor groups, to appear in Canadian J.of Math.

[2] S. Bocherer, Bemerkungen uber due Dirichletreihen von Koecher und Maaβ,Preprint Math. Gottingensis Heft 68(1986).

[3] S. Bocherer and R. Schulze-Pillot, The Dirichlet series of Koecher and Maassand modular forms of weight 3

2 , Math. Z. 209(1992), 273-287.[4] A. Borel, Automorphic L-functions, Automorphic forms, representations and

L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore.,1977), Part 2, pp. 27–61, Proc. Sympos. Pure Math., XXXIII, Amer. Math.Soc., Providence, R.I., 1979.

[5] D. Bump, Rakin-Selberg Method: A Survey, In Number Theory, Trace Formu-las, and Discrete Groups (K. E. Bombieri, and D. Goldfed, eds.), AcademicPress, San Diego, CA, 1989, 49-109.

[6] W. Casselman, and J. A. Shalika, The unramified principal series of p-adicreductive groups II: The Whittaker function, Compositio Math. 41 (1980),207-231.

[7] M. Furusawa, On L-functions for GSp(4)×GL(2) and their special values, J.Reine Angew. Math. 438 (1993), 187-218.

[8] M. Furusawa and J. A. Shalika, The fundamental lemma for the Besseland Novodvorsky subgroups of GSp(4), C. R. Acad. Sci. Paris Ser. I Math.328(1999). no. 2, 105-110.

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[9] S. Gelbart, Automorphic forms on adle groups, Annals of Mathematics Studies,No. 83. Princeton University Press, Princeton, N.J.; University of Tokyo Press,Tokyo, 1975. x+267 pp.

[10] S. Gelbart and F. Shahidi, Analytic Properties of Automorphic L-Functions,Prospectives in Mathematics, Academic Press, 1988.

[11] M. Harris and S. S. Kudla, Arithmetic automorphic forms for the nonholomor-phic discrete series of GSp(2), Duke Math. J. 66(1) (1992), 59-121.

[12] M. Harris, D. Soudry, and R. Taylor, l-adic representations associated to modu-lar forms over imaginary quadratic fields, I. Lifting to GSp4(Q), Invent. Math.112(1993), 377-411.

[13] H. Jacquet, Sur un resultat de Waldspurger, Ann. Scient. Ec. Norm. Sup., 4serie, t. 19 (1986), 185-229.

[14] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lec-ture Notes in Mathematics 114, (1970).

[15] M. Novodvorsky, Automorphic L-functions for the symplectic group GSp4,In Automorphic Forms, Representations, and L-Functions, AMS Proc. Sym.Pure. Math. 33(2)(1979).

[16] P. Sally and M. Tadic, Induced representations and classifications for GSp(2,F) and Sp(2, F), Mem. Soc. Math. France 52(1993), 75-133.

[17] F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-355.[18] ——-, A proof of Langlands’ conjecture on Plancherel measure; Complemen-

tary series for p-adic groups, Ann. Math. 132(1990), 273-330.[19] J. A. Shalika, The multiplicity one theorem on GL(n), Ann. Math. 100 (1974),

171-193.[20] R. Takloo-Bighash, L-functions for the p-adic group GSp(4), Amer. J. Math.

122(6) (2000), 1085-1120.[21] ———-, L-functions for the real group GSp(4), in preparation.[22] J. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions. 1967

Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305–347 Thompson, Washington, D.C.

[23] J.-L. Waldspurger, Sur les valeurs de certaines fonction L automorphes en leurcentre de symetrie, Compositio Math. 54 (1985), 173-242.

Department of Mathematics, Fine Hall, Princeton, NJ 08544

E-mail address: rtakloo@math.princeton.edu

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