The Koecher-Maass series for real analytic Siegel-Eisenstein series of degree two (the case for positive definite)

The Koecher-Maass series for real analytic Siegel-Eisenstein series of degree two

(the case for positive definite)

Yoshinori Mizuno

1 Introduction

Koecher-Maass series are certain Dirichlet series attached to modular forms of general degree. If a Siegel modular form is holomorphic, the an- alytic properties of its Koecher-Maass series is well known. In this paper, we treat the non-holomorphic Siegel-Eisenstein series. We give a meromor- phic continuation to the whole complex plane and a functional equation of the Koecher-Maass series for real analytic Siegel-Eisenstein series of degree two associated with the Fourier coefficients indexed by posotive definite half integral symmetric matrices of size two.

We first recall the holomorphic case only for the case its degree is two, for simplicity. Let

f(Z) = ^X

T∈L2,T≥O

af(T)e^{2πitr(T Z)}

be a holomorphic Siegel modular form of degree two, where the summation extends over all semi-positive definite half integral symmetric matrices T of size two,

L2 =

(

T = a b

b d

!

;a, d∈Z,2b ∈Z

)

. Then its Koecher-Maass series ηf(s) is defined by

ηf(s) = ^X

T∈L⁺/SL (Z)

af(T) ǫ(T)(detT)^s,

where the summation extends over all positive definite half integral symmet- ric matrices T of size two modulo the usual action T → T[U] = ^tUT U of the group SL2(Z) and ǫ(T) = ♯{U ∈SL2(Z);T[U] = T} is the order of the unit group of T.

Basic analytical properties ofηf(s) are well known. It has a meromorphic continuation to the whole s-plane and a functional equation. The starting point of the proof is an integral representation for the Koecher-Maass series,

η_f^∗(s) = 2(2π)^−2sπ^1/2Γ(s)Γ(s−1/2)ηf(s)

=

Z

R2

X

T∈L⁺₂

af(T)e^{−2πtr(T Y)}(detY)^s−3/2dY,

where Γ(x) is the usual gamma function,R2is the Minkowski reduced domain and L⁺₂ is the set of allT ∈L2 so thatT > O. It satisfies

η_f^∗(s) =η^∗_f(k−s).

In his lecture note ([13] p. 307), Maass raised the question ”whether it is possible to attach Dirichlet series by means of integral transforms to the non analytic Eisenstein series” and also said that ”already in the case degree is two difficulties come up which show that one can not proceed in the usual way.”

2 Results

We take a real analytic Siegel-Eisenstein series of degree two and weight k defined by

E_2,k(Z, σ) = ^X

{C,D}

det(CZ+D)^−k|det(CZ+D)|^−2σ. It has the Fourier expansion

E2,k(Z, σ) = ^X

T∈L2

C(T, σ, Y)e^{2πitr(T X)}, Z =X+iY.

If detT is not zero, then the Fourier coefficientsC(T, σ, Y) can be written as a product of the Siegel series and the confluent hypergeometric function of degree two by

C(T, σ, Y) =b(T, k+ 2σ)ξ(Y, T, σ+k, σ),

b(T, σ) = ^X

R∈S2(Q)/S2(Z)

[RZ²+Z² :Z²]^−σe^{2πitr(T R)}, ξ(Y, T, α, β) =

Z

S2(R)e−2πitr(T X)det(X+iY)^−αdet(X−iY)^−βdX, where S2(K) is the set of all symmetric matrices of size two whose compo- nents are elements in K.

Let a(T, σ) be arithmetic parts of C(T, σ, Y) defined by a(T, σ) =e^πi(k+2σ) 4π2(k+2σ)−1/2

Γ(k+ 2σ)Γ(k+ 2σ−1/2)|det(2T)|^k+2σ−3/2b(T, k+ 2σ).

Then following Ibukiyama and Katsurada, the Koecher-Maass series of sig- nature (2,0) is defined by

L(s, σ) = ^X

T∈L⁺₂/SL2(Z)

a(T, σ) ǫ(T)(detT)^s,

L^∗(s, σ) = (2π)^−2sΓ(s)Γ(s−2σ−k+ 3/2)L(s, σ).

Our results are the following analytical properties of the Koecher-Maass series.

Theorem 1 The Koecher-Maass seriesL^∗(s, σ)can be meromorphically con- tinued to the whole s-plane. It satisfies a functional equation

L^∗(k+ 2σ−s, σ) = L^∗(s, σ) + (−1)^k/2 2^−k−2σ

π^4σ+2k−3/2

sin 2πσ cos 2πσ

ζ(2σ+k−1)Γ(2k+ 4σ−3) ζ(1−k−2σ)ζ(3−2k−4σ)

× Γ(s)Γ(s−2σ−k+ 2/3) Γ(s−1/2)Γ(s−2σ−k+ 1)

cosπ(s−2σ) sinπs cosπ(s−σ)

1

cosπssinπs+ 1 sin²π(s−2σ)

!

× ζ^∗(2s−1)ζ^∗(2s−4σ−2k+ 2),

where ζ^∗(s) = π^−s/2Γ(s/2)ζ(s)is the Riemann zeta function with the gamma factor.

Although we defined the Koecher-Maass series only for the Fourier coef- ficients indexed by half integral positive definite symmetric matrices of size two, there exist the Fourier coefficients indexed by indefinite half integral

symmetric matrices of size two. We would like to define a Koecher-Maass series for indefinite case. We should replaceǫ(T)⁻¹ by a certain volumeµ(T) associated with T introduced by Siegel. However it is known that if −detT is a square of a rational number then µ(T) is not finite.

The same difficulty comes up when we treat the prehomogeneous zeta function associated with the space of two by two symmetric matrices. This case was solved by Shintani [15]. There are other different approaches due to Sato [14] and Ibukiyama [7] to get Shintani’s result. Ibukiyama proved all of the Shintani’s results by using certain Eisenstein series of half integral weight.

Ibukiyama’s method gives us a reasonable definition of the zeta function very naturally. Our treatment of the Koecher-Maass series is a ”convolution version” of Ibukiyama’s method. The idea is due to Professor Ibukiyama. We will use the results in this paper to get a reasonable definition of the Koecher- Maass series for the indefinite case by following Ibukiyama’s idea.For the case the degreen≥3, the Koecher-Maass series for real analytic Siegel-Eisenstein series of each signature was first introduced by Arakawa [1]. He got a vector type functional equation between them. Combining Ibukiyama-Katsurada’s explicit formula and the method given in this paper, we can reprove and simplify Arakawa’s result as appeared in [6]. See [12] for details.

At the conference, Professor Boecherer asked whether my method works or not for the Koecher-Maass series with Grossencharacters. I feel that this is possible for the case the degree is two. To get an explicit formula of the Koecher-Maass series for real analytic Siegel-Eisenstein series with Grossen- characters, we should use Boecherer’s result [4] and the Maass type relation for the Fourier coefficients of real analytic Siegel-Eisenstein series obtained by Kohnen [9]. To get a meromorphic continuation and a functional equation of it, we should use the techniques developed in this paper, [11], [12] and [13].

3 Outline for the proof

Our starting point for the proof is the explicit formula due to Ibukiyama and Katsurada [6].

Theorem 2 (very special case of Ibukiyama-Katsurada’s formula) L(s, σ)

= E(σ, k)2^2sζ(2s−k−2σ+ 1)^X

d>0

c(d,2σ+ 2,−2k+ 5)c(d,0,−3)d^{2σ+k−1−s}

where E(σ, k) is an explicit constant, c(d, σ, k)are the Fourier coefficients of a linear combination of two standard Eisenstein series of half integral weight on Γ0(4) defined by

F(k, σ, τ) = E(k, σ, τ) + 2^k/2−σ(e^2πik8 +e^−2πik8)E^∗(k, σ, τ)

= v^σ/2+v^σ/2

∞

X

d=−∞

c(d, σ, k)e^2πiduτd(v,σ−k 2 ,σ

2) where τ =u+iv,

E(k, σ, τ) =v^σ/2

∞

X

d=1,odd

∞

X

c=−∞

(4c

d )ǫ^−k_d (4cτ +d)^k/2|4cτ +d|^−σ, E^∗(k, σ, τ) =E(k, σ,−1/4τ)(−2iτ)^k/2

with the well known automorphic factor (^4c_d)ǫ⁻¹_d (4cτ +d)^1/2 on Γ0(4) θ(γτ)

θ(τ) = (4c

d)ǫ⁻¹_d (4cτ +d)^1/2, a b 4c d

!

∈Γ0(4)

andτn(y, α, β)is the Whittaker function defined byτn(y, α, β) =^R_−∞^{∞ e−2πinx}_zαz^β dx.

F(k, σ, τ) was first introduced by Cohen [5] for holomorphic case (σ = 0) and the case with the parameter σ was introduced by Ibukiyama and Saito [7]. We call it a Cohen type Eisenstein series.

The explicit formula says that the Koecher-Maass series is the Rankin- Selberg convolution of two Cohen type Eisenstein series. Hence we can expect to apply the Rankin-Selberg method to get the anaytic properties of the convolution product.

The usual Rankin-Selberg method works as follows. Let f(τ) =

∞

X

n=0

ane^2πinτ, g(τ) =

∞

X

n=1

bne^2πinτ

be two holomorphic automorphic functions on the congruence subgroup Γ, where we assume that g(τ) is of rapid decay as the variable τ in the up- per half-plane tends to the cusp i∞. Let R∞(s, f g) be the Rankin-Selberg

convolution of f and g associated to the cusp i∞ defined by R∞(s, f g) =

Z ∞ 0

Z 1

0 f(τ)g(τ)v^s−2dudv

= Γ(s−1) (4π)^s−1

∞

X

n=1

anbn

n^s−1. (1)

Then the Rankin-Selberg method gives us an integral expression of the con- volution R∞(s, f g) as the scalar product of f g with an Eisenstein series E_∞(s, τ) on Γ i.e.

R_∞(s, f g) =

Z Z

Df(τ)g(τ)E_∞(s, τ)dudv

v² , (2)

where we assume that the stabilizer Γ∞of the cuspi∞in Γ is given by Γ∞=

(

± 1 Z 0 1

!)

and D is a fundamental domain of Γ, E∞(s, τ) is the Eisen- stein series associated to the cuspi∞defined byE∞(s, τ) =^P_γ∈Γ∞\Γℑ(γτ)^s. Hence the well known analytical properties of E∞(s, τ) give the same prop- erties of the convolution R∞(s, f g).

Returning to our case, there are three main problems to apply this method.

(i) We treat the convolution of two Cohen type Eisenstein series f(τ) = F(k1, σ1, τ) and g(τ) = F(k2, σ2, τ). Since Cohen type Eisenstein series are not of rapid decay, the integral in (2) does not converge.

(ii)Since Cohen type Eisenstein series F(k, σ, τ) are modular forms on Γ0(4), the Eisenstein series E∞(s, τ) appearing (2) is also a form on Γ0(4).

Because there exist three inequivalent cusps i∞,0,1/2 of Γ0(4), the func- tional equation of E∞(s, τ) is a vector type which involves other types of Eisenstein series E0(s, τ) and E1/2(s, τ) associated to the cusp 0 and the cusp 1/2:

E∞(s, τ) =φ∞(s)E∞(1−s, τ) +φ0(s)E0(1−s, τ) +φ1/2(s)E1/2(1−s, τ).

Hence we can get only the vector type functional equation of convolutions which involves not only R∞(s, f g) but also R0(s, f g) and R1/2(s, f g), the convolutions associated to the Fourier expansion off g at the cusp 0 and the cusp 1/2:

R_∞(s, f g) =φ_∞(s)R_∞(1−s, f g)+φ₀(s)R₀(1−s, f g)+φ_1/2(s)R_1/2(1−s, f g).

We want to get a functional equation of the convolutionR∞(s, f g) associated to the cusp i∞ itself.

(iii)Since Cohen type Eisenstein series F(k, σ, τ) are real analytic, its Fourier expansion involves Whittaker functions τn(v, α, β) and contains the Fourier coefficients indexed by all integers. Hence the convolution is the sum of two subseries consists of the terms indexed by all positive integers and the terms indexed by all negative integers. We want to get a functional equation of each subseries of the convolution itself if possible. To do this is not easy because the gamma like factor of the convolution is the Mellin transform of a product of two Whittaker functions which involves Barnes’ generalized hypergeometric functions 3F2. Unlike the usual gamma function appeared in (1), Barnes’ generalized hypergeometric functions 3F2 are not familiar for us. I don’t know whether this gamma like factor can be written in terms of the usual gamma functions or not in the present case.

The problems (i) and (ii) can be solved by applying Zagier’s Rankin- Selberg method [17] and Kohnen’s technique [8] for modular forms belonging to the plus space. This method was published in [11].

In this paper we give a new method to solve the problems (i) and (ii).

The key idea is a use of the theory of Jacobi forms. It is known that there is a correspondence between the space of holomorphic Jacobi forms of index one and the Kohnen’s plus space. We give an example of a real analytical version of such correspondence. We show that a real analytic Jacobi Eisen- stein series of index one corresponds to a real analytic Cohen type Eisenstein series by a similar manner as holomorphic case. Then we apply this cor- respondence and Zagier’s Rankin-Selberg method to solve the problems (i), (ii) and we get a functional equation of the convolution. This new method clarifies the reason why the convolution of two Cohen type Eisenstein series has the functional equation itself. Next we apply this functional equation of the convolution to the case relating the Koecher-Maass series. We solve the problem (iii) by using two types of the functional equation of Barnes’ gen- eralized hypergeometric functions 3F2. We can remove Barnes’ generalized hypergeometric functions 3F2 in the final form of the functional equation.

Only the usual gamma functions remain as the gamma factors in the final form of the functional equation as stated in Theorem 1.

4 Proof(the first step)

It is known that there is a correspondence between the space of holomor- phic Jacobi forms of index one and the Kohnen’s plus space. We give an example of a real analytical version of such correspondence as follows.

Let Ek,1(τ, z, s) be a real analytic Jacobi Eisenstein series of index one defined by

Ek,1(τ, z, s) = (ℑτ)^s 2

X

c,d∈Z (c,d)=1

X

λ∈Z

e^{2πi(λ2aτcτ+d+b+2λcτ+dz − cz}

2 cτ+d)

(cτ +d)^k|cτ +d|^2s This was introduced and studied by Arakawa [2].

Theorem 3 Ek,1(τ, z, s) can be written as

Ek,1(τ, z, s) =h0(k, s, τ)θ0(τ, z) +h1(k, s, τ)θ1(τ, z), where we put

θµ(τ, z) = ^X

λ∈Z λ≡µ (mod 2)

q^λ

2 4 ζ^λ with q =e^2πiτ and ζ =e^2πiz.

If k≡1 (mod 4) then we have h0(1−k

2 ,σ

2,4τ) +h1(1−k 2 ,σ

2,4τ) = 2^σF(k, σ, τ) and if k ≡3 (mod 4) then we have

(4v)^k2

(

h₀(1 +k

2 ,σ−k

2 ,4τ) +h₁(1 +k

2 ,σ−k 2 ,4τ)

)

= 2^σF(k, σ, τ).

This theorem has some applications for Cohen type Eisenstein series.

(a)Arakawa showed that Ek,1(τ, z, s) can be continued meromorphically to the whole s-plane and satisfies a functional equation. Hence Cohen type Eisenstein series have the same analytic properties.

(b) We assume thatk is congruent to 1 modulo 4. The casekis congruent to 3 modulo 4 is similar. The following fact is important for us.

For the sake of simplicity we write Fj(τ) =F(kj, σj, τ) and set fµ(τ) =hµ(1−k1

2 ,σ1

2 , τ), gµ(τ) =hµ(1−k2

2 ,σ2

2 , τ), ξ(τ) =f0(τ)g0(τ) +f1(τ)g1(τ).

Then we have

Proposition 1 ξ(τ) transforms like a modular form on SL2(Z):

ξ(γτ) = (cτ +d)^(k2−k1)/2

|cτ +d|^k2 ξ(τ), γ = a b c d

!

∈SL2(Z).

We return to our case. Our target is the convolution associated with the cusp i∞, essentially equals the Koecher-Maass series, defined by

R∞(s, F1F2) =

Z _∞

0

Z ₁

0 (F1(τ)F2(τ)−ψ(v))v^s−2dudv,

where ψ(v) is the product of constant terms of the Fourier expansion for F1(τ) and F2(τ), which breaks the convergence of the usual Rankin-Selberg integral.

Theorem 4 Assume that odd integers k1, k2 satisfy k1 ≡k2 (mod 4). We put

Ω(s) = 2^2sπ^−sΓ s+ k1

2

!

ζ 2s+ k1

2 +k2

2

!

R∞(s, F1F2).

Then Ω(s) can be meromorphically continued to the whole s-plane and Ω(s) satisfies the functional equation

Ω(s) = Ω 1− k1

2 − k2

2 −s

!

. Proof. Let

R∞(s, ξ) =

Z ∞ 0

Z 1

0 (ξ(τ)−ψ(v

4))v^s−2dudv be the Rankin-Selberg transformation of ξ(τ). It is easy to see

R_∞(s, F₁F₂) = 2^{2−σ1−σ2−2s}R_∞(s, ξ).

Applying Zagier’s Rankin-Selberg method (see [17], [11]), we obtain R∞(s, ξ) =

Z Z

Dξ(τ)(E(τ, s)−a0(v, s))dudv v² +

Z Z

D(ξ(τ)−ψ(v

4))a0(v, s)dudv v² +ϕ(s+k1+k2

4 )

Z Z

Dψ(v

4)v^{1−s−k1+2k2}dudv v²

−

Z Z

Lψ(v

4)v^sdudv v² ,

where we take the standard fundamental domainDfor the action ofSL2(Z), L is defined by L = {u+iv : −1/2 ≤ u ≤ 1/2, v > 0} \D, E(τ, s) is the Eisenstein series defined by

E(τ, s) = v^s 2

X

c,d∈Z (c,d)=1

(cτ +d)^{k2−k2 1}

|cτ +d|^2s+k2

and a₀(v, s) =v^s+ϕ(s+^k1+k₄ ²)v^{1−s−k1+2k2} is the constant term ofE(τ, s).

By the standard facts for E(τ, s), this expression gives a meromorphic continuation of R∞(s, ξ) and a functional equation

R∞(s, ξ) =ϕ(s+k1 +k2

4 )R∞(1−s−k1+k2

2 , ξ).

Hence the same facts are valid for our targetR∞(s, F1F2).

The good point is that the Eisenstein series E(τ, s) is a form on SL2(Z) because of Proposition 1. Hence it has a functional equation itself. This is the reason why we introduced the function ξ(τ).

5 Proof(the second step)

We apply Theorem 4 to the Koecher-Mass series. We take the parameters by

k1 =−3, σ1 =η, k2 =−2k+ 5, σ2 = 2σ+ 2

and later we will letη →0. Then Ω(s) in Theorem 4 has the Dirichlet series expression as

Ω(s) = 2^2sπ^−sΓ

s− 3 2

ζ(2s−k+ 1)

× ^X

d6=0

c(d, η,−3)c(d,2σ+ 2,−2k+ 5)Id(s, η,2σ+ 2,−3,−2k+ 5) where

Id(s, σ1, σ2, k1, k2) =

Z ∞

0 τd(y,σ1−k1

2 ,σ1

2 )τd(y,σ2−k2

2 ,σ2

2 )y^{σ1+2σ2+s−2}dy is the Mellin transform of the product of two Whittaker functions.

Let Ω⁺(s) be the subseries of Ω(s) consisting of the terms indexed by all positive integers with the gamma factor and Ω⁻(s) be the subseries of Ω(s) consisting of the terms indexed by all negative integers with the gamma factor.

Ω⁺(s) = 2^2sπ^−sΓ

s− 3 2

ζ(2s−k+ 1)

× ^X

d>0

c(d, η,−3)c(d,2σ+ 2,−2k+ 5)Id(s, η,2σ+ 2,−3,−2k+ 5)

Ω⁻(s) = 2^2sπ^−sΓ

s−3 2

ζ(2s−k+ 1)

× ^X

d<0

c(d, η,−3)c(d,2σ+ 2,−2k+ 5)Id(s, η,2σ+ 2,−3,−2k+ 5).

Then by Theorem 4 we have

Ω⁺(s) + Ω⁻(s) = Ω⁺(k−s) + Ω⁻(k−s).

Let η → 0. Then Ibukiyama-Katsurada’s formula says that Ω⁺(s) is essen- tially the Koecher-Maass series L(s, σ) associated with real analytic Siegel- Eisentein series of degree two and weight k.

By some calculations we can see that Ω⁻(s) is essentially a product of two shifted Riemann zeta functions and Ω⁻(s) has a meromorphic continuation to all s. Hence our remaining task is to simplify Ω⁻(s)−Ω⁻(k−s) when η→0. By some calculations we get

Ω⁻(s)−Ω⁻(k−s) = C(σ, k)π^−k+1/2ζ^∗(2s+ 2σ−1)ζ^∗(2s−2σ−2k+ 2)

×

( Γ(s−³₂)

Γ(s+σ− ¹₂)Γ(s−σ−k+ 1)K(s)− Γ(k−s−³₂)

Γ(k−s+σ− ¹₂)Γ(1−s−σ)K(k−s)

)

where C(σ, k) is an explicit constant andK(s) can be written as K(s) =K1(s) +K2(s),

K1(s) = Γ(s+σ)Γ(s+σ−1/2)Γ(−2σ−k+ 3/2) Γ(−σ)Γ(s+σ+ 1)

× 3F2

"

s+σ, s+σ−1/2, k+σ−3/2 2σ+k−1/2, s+σ+ 1

#

,

K2(s) = Γ(s−σ−k+ 3/2)Γ(s−k−σ+ 1)Γ(2σ−k−3/2) Γ(k+σ−3/2)Γ(s−σ−k+ 5/2)

× ₃F₂

"

s−σ−k+ 3/2, s−k−σ+ 1, −σ

−2σ−k+ 5/2, s−σ−k+ 5/2

#

,

where 3F2 is the generalized hypergeometric function at unit argument ini- tially defined by

3F2

"

a, b, c e, f

#

=

∞

X

n=0

(a)n(b)n(c)n

(e)n(f)nn! , (x)n= Γ(x+n)/Γ(x) and its meromorphic continuation with respect to all s.

To simplify the avobe terms is the problem (iii) mentioned in Introduc- tion. The avobe terms contains four Barnes’ generalized hypergeometric functions all of which have different parameters each other.

By using two types of functional equation of Barnes’ generalized hyper- geometric function 3F2 (see [3], [16]), we can see that

Ω⁻(s)−Ω⁻(k−s) = C(σ, k)π^−k+1/2ζ^∗(2s+ 2σ−1)ζ^∗(2s−2σ−2k+ 2)

× {(Γ[∗]−Γ[∗] + Γ[∗]−Γ[∗]) + (Γ[∗]−Γ[∗] + Γ[∗]−Γ[∗])3F2[∗]}

where Γ[∗] are functions of the form Γ[∗] = Γ(∗)Γ(∗)....

Γ(∗)Γ(∗).... We can show that (Γ[∗]−Γ[∗] + Γ[∗]−Γ[∗])

appeared as the coefficient of 3F2is zero. Hence we can remove Barnes’ gener- alized hypergeometric functions from the gamma like factor of the functional equation in the final form. This gives Theorem 1.

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IA Math. 28 (1981), 415–437.

Yoshinori Mizuno

Department of Mathematics, Graduate School of Science, Osaka University,

Machikaneyama 1-16,

Toyonaka, Osaka, 560-0043, Japan

E-mail: mizuno@gaia.math.wani.osaka-u.ac.jp