Analytic Jacobi Eisenstein series and the Shimura method

Von Bernhard E. Heim

Abstract

In this paper it is proven that analytic Jacobi Eisenstein series always admit a meromorphic continuation on the whole complex plane and statements about the location of possible poles are given. Moreover a new interpretation of Shimura’s approach to the standard L-function of a Siegel modular form is presented.

The purpose of this paper is twofold. First and mainly we study the analytic continuation and the location of poles of analytic Jacobi Eisenstein series. Secondly we present a new interpretation of Shimura’s approach to automorphic L-functions and indicate several improvements and applications.

In [Sh75] Goro Shimura introduced a method to study in a precise way the poles of the Rankin L-function D_{f}(s) attached to an elliptic cusp form f by integrating the cusp form against an Eisenstein series of half-integral weight multiplied with a theta series. Later on this method was generalized to standard L-functions of Siegel modular forms ([An79][A-K79],[Sh94]).

In this paper we show that Shimura’s integral can be interpreted as the Petersson scalar product of an analytic Jacobi Eisenstein series (introduced by Arakawa [Ar90], [Ar92], [Ar94]) restricted to the Siegel variable against the modular form.

The reason for this is the following insight: an analytic Jacobi Eisenstein series can be interpreted as the product of a Siegel Eisenstein series (of integral or half integral weight) with a theta series, up to some trace. As a by-product we get the meromorphic continuation and a precise description of possible poles of analytic Jacobi Eisenstein series of every degree and index. To get the functional equation one may has to proceed in a different way. This has been done in the elliptic case by Tsuneo Arakawa in [Ar90] and [Ar92], using methods of Roelcke and Selberg. He worked with vector-valued half-integral Eisenstein series. But if one is satisfied with Jacobi-Eisenstein series attached to congruence subgroups of the Jacobi group (actually this is sufficient for the Shimura method), then the functional equation follows from the theory of Siegel Eisenstein series (of half-integral weight).

It should be mentioned that the Fourier-Jacobi expansion of an analytic Siegel Eisenstein series is not a Jacobi Eisenstein series in the sense of Arakawa. This is in contrast to the holomorphic situation. Acknowledgment This work was inspired by Tsuneo Arakawa’s papers [Ar90] and [Ar94]. The author would like to thank him, Siegfried Böcherer and Paul Garrett for useful discussions and encouragement. But without the hospitality of the Max-Planck Institut für Mathematik in Bonn and a Max-Planck fellowship, this work would have not been possible.

Table of content

1. Modular forms 1.1 Siegel modular forms 1.2 Jacobi Eisenstein series

2. The Main formula

3. Jacobi Eisenstein series and theta series

4. Applications 4.1 Analytic properties of Jacobi Eisenstein series 4.2 Properties of the standard L-function 4.3 Analytic properties of Jacobi Eisenstein series of Klingen type.

Notation: For an associative ring R with identity element, we denote by R^{×} the group of all invertible elements of R and by R^{m,n} the module of all m × n matrices with entries in R. If M is a matrix, M^{t}, det(M), and tr(M) stand for its transpose, determinant, and trace. We put M_{n}(R) = R^{n,n}, Gl_{n}(R) = M_{n}(R)^{×}, Sl_{n}(R) = {M ∈ Gl_{n}(R)| det(M) = 1} (when R is commutative). The identity and zero elements of M_{n}(R) are denoted by 1_{n} and 0_{n} respectively (when n needs to be stressed). Let J_{n} = \binom{0 1}{−1 0} then Sp_{n}(R) = \{M ∈ Gl_{2n}(R)|M^{t}J_{n}M = J_{n}}.

P_{n,r}(R) = \{( \begin{matrix}∗ & ∗ \\ 0_{n+r,n−r} & ∗ \end{matrix} ) ∈ Sp_{n}(R)\} ,
P_{n,r}^{J}(R) = \{( \begin{matrix}1_{n−r} & ∗ \\ 0_{n+r,n−r} & ∗ \end{matrix} ) ∈ Sp_{n}(R)\} .

For convenience let Γ_{n} = Sp_{n}(ℤ), P_{n,r} = P_{n,r}(ℤ), Γ = Γ_{1} and Γ_{∞} = P_{1,0}. For real symmetric matrices A and B, we write A > B (A ≥ B) to indicate that A − B is positive definite (positive semi-definite) and A[B] = B^{t}AB if A, B are suitable. If A_{1}, A_{2}, …, A_{n} are square matrices, [A_{1}, A_{2}, …, A_{n}] denotes the matrix with A_{1}, A_{2}, …, A_{n} in the diagonal blocks and 0 in all other blocks. Let A_{n}^{+} denote the set of all semi-integral positiv-definit matrices of size n. Let Z ∈ ℂ^{n,n}, then we put e{Z} = e^{2πi (tr Z)} and Re(Z), Im(Z) for the real and imaginary part of Z and for M ∈ ℂ^{m,n} we put occasionally x_{M} = Re(M) and y_{M} = Im(M). Further δ(Z) = det (Im (Z)). By \overline{a} we denote the complex conjugation of an element a ∈ ℂ.

1 Modular forms

1.1 Siegel modular forms

The group Sp_{n}(ℝ) acts on the Siegel upper half space IH_{n} of degree n by

(M, Z) ↦ M(Z) = (AZ + B)(CZ + D)^{−1},

where M = \binom{A B}{C D} with A, B, C, D ∈ ℝ^{n,n}. We denote by M_{n}^{k} be the space of Siegel modular forms and S_{n}^{k} the subspace of cusp forms of degree n and weight k for the Siegel modular group Γ_{n}. Let J(M, Z) = (CZ + D) and j(M, Z) = det (CZ + D). Then every F ∈ M_{n}^{k} satisfies the transformation law

F (Z) = F (M(Z)) j(M, Z)^{−k} for all M ∈ Γ_{n}, Z ∈ IH_{n}.  (1)

The right side of the equation (1) is usually abbreviated by (F |_{k}M)(Z). For arbitrary complex valued functions F and G on IH_{n}, which satisfy (1) for a congruence subgroup Γ′ of Γ_{n}, we define the Petersson integral, convergence assumed, by

⟨F, G⟩_{Γ′} = ∫_{Γ′\IH_{n}} F (Z) \overline{G(Z)} δ(Z)^{k}d^{∗}Z.  (2)

Here Z = X + i Y and d^{∗}Z = det (Y)^{−(n+1)}dXdY denotes the symplectic volume element. Every F ∈ S_{n}^{k} has a Fourier expansion of the form F (Z) = ∑_{T ∈A_{n}^{+}} a^{F} (T)e{T Z}. If σ is an automorphism of ℂ, then

F^{σ}(Z) = ∑_{T ∈A_{n}^{+}} a^{F} (T)^{σ}e{T Z} ∈ S_{n}^{k}.  (3)

We denote by # the complex-conjugation in ℂ. If we assume that F ∈ S_{n}^{k} is a Hecke eigenform with Satake-parameters (α_{0,p}; α_{1,p}; …; α_{n,p}). Then the standard zeta function D_{F}^{n} (s) of F is given by

D_{F}^{n} (s) = ∏_{p} {(1 − p^{−s}) ∏_{j=1}^{n}(1 − α_{j,p}p^{−s}) (1 − α_{j,p}^{−1}p^{−s})}^{−1} .  (4)

For more details the reader is referred to Andrianov [An87] or Klingen [Kl90].

1.2 Jacobi Eisenstein series

Every element g ∈ P_{n+l,n}^{J}(ℝ) can be parameterized by

((λ, μ; ρ), M) ∈ (ℝ^{l,n} × ℝ^{l,n} × ℝ^{l,l}) × Sp_{n}(ℝ)  (5)

determined by

g = ( \begin{matrix}1_{l} & 0 & ρ & μ \\ 0 & 1_{n} & μ^{t} & 0 \\ 0 & 0 & 1_{l} & 0 \\ 0 & 0 & 0 & 1_{n} \end{matrix} ) ( \begin{matrix}1_{l} & λ & 0 & 0 \\ 0 & 1_{n} & 0 & 0 \\ 0 & 0 & 1_{l} & 0 \\ 0 & 0 & −λ^{t} & 1_{n} \end{matrix} ) (1_{2l} × M) . (6)

An important normal subgroup is given by the Heisenberg group H_{n,l}(ℝ), consisting of all elements h of the form h = (λ, μ; ρ) = ((λ, μ; ρ), 1_{2n}). The group P_{n+l,n}^{J}(ℝ) acts on D_{n,l} = IH_{n} × ℂ^{l,n} by

P_{n+l,n}^{J}(ℝ) × D_{n,l} ⟶ D_{n,l}
g(τ, z) = (M(τ), zJ(M, τ)^{−1} + λM(τ) + μ),  (7)

where g = ((λ, μ; ρ), M) and (τ, z) ∈ D_{n,l}. It is sometimes convenient to regard Sp_{n}(ℝ) as a subgroup of P_{n+l,n}^{J}(ℝ).

Definition 1.1 Let k, l, n ∈ ℕ and S ∈ A_{l}^{+}. The factor of automorphy j_{k,S} of index S and weight k for P_{n+l,n}^{J}(ℝ)is the map

j_{k,S} : \{ \begin{matrix}P_{n+l,n}^{J}(ℝ) × D_{n,l} & ⟶ & D_{n,l} \\ (g, (τ, z)) & ↦ & j_{k,S}(g, (τ, z)) \end{matrix} ,  (8)

where

j_{k,S}(g, (τ, z)) = j(M, τ)^{k}e{−Sρ}  (9)
e{−S[λ] M(τ) − 2λ^{t}SzJ(M, τ)^{−1} + S[z]J(M, τ)^{−1}c}

for g = ((λ, μ; ρ), M) and M = \binom{a b}{c d}. Let F : D_{n,l} ⟶ ℂ, then the generalized Petersson slash operator |_{J,k} is given by

(F |_{J,k}g) (τ, z) := j_{S,k}(g, (τ, z))^{−1}F (g(τ, z)).  (10)

Let J_{k,S}^{n} and J_{k,S}^{n,cusp} denote the space of Jacobi forms and Jacobi cusp forms, respectively, of weight k, index S and degree (n, l) for P_{n+l,n}^{J}(ℤ). We abbreviate P_{n+l,n}^{J}(ℤ) by Γ_{n,l}^{J}. For more details see Arakawa [Ar94] (section 1.1 and 1.2). Similar as Eisenstein series on the Siegel upper half plane, one defines Jacobi Eisenstein series. For each r with 0 ≤ r ≤ n, let

Γ_{n,l,r}^{J} = {((0λ_{2}, μ; ρ), M) | λ_{2} ∈ ℤ^{l,r}, μ ∈ ℤ^{l,n}, ρ ∈ ℤ^{l,l} and M ∈ P_{n,r}} (11)

and

∗_{J} : \{ \begin{matrix}D_{n,l} & ⟶ & D_{r,l} \\ (τ, z) & ↦ & (τ [\binom{0}{1_{r}}] , z\binom{0}{1_{r}}) \end{matrix} .  (12)

Definition 1.2 Let k, l, n ∈ ℕ with k even, 0 ≤ r ≤ n and S ∈ A_{l}^{+}. To ϕ ∈ J_{k,S}^{r,cusp} we attach an analytic Jacobi Eisenstein series of Klingen type on D_{n,l} defined by:

E_{n,r}^{S,k}((τ, z), ϕ); s) = ∑_{γ∈Γ_{n,l,r}^{J}\Γ_{n,l}^{J}}ϕ(γ(τ, z)_{∗_{J}}) j_{k,S}(γ, (τ, z))^{−1} (\frac{δ(M(τ))}{δ(M(τ)_{∗})})^{s} , (13)

here γ = (h, M), s ∈ ℂ with Re(s) sufficiently large and ( \begin{matrix}τ̃_{1} & τ̃_{2} \\ ∗ & τ̃_{4} \end{matrix} )_{∗} = τ̃_{4} ∈

IH_{r}. If r = 0, we put E_{n}^{S,k}((τ, z); s) = E_{n,0}^{S,k}((τ, z), 1; s).

Let ϕ ∈ J_{k,S}^{n}, then (D_{0}ϕ)(τ) := ϕ(τ, 0) ∈ M_{n}^{k}.

Remark 1.3

a) The Eisenstein series is absolutely convergent for k + 2 Re(s) > n + r + l + 1. If k > n + r + l + 1, then E_{n,r}^{S,k}(ϕ, (τ, z); 0) ∈ J_{k,S}^{n}.

b) Let us put

χ_{μ}^{k,s;J}(g, (τ, z)) = j_{k,S}(g, (τ, z))^{−1}|j(M, τ)|^{−2s},  (14)

then we have the cocycle relation

χ_{μ}^{k,s;J}(gh, (τ, z)) = χ_{μ}^{k,s;J}(g, h(τ, z)) χ_{μ}^{k,s;J}(h, (τ, z))  (15)

for g, h ∈ P_{n+l,n}^{J}(ℝ) and

E_{n}^{S,k}((τ, z), s) = δ(τ)^{s} ∑_{γ∈Γ_{(n,l),0}^{J}\Γ_{n,l}^{J}} χ_{μ}^{k,s;J}(γ, (τ, z)).

2 The MAIN FORMULA

The Jacobi Eisenstein series introduced in the last section, can be used to study analytic and arithmetic properties of the standard L-function attached to Siegel modular forms. In this section we present a formula, which can be used as a starting point in this direction.

Let S ∈ A_{l}^{+}. Then we attach to S the theta series

Θ_{S}^{o}(τ) = ∑_{λ∈ℤ^{l,n}} e{S[λ]τ},  (τ ∈ IH_{n}).  (16)

It is obvious, that one can choose

R_{l,n} = {((λ, 0; 0), M) | λ ∈ ℤ^{l,n} and M ∈ P_{n,0}\Γ}  (17)

for a representative system of Γ_{n,l,0}^{J} \ Γ_{n}^{J}. Thus a simple pullback formula of the Jacobi Eisenstein series via the natural embedding

IH_{n} ⟶ IH_{n} × ℂ^{l,n}, τ ↦ (τ, 0)

is given by

E_{n}^{S,k} ((τ, 0), s) = δ(τ)^{s} ∑_{M∈P_{n,0}\Γ_{n}} χ_{μ}^{k,s}(M, τ) Θ_{S}^{o}(M(τ)).  (18)

Because the generalized automorphic factor χ_{μ}^{k,s;J} splits into

χ_{μ}^{k,s}(M, τ) e{S[λ]M(τ)}.  (19)

For further simplifications we put: Γ_{n}(s) = ∏_{ν=1}^{n} Γ(s − \frac{ν−1}{2}) and

D_{0}E_{n}^{S,k}(s) := E_{n}^{S,k}((τ, 0), s).

Theorem 2.1 (Main formula) Let k, l, n ∈ ℕ and k even. Let S ∈ A_{l}^{+} and F ∈ S_{n}^{k}. Let s ∈ ℂ with 2 Re(s) + k > n + l + 1. Then

\begin{aligned}⟨D_{0}E_{n}^{S,k}(s) ; F ⟩_{Γ_{n}} & = (4π)^{−n(k+s− \frac{n+1}{2} )}π^{n(n−1)/4} Γ_{n}(s + k − \frac{n+1}{2}) \\ & × D_{S}^{n,l}(F^{#}, k + s − \frac{n+1}{2} ). \end{aligned} (20)

Here

D_{S}^{n,l}(G, s) := ∑_{λ∈ ℤ^{l,n} / Gl_{n}(ℤ)} a^{G}(S[λ]) (det(S[λ]))^{−s}

for G ∈ S_{n}^{k}.

Proof: After all our previous observations, this can be done in a straightforward manner. Let P_{n} = {Y ∈ ℝ^{n,n} | iY ∈ IH_{n}} and R_{n} a fundamental domain of the action of Gl_{n}(ℤ) on P_{n}, via Y ↦ Y [U^{t}].

⟨D_{0}E_{n}^{S,k}(s) ; F ⟩_{Γ_{n}} = ∫_{Γ_{n,0}\IH_{n}} Θ_{S}^{o}(τ)\overline{F (τ)}δ(τ)^{k+s} d^{∗}τ  (21)

= ∫_{X∈(ℝ/ℤ)^{n,n}} ∫_{R_{n}} Θ^{o}(X + iY) \overline{F (X + iY)}δ(Y)^{k+s−n−1} dX dY (22)

= ∑_{λ∈ℤ^{l,n} /Gl_{n}(ℤ)} \overline{a(S[λ])} ∫_{y>0} det(y)^{k+s−n−1} e^{−4πtr(S[λ]y)} dy.  (23)

Obviously, we have only to care for S[λ] > 0. But if this is the case, then ∫_{y} can be calculated by applying Lemma 2 given in [Kl90](Chapter 6), which proves the theorem.

Thus ⟨D_{0}E_{n}^{S,k}(s) ; F ⟩_{Γ_{n}} ≡ 0, if l < n. Let now F be a Hecke eigenform. It had been shown by Andrianov [An79], Böcherer [Boe86] and Shimura [Sh94], that D_{S}^{n,n}(F, s) is essentially an Euler product obtained from the eigenvalues, i.e the standard L-function of F.

Moreover, Andrianov and Shimura used a similar expression as given in the box, given in (21), to study analytic properties of the L-function. Actually, this approach had been introduced by Shimura [Sh75] to give a precise description of the poles of the Rankin L-function related to D_{1}^{1,1}(F, s). Hence the Shimura method can be interpreted as the Petersson scalar product of a cusp form with an on the Siegel variable restricted analytic Jacobi Eisenstein series. This view point, avoids somehow, the theory of modular forms of half-integral weight. But nevertheless, in the next section we will go somehow in the opposite direction to study analytic properties of the Jacobi Eisenstein series.

3 Jacobi Eisenstein series and theta series

Shimura established an isomorphism between J_{k,S}^{n} and a certain space of vectorvalued Siegel modular forms [Zi89] (Theorem 3.3). Main ingredients of the isomorphism are theta series with characteristic (a, b) on D_{n,l} of the following type:

Θ_{S,a,b}(τ, z) := ∑_{λ∈ℤ^{l,n}} e{S[λ + a]τ + 2(λ + a)^{t}S(z + b)}  (24)

with (a, b) ∈ ℚ^{l,n} × ℚ^{l,n}. For n = l = 1 see [E-Z85] and for simplification of the notation let Θ_{S} = Θ_{S,0,0}. In a non-holomorphic setting (e.g. Jacobi Eisenstein series E_{n,r}^{S,k}( ∗ , s)) the situation becomes much more complicated. A first approach to connect nonholomorphic Jacobi Eisenstein series with Siegel Eisenstein series has been established by Arakawa [Ar90] (Proposition 3.1) and [Ar92] (Theorem 6.2), at least in the case n = 1. He related the Jacobi Eisenstein series to Eisenstein series with theta multiplier system on IH_{1}. By making use of a general theory for Eisenstein series of SL_{2}(ℤ) due to Selberg and Roelcke, he was able to establish the meromorphic continuation and the functional equation of the Jacobi Eisenstein series. Here we would like to state a slidly different approach, which reduces the problem of meromorphic continuation to that of certain Siegel Eisenstein series (of half-integral weight).

Remark 3.1 Let g = ((λ, 0; 0), M) ∈ Γ_{n,l}^{J}. Then we have

\begin{aligned}J_{k,S}(g, (τ, z)) & = j(M, τ)^{k} e\{S[z]J(M, τ)^{−1} c\} \\ & × e\{−S[λ]M(τ) − 2λ^{t}SzJ(M, τ)^{−1}\}. \end{aligned}  (25)

This gives us the following description of analytic Jacobi Eisenstein series.

\begin{matrix}E_{n}^{S,k}((τ, z), s) & = δ(τ)^{s} ∑_{M=\binom{a b}{c d}∈Γ_{n,0} \Γ_{n}}χ_{μ}^{k,s}(M, τ) e\{S[z]J(M, τ)^{−1} c\} \\ × ∑_{λ∈ℤ^{l,n}} e\{S[λ]M(τ) + 2λ^{t}SzJ(M, τ)^{−1}\} \\ = δ(τ)^{s} ∑_{M=\binom{a b}{c d}∈Γ_{n,0} \Γ_{n}} χ_{μ}^{k,s}(M, τ) e\{−S[z]J(M, τ)^{−1} c\} Θ_{S} (M(τ, z)) . \end{matrix} (26)

To employ the transformation law of the theta series Θ_{S} we decompose Γ_{n,0}\Γ_{n} into

(P_{n,0} \ Γ_{0}^{n}(q)) × (Γ_{0}^{n}(q) \ Γ_{n}) .

Here q denotes the level of the quadratic form 2S, and

Γ_{0}^{n}(q) = \{γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{n} | C ≡ 0 (q)\}.

By χ_{S}^{(n)}(M) we denote the multiplier, an eighth root of unity, attached to S ∈ A_{l}^{+} and Γ_{0}^{n}(q) ([A-M75]). For example in the case l = n = 1 and S = 1 we have

χ_{1}^{(1)}(M) = (\frac{c}{d}) ϵ_{d}^{−1}  (M = \binom{a b}{c d} ∈ Γ_{0}(4)) ,  (27)

here ϵ_{d} = 1 if d ≡ 1(4) and i if d ≡ 3(4) and (\frac{c}{d}) is a generalized quadratic residue symbol (Jacobi symbol, for more details [Sh73]). Let τ ∈ ℂ^{×}. For the square root τ^{1/2} we always choose the branch with −π/2 < arg(τ^{1/2}) ≤ π/2 and put τ^{r/2} = (τ^{1/2})^{r} for all r ∈ ℤ.

((\frac{c}{d}) ϵ_{d}^{−1}(cτ + d)^{1/2})^{2} = (\frac{−1}{d}) j(M, τ).  (28)

More general (e.g. [Boe83], Proposition 9) we have

Lemma 3.2 Let S ∈ A_{l}^{+} and let q be the level of S. Then for M ∈ Γ_{0}^{n}(q) we have

Θ_{S}(M(τ, z)) = e{S[z] J(M, τ)^{−1}c} h_{S}(M, τ) Θ_{S}(τ, z),  (29)

with h_{S}(M, τ) = j(M, τ)^{l/2}χ_{S}^{(n)}(M) and (τ, z) ∈ D_{n,l}.

Definition 3.3 Let k, l, n ∈ ℕ and k even and let S ∈ A_{l}^{+} with level q. Let s ∈ ℂ with 2 Re(s) + k > n + l + 1. An (Siegel-)Eisenstein series of general type is defined by

\begin{aligned}E_{n}^{S,k−l/2,q}(τ, s) & = δ(τ)^{s} ∑_{γ∈P_{n,0}\Γ_{0}^{n}(q)} χ_{μ}^{k,s}(γ, τ) h_{S}(γ, τ) \\ & = δ(τ)^{s} ∑_{γ∈P_{n,0}\Γ_{0}^{n}(q)} χ_{S}^{(n)}(γ) χ_{μ}^{k−l/2,s}(γ, τ). \end{aligned}  (30)

Let us recall some well known results on multiplier systems coming from theta series ([An87], Chapter 1). Let S ∈ A_{l}^{+} with level q.

• Let l = 2g with g ∈ ℕ. Then χ_{S}^{(n)}(M) = ψ_{S}(det(D), where ψ_{S} is a certain real Dirichlet character modulo q.

• Let l = 2g + 1 with g ∈ ℕ. Then χ_{S}^{(n)}(M) = χ_{1_{n}}^{(n)}(M) ψ′_{S}(det(D), where ψ′_{S} is again a certain real Dirichlet character modulo q. Further let

(χ_{1_{n}}^{(n)}(M))^{2} = sgn(detD) (\frac{−4}{|det(D)|}) and (χ_{1_{n}}^{(n)}(M))^{4} = 1,

here sgn(x) = x/|x| and ( ) denotes the Jacobi symbol.

Remark 3.4 Let l = 2g then the function defined in (30) is an ordinary Eisenstein series (of integer weight) with real Dirichlet character ψ_{S}. If l = 2g + 1, then it is a Eisenstein series of half-integral weight with Dirichlet character ψ′_{S}. Both Eisenstein series posses a meromorphic continuation on the whole s−plane (Langlands [La76], see also Shimura [Sh94], the remark given in the last paragraph on page 562).

Thus the E_{n}^{S,k−l/2,q}(τ, s) is absolute convergent and behaves like an automorphic form with respect to Γ_{0}^{n}(q). Further it admits a meromorphic continuation on the whole s−plane.

Definition 3.5 Let F be a complex-valued function on D_{n,l}, which transforms like a Jacobi form (of weight k and index S) with respect to a congruence subgroup Γ′ of Γ_{n}. Then the n^{th}-trace of F is defined by

(Tr_{Γ′}^{Γ_{n}}F ) (τ, z) = ∑_{M∈Γ′\Γ_{n}} (F |_{J,k}M) (τ, z).  (31)

This new obtained function transforms like a Jacobi form with respect to Γ_{n}. Of course, we can also apply this construction to functions which are restricted to the symplectic variable τ (i.e. z = 0).

Theorem 3.6 Let k, l, n ∈ ℕ and S ∈ A_{l}^{+}. Let k be even and q be the level of S and s ∈ ℂ with Re(s) + k > n + l + 1. Then we have for (τ, z) ∈ D_{n,l}:

E_{n}^{S,k} ((τ, z), s) = Tr_{Γ_{0}^{n}(q)}^{Γ_{n}} (E_{n}^{S,k−l/2,q}(τ, s) Θ_{S}(τ, z)) .  (32)

Theorem 3.6 gives us a tool to study analytic properties of the Jacobi Eisenstein series, by reducing everything to the theory of (Siegel-)Eisenstein series of integral or half-integral weight and properties of the trace map Tr. not known to everybody. Let Γ′ be a congruence subgroup of Γ_{n} and let us assume that F behaves like automorphic forms with respect to Γ′ and G with respect to Γ. Then we have, if the scalar products exist,

⟨Tr_{Γ′}^{Γ_{n}}F, G⟩_{Γ_{n}} = ⟨F, G⟩_{Γ′}.  (33)

This principle works of course in much more general situations. A first application is the following. If we make the same assumptions as in Theorem 2.1, then we have

I(F, s) := ⟨D_{0}E_{n}^{S,k}(s) ; F ⟩_{Γ_{n}} = ⟨E_{n}^{S,k,q}(∗, s) (D_{0}(Θ_{S})) ; F)⟩_{Γ_{0}^{n}(q)} . (34)

Thus the meromorphic continuation (and functional equation) of D_{0}E_{n}^{S,k}(s) and E_{n}^{S,k,q}(τ, s) transfer directly to I(F, s). Moreover one gets a sharp picture about the possible poles of I(F, s). It is of course also interesting to compare directly the properties of the corresponding Eisenstein series of Jacobi and Siegel type.

4 Applications

4.1 Analytic properties of Jacobi Eisenstein series

Let U ⊆ ℂ be an open subset and F (τ, z, s) a function F : D_{n,l} × U ⟶ ℂ holomorphic in s. Further, let Γ′ a congruence subgroup of Γ and F |_{J,k}γ = F for S ∈ A_{l}^{+} and γ ∈ Γ′. Then Tr_{Γ′}^{Γ}F remains holomorphic. Hence we have the following Theorem 4.1 Let l, n ∈ ℕ and k ∈ 2ℤ and S ∈ A_{l}^{+}. Then E_{n}^{S,k}((τ, z), s) has a meromorphic continuation to the whole s−plane with possible poles coming from Eisenstein series of integral or half-integral weight E_{n}^{S,k−l/2,q}((τ, z), s) (the latter can be studied by [Sh75],[Sh94] (Theorem 7.3)).

Let us be more precise, at least in the simplest (non-trivial) case n = l = 1. Let K ∈ ℕ odd, N ∈ 4ℕ and ω be a Dirichlet character modulo N which satisfies ω(−1) = 1. Further, let L(s, ω) = ∑ ω(n) n^{−s}, j̃(γ, z) = Θ_{1}^{o}(γ(z)) / Θ_{1}^{o}(z) for γ ∈ Γ_{0}^{1}(4) and

E(z, s, K, ω) = δ(z)^{s/2} ∑_{γ∈P_{1,0}\Γ_{0}^{1}(N)} ω(d) j̃(γ, z)^{−K} |j̃(γ, z)|^{−2s}. (35)

We put λ_{0} = 0 or 1 according λ = \frac{1−K}{2} is even or odd. Then it is well known [Sh75] (Proposition3), that

\begin{aligned}F (z, s, K, ω) = & Γ(\frac{s+K}{2} )Γ(\frac{s−λ+λ_{0}}{2} ) \\ & L(2s − 2λ, ω^{2})E(z, s, K, ω), \end{aligned}  (36)

multiplied with (s − λ − 1) can be continued to a holomorphic function in the whole s−plane. The factor (s − λ − 1) is unnecessary either if \frac{|K|+1}{2} is even or if ω^{2} is non-trivial. Hence we get

Corollary 4.2 Let k ∈ 2ℕ and χ_{0}(N) be the principal Dirichlet character modulo N, then

Γ (s + k − 1/2) Γ (s + k/2) L(4s + 2k − 2, χ_{0}(4)) E_{1}^{1,k}((τ, z), s) (37)

can be continued to a holomorphic function in the whole s−plane. In particular E_{1}^{1,k}((τ, z), \frac{3−2k}{4}) ≡ 0.

Next we would like to compare theses results with [Ar90]. For simplicity let κ = \frac{k}{2} − \frac{1}{4} and E^{A}(s) = E_{1}^{1,k}((τ, z), s − κ). From the work of Arakawa one can deduce the following

Proposition 4.3 The Eisenstein series E^{A}(s) admits a meromorphic continuation on the whole s−plane with functional equation

E^{A}(1 − s) = Φ(1 − s)E^{A}(s),  (38)

here

Φ(s) = 2^{3/2−2s}e^{−πik/2} \frac{Γ(2s − 1)}{Γ(s + κ)Γ(s − κ)} \frac{ζ(4s − 2)}{ζ(4s − 1)}.  (39)

Poles of the Eisenstein series can only occur in the stripe \frac{1}{4} < Re(s) < \frac{1}{2}, when ζ(2 − 4s) has (non-trivial) zeroes.

The result about possible poles is equivalent to the statement, that the function

Ẽ^{A}(s) = Γ(2s + \frac{1}{2}) ζ(4s − 1) (s − \frac{1}{2}) E^{A}(s)

is holomorphic. Which is again the same as to say that

Γ(2s + k) ζ(4s + 2k − 2) (2s + k − 3/2) E_{1}^{1,k}((τ, z), s)  (40)

is holomorphic.

Remark 4.4 Corollary 4.2 and Proposition 4.3 give the same candidates for possible poles of the analytic Jacobi Eisenstein series. But of course, Arakawa’s approach has the advantage of giving a functional equation. Supplementary the Corollary determines a value of s where the Eisenstein series vanishes, which is not obvious from the Proposition 4.3. Hence (40) can be improved to the statement that

Γ(2s + k) ζ(4s + 2k − 2) E_{1}^{1,k}((τ, z), s)  (41)

is holomorphic.

4.2 Properties of the standard L-function

As already stated, Theorem 2.1 can be used to study properties of the standard L-function D_{F}^{n} (s) attached to a Hecke eigenform F ∈ S_{n}^{k}. For simplicity and because Arakawa’s result is at the moment only available for Eisenstein series on IH_{1} × ℂ^{l} we demonstrate this in the case of elliptic cusp forms.

Let us recall, that D_{1}^{1,1}(F, s)/2 = ∑_{λ∈ℕ} a^{F} (λ^{2}) λ^{−2s}, which is equal to

ζ(4s − 2k + 2)^{−1}D_{F}^{1} (2s),

here D_{F} (s) is the Euler product of degree 3 attached to a normalized Hecke eigenform F ∈ S_{1}^{k}. The Main formula, given in Theorem 2.1, specialized to this case leads to

⟨D_{0}E_{1}^{1,k}(s) , F ⟩_{Γ} = (4π)^{−(k+s−1)} Γ(s + k − 1) D_{1}^{1,1}(F, k + s − 1)
= (4π)^{−(k+s−1)}Γ(s + k − 1)ζ(4s + 2k − 2)^{−1}
×D_{F}^{1} (2s + 2k − 2).

This integral representation of D_{F}^{1} (s) and the holomorphy of the expression given in (41) leads to the holomorphy of

Γ(s − k + 2) Γ(s/2 − k) D_{F}^{1} (s).  (42)

Thus D_{F}^{1} (s) is a holomorphic function. Proposition 4.3 allows the standard zeta function to have only one possible pole (at s = k − 1/2), but from Corollary 4.2, we know that ‘there’ the Eisenstein series has a zero, which gives the desired result. Hence the holomorphy of the L-function can be deduced directly from the analytic behavior of an (Jacobi) Eisenstein series, which can be seen as a slid refinement of the Shimura method (e.g.[Sh75]).

Remark 4.5 These observations suggest, that generalizing Proposition 4.3 to Jacobi Eisenstein series of degree n should lead to a better understanding of the analytic behavior of the standard zeta function attached to a Siegel Hecke eigenform of arbitrary degree ([Mi91]).

4.3 Analytic properties of Jacobi Eisenstein series of Klingen type

In this last subsection we would like to indicate briefly, how one can also study analytic properties of Jacobi Eisenstein series of Klingen type (attached to Jacobi cusp forms). The recipe, which worked already for Eisenstein series of Klingen type attached to the Siegel modular group, could be described as follows.

It starts with the fact, that a result in the style of the Theorem of Garrett and Böcherer [Boe85b] (Theorem 1.8), [He96] (Theorem 1.11), an important tool in the arithmetic theory of Siegel modular forms, is available. A proof and a precise description of this is given in [Ar94] (Theorem 2.9), see also the introduction of [Du97]. Thus let Φ ∈ J_{k,S}^{r,cusp} be a Hecke-Jacobi eigenform and Z_{n}(s, Φ) the standard zeta function related to Φ (cf. [Ar94], section 2.2 and Theorem 2.9). Then the analytical properties of

Z_{n}(2s + k, Φ) E_{m,n}^{S,k}(⋆, Φ, s)

are directly related with the ones of E_{n+m}^{S,k}(⋆, s): notice that Z_{n}(s, Φ) does not depend on m. Thus one can also study this function alone, because E_{n,n}^{S,k}(z, 1, s) = f(z).

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