Real analytic Jacobi Eisenstein series and Dirichlet series attached to three Jacobi forms

By T. Arakawa and B. Heim

Abstract In this paper, the iterated Petersson scalar product of a diagonal-restricted real analytic Jacobi Eisenstein series of degree (3, 1) against three elliptic Jacobi cusp form is determined. This scalar product gives an integral representation of a certain Dirichlet series, which involves the Fourier coeffiencts of the three forms. Moreover new results and some partly proven conjectures towards analytic properties of real analytic Jacobi Eisenstein series of degree (n, 1) are given.

Introduction

P. Garrett studied the iterated Petersson scalar product of a diagonal-restricted Siegel Eisenstein series of degree 3 against three elliptic cusp forms [Ga87], which eventually led him to an integral representation of the Rankin triple L-function. Then analytic and arithmetic properties of the Siegel Eisenstein series had been used to get the meromorphic continuation, funqtional equation and ‘special value’ properties of the L-function from his approach. But to generalize and to use Garrett’s interpretation of the Rankin-Selberg method for other L-functions or at least Dirichlet series, seems to be a very hard and delicate problem in modern number theory. Nevertheless I. Piatetski-Shapiro and S. Rallis [P-R87] succeded in refining Garrett’s work to get an integral representation of the “internal tensor product L-function” of an automorphic form on a semi-simple cubic abelian algebra over a number field (see also [Ga85]). Recently, B. Heim [He97] used a diagonal restricted Eisenstein series of degree 5 to get an integral representation of the twisted spinor L-function.

In this paper we start with a new generalization. We study the iterated Petersson scalar product of a diagonal-restricted real analytic Jacobi Eisenstein series of degree (3, 1) against three elliptic Jacobi cusp form. We show that this leads to an integral representation of a certain Dirichlet series, which involves the Fourier coeffiencts of the three forms.

We explain this more precisely. Let k, n, t be positive integers and let IH_{n} be Siegel’s upper-half-space of degree n. We denote by J_{k,t}^{cusp} the space of Jacobi cusp forms of weight k and index t on IH_{1} × ℂ. Let Φ ∈ J_{k,t}^{cusp}. Then it has the Fourier expansion

Φ(τ, z) = ∑_{{r∈ℤ, D∈ℕ \atop −D≡r^{2} mod(4t)}} c_{r}^{Φ}(D) e(\frac{r^{2}+D}{4t} τ + rz), (1)

where (τ, z) ∈ IH_{1} × ℂ and e(z) = e^{2π i z}. For (τ̃,z̃) ∈ IH_{n} × ℂ^{1,n} and s ∈ ℂ, we denote by E_{n}^{k,t}((τ̃,z̃); s) the real analytic Jacobi Eisenstein series (for more details see Definition 1.4 and Remark 1.5). Let further ⟨; ⟩_{J} denote the Petersson scalar product of Jacobi forms.

Theorem Let k be an even integer with k + 2 Re (s) > 4 and let t be a natural number. Let Φ_{1}, Φ_{2}, Φ_{3} ∈ J_{k,t}^{cusp} and let Φ_{3} also be a Hecke Jacobi eigenform. Then

(2) ⟨⟨⟨E_{3}^{k,t} ((\begin{matrix}τ_{1} & & 0 \\ & τ_{2} \\ 0 & & τ_{3} \end{matrix}) , (z_{1}z_{2}z_{3}); s) ; Φ_{1}(τ_{1}, z_{1}) Φ_{2}(τ_{2}, z_{2})Φ_{3}(τ_{3}, z_{3})⟩_{J}⟩_{J}⟩_{J}

= γ(k, t, s) Γ_{∞}(s) Z_{1}^{J}(2s + k, Φ_{3}) D_{1,2;3}^{∗}(s).

Here γ(k, t, s) depends only on k, t, s and is never vanishing, Z_{1}^{J}(s, Φ_{3}) is the standard L-function of Φ_{3} (c.f. section 1.3) and

Γ_{∞}(s) = \frac{Γ(2k + s − 3) Γ(s + k − 3/2)^{3}}{Γ(2k + 2s − 3) Γ(s + k − 1/2)}

D_{1,2;3}^{∗}(s) = ∑_{r_{0}=0}^{2t−1} ∑_{{D∈ℕ \atop −D≡r^{2}_{0} mod 4t}} ∑_{{ε,ν=1 \atop (ε,ν)=1}}^{∞} \overline{c_{r_{0}}^{Φ_{3}}(D_{0})} \overline{c_{r_{0}ε}^{Φ_{1}}(D_{0}ε^{2})} \overline{c_{r_{0}ν}^{Φ_{2}}(D_{0}ν^{2})}

(εν)^{−(2s+2k−3)}D_{0}^{−(s+2k−3)}.

Now the meromorphic continuation of the Dirichlet D_{1,2;3}^{∗}(s) series on the whole complex plane follows from [Ar94] and [He97]. We also would like to obtain a functional equation of this Dirichlet series, but this would require more knowledge about the analytic properties of real analytic Jacobi Eisenstein series, which is of course of interest in its own right. However, in §2 of this paper, we develop a strategy to obtain the functional equation of real analytic Jacobi Eisenstein series E_{n}^{k,S}((τ, z); s) of general degree n and of index S (see Definition 1.4 again). In this connection our goal will be to prove the following conjecture (Conjecture 2.4) which states the explicit description of the functional equation satisfied by those Eisenstein series.

Conjecture [Main Conjecture] Under a certain maximality condition on S (see (2.9)) the Eisenstein series E_{n}^{k,S}((τ, z); s) will satisfy the functional equation

E_{n}^{k,S}((τ, z); s − κ) = Φ^{(n)}(s)E_{n}^{k,S} ((τ, z); \frac{n + 1}{2} − s − κ) ,

where κ = (k − l/2)/2 and where Φ^{(n)}(s) is a meromorphic function of s which is constructed from the ’constant’ term of the Eisenstein series. Moreover Φ^{(n)}(s) will satisfy the functional equation

Φ^{(n)}(s)Φ^{(n)} (\frac{n + 1}{2} − s) = 1.

In the case of l = 1 and S = 1 the explicit form of Φ^{(n)}(s) has been almost determined in Remark 2.7.

Acknowledgement The second author would like to thank G. Faltings for his support and the Max-Planck Institut für Mathematik in Bonn for hospitality.

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) and S_{n}(R) = {M ∈ M_{n}(R) | M = M^{t}}. 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}, A_{n}^{sp}, and A_{n}^{+} denote the set of all semi-integral symmetric matrices, the set of all semi-integral positive semi-definite symmetric matrices, and the set of all semi-integral positive definite symmetric matrices of size n, respectively. The disjoint union of sets is denoted by ⨆. 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 let δ(Z) = det (Im (Z)) and S_{n}(ℝ)^{+} be the set of all positive definite real symmetric matrices of size n.

1 Background on Jacobi forms

In this section we first fix notation regarding Jacobi forms, and recall some important facts on analytic Eisenstein series. Then a Jacobi-version of Garrett-Böcherer’s Theorem is given.

1.1 The space of Jacobi forms

It is well known that the symplectic group Sp_{n}(ℝ) acts on Siegel’s upper-half-space IH_{n} of degree n as a group of biholomorphic automorphism by

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

where M = \binom{A B}{C D} ∈ Sp_{n}(ℝ) with A, B, C, D ∈ ℝ^{n,n}. Let us put J(M, Z) = CZ + D and j(M, Z) = det(CZ + D). More general, let (n_{j})_{j=1}^{q} be a partition of n and M_{j} = \binom{a_{j} b_{j}}{c_{j} d_{j}} ∈ Sp_{n_{j}}(ℝ), then we denote by M_{1} × … × M_{q} the matrix M = \binom{A B}{C D} ∈ Sp_{n}(ℝ) with A = [a_{1}, …, a_{q}], B = [b_{1}, …, b_{q}], C = [c_{1}, …, c_{q}], D = [d_{1}, …, d_{q}]. These matrices form a subgroup of Sp_{n}(ℝ), which acts on IH_{n_{1}} × … × IH_{n_{q}}, identified with [IH_{n_{1}} × … × IH_{n_{q}}], via

((M_{1} × … × M_{q}), (z_{1} × … × z_{q})) ⟼ [M_{1}(z_{1}), …, M_{q}(z_{q})] . (3)

Hence J(M, [z_{1}, …, z_{q}]) = [J(M_{1}, z_{1}), …, J(M_{q}, z_{q})]. Let l be a natural number, then we regard the Jacobi group G_{n,l}^{J}(ℝ) := P_{n+l,l}^{J}(ℝ) as a subgroup of Sp_{n+l}(ℝ). Let g ∈ G_{n,l}^{J}(ℝ), then

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

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), (4)

gives a convenient parameterization of g. If we regard H_{n,l}(ℝ) := ℝ^{l,n} × ℝ^{l,n} × ℝ^{l,l} and Sp_{n}(ℝ) as subgroups of G_{n,l}^{J}(ℝ) with respect to the embeddings

(λ, μ, ρ) ↦ ((λ, μ, ρ), 1_{2n}) and M ↦ ((0, 0, 0), M), (5)

then the Jacobi group can be interpreted as the semi-direct product of the normal subgroup H_{n,l}(ℝ), the Heisenberg group, and Sp_{n}(ℝ). The operation of Sp_{n+l}(ℝ) on IH_{n+l} leads to an operation of G_{n,l}^{J}(ℝ) on D_{n,l} = IH_{n} × ℂ^{n,l}, given by

g(τ, z) = (M(τ), zJ(M, τ)^{−1} + λM(τ) + μ) , (6) where g = ((λ, μ, ρ), M) and (τ, z) ∈ D_{n,l}. For each r ∈ ℕ_{0} with r ≤ n, we define a subgroup of G_{n,l}^{J}(ℝ) given by

(7) G_{n,l,r}^{J}(ℝ) = {((0λ_{2}, μ, ρ), M) | λ_{2} ∈ ℝ^{l,r}, μ ∈ ℝ^{l,n}, ρ ∈ ℝ^{l,l} and M ∈ P_{n,r}(ℝ)}

and the map

∗_{J,r} : \{ \begin{matrix}D_{n,l} & ⟶ & D_{r,l} \\ (τ, z) & ⟼ & (τ [\begin{matrix}0 \\ 1_{r} \end{matrix}] , z (\begin{matrix}0 \\ 1_{r} \end{matrix})) \end{matrix} , (8)

which is a generalization of the projection ∗ : IH_{n} → IH_{r}, τ ↦ τ_{∗} = τ[{0 \atop 1_{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 G_{n,l}^{J}(ℝ) is the map

j_{k,S} : \{ \begin{matrix}G_{n,l}^{J}(ℝ) × D_{n,l} & ⟶ & D_{n,l} \\ (g, (τ, z)) & ↦ & j_{k,S}(g, (τ, z)) \end{matrix} , (9)

where

(10) j_{k,S}(g, (τ, z)) = j(M, τ)^{k}e{−Sρ}
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 |_{k,S}) (τ, z) := j_{k,S}(g, (τ, z))^{−1}F (g(τ, z)). (11)

It follows from our construction that Γ_{n,l,r}^{J} := G_{n,l,r}(ℤ) and H_{n,l} := H_{n,l}(ℤ) are groups. For convenience we put Γ_{n,l}^{J} = Γ_{n,l,0}^{J}.

Definition 1.2 Let k, l ∈ ℕ and S ∈ A_{l}^{+}. We denote by J_{k,S}^{n} the ℂ−vector space consisting of holomorphic functions F (τ, z) on D_{n,l} satisfying:

(i) (F |_{k,S} g) = F for all g ∈ Γ_{n,l}^{J},

(ii) F has a Fourier expansion of the form

F (τ, z) = ∑_{(\begin{matrix}N & R^{t}/2 \\ R/2 & S \end{matrix})∈A_{n+l}} c(N, R) e\{Nτ + R^{t}z\}. (12)

Each element of J_{k,S}^{n} is called Jacobi form of weight k and index S with respect to the Jacobi group Γ_{n,l}^{J}. Let (τ, z) ∈ D_{n,l} and Im(τ, z) = (τ_{y}, z_{y}), further we put

C_{k,S}(τ, z) = δ(τ)^{k} e{2iS[z_{y}]τ_{y}^{−1}}, (13)

then F ∈ J_{k,S}^{n} is said to be a cusp form, if it satisfies the following condition:

(14) C_{k,S}(τ, z)^{1/2}|F (τ, z)| is bounded on D_{n,l}.

Denote by J_{k,S}^{n,cusp}the subspace of J_{k,S}^{n}, consisting of all cusp forms. Supplementary we put J_{k,S}^{0,cusp} = ℂ.

Remark 1.3 Let F be a cusp form, then c(N, R) = 0 if T ∉ A_{n+l}^{+}.

1.2 Jacobi Eisenstein series and Petersson scalar products

Let us begin with the definition of an analytic Jacobi Eisenstein series.

Definition 1.4 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:

(15) E_{n,r}^{k,S}((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} ,

here γ = (h, M) and s ∈ ℂ with Re(s) sufficiently large. If r = 0, we denote by E_{n}^{k,S}((τ, z); s) = E_{n,0}^{k,S}((τ, z), 1; s) the Siegel-Jacobi Eisenstein series.

Remark 1.5

a) The Eisenstein series is absolutely convergent for k + 2Re(s) > n + r + l + 1. If k > n + r + l + 1, then E_{n,r}^{k,S}(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}, (16)

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)) (17)

and

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

c) The series E_{n,r}^{k,S}((τ, z), Φ); s) is absolutely convergent. Moreover, from [He97] (Theorem 4.1) we know that E_{n}^{k,S}((τ, z), s) admits a meromorphic continuation on the whole s−space. From there we have a precise knowledge about the location of possible poles. In the case n = l = 1 also a functional equation is available [Ar90].

Let (τ, z)∈ D_{n,l}. Then we denote by dω_{n,l}(τ, z) the Γ_{n,l}^{J}- invariant measure on D_{n,l} (for an explicit definition we refer to [Ar94](2.2)). For continues functions F, G : D_{n,l} → ℂ satisfying item (i) in Definition 1.2, we define the Petersson scalar product ⟨F , G⟩_{J} by

⟨F , G⟩_{J} = ∫_{Γ_{n,l}^{J}\D_{n,l}} F (τ, z)\overline{G(τ, z)} C_{k,S}(τ, z) dω_{n,l}(τ, z) , (18)

if the integral on the right side exist. This is for example the case if F G is a cusp form.

1.3 The Theorem of Garrett and Böcherer for Jacobi forms

As already mentioned, we can consider Sp_{n}(ℝ) as a subgroup of G_{n,l}^{J}(ℝ). Then it is easy to see, that every Γ_{n,l}^{J}-double coset attached to α ∈ Sp_{n}(ℚ) decomposes into finitely many Γ_{n,l}^{J}- left cosets:

Γ_{n,l}^{J} α Γ_{n,l}^{J} = ⊔ Γ_{n,l}^{J} g_{j}. (19)

Hence for Φ on D_{n,l} satisfying the condition Φ|_{k,S} g = Φ for all g ∈ Γ_{n,l}^{J} and α ∈ Sp_{n}(ℚ), we can define the Hecke-Jacobi operators

Φ|[α]_{J} := ∑_{j=1}^{h} Φ|_{k,S} g_{j}. (20)

A Jacobi form Φ which is a common eigen form under all the Hecke-Jacobi operators [α]_{J} is simply called a weak Hecke-Jacobi eigen form. The eigen values are denoted by λ(Φ, α). We know from the elementary divisor theorem that Γ_{n}αΓ_{n} = Γ_{n}[D, D^{−1}]Γ_{n}, where D = [d_{1}, …, d_{n}], d_{j} ∈ ℕ and d_{j}|d_{j+1}. Thus ‘λ(Φ, D) = λ(Φ, α)’ by abuse of notation. Let D_{n} be the set of all elementary matrixes and D_{n}^{(p)} the subset, where the elements have to form [p^{e_{1}}, …, p^{e_{n}}]. From Arakawa [Ar94] we know that

Z_{n}^{J}(s, Φ) := ∑_{D∈D_{n}(ℤ)}λ(Φ, D) det(D)^{−s} (21)

is absolutely convergent for Re(s) > 2n + l + 1 and has the Euler product expansion

Z_{n}^{J}(s, Φ) = ∏_{p} Z_{n,p}^{J}(s, Φ), (22)

where

Z_{n,p}^{J}(s, Φ) = ∑_{D∈D_{n}^{(p)}(ℤ)}λ(Φ, D) det(D)^{−s}. (23)

To state the Theorem of Garrett and Böcherer in the frame work of Jacobi forms ([Ar94], Theorem 2.9) we still need some more notation, in peculiar we need a generalization of the operation given in (3). Let (n_{j})_{j=1}^{q} be a partition of n and g_{j} = ((λ_{j}, μ_{j}, ρ_{j}), M) ∈ G_{n_{j},l}^{J}(ℝ). Then we denote by g_{1} × … g_{q} the element

g = ( ((λ_{1} … λ_{q}), (μ_{1} … μ_{q}), ∑ ρ_{j}) , M) ∈ G_{n,l}^{J}(ℝ). (24)

All elements of this type form a subgroup, which operates on D_{n_{1}} × … D_{n_{q}} (identified with (IH_{n_{1}} × … IH_{n_{q}}) × (ℂ^{l,n_{1}} … ℂ^{l,n_{q}}) ⊆ D_{n,l}), by

((g_{j})_{j}, (τ_{j}, z_{j})) ⟼ (g_{j}(τ_{j}, z_{j}))_{j}.

Theorem 1.6 (Version of [Ar94], Theorem 2.9) Let m ≥ n and k be an even natural number with k > m + n + l + 1. Let Φ ∈ J_{k,S}^{n,cusp} and k + 2Re(s) > m + n + l + 1, then for any (τ, z)∈ D_{m,l}:

⟨E_{m+n}^{k,S} ((τ, z) × (τ′, z′), s) , Φ(τ′, z′)⟩_{J} (25)

(26)  = μ(k, S, n, s) Z_{n}^{J}(2s + k, Φ) E_{m,n}^{k,S}(Φ^{♯}, (τ, z), s).

In particular if m = n, then the right side of this equation is equal to

μ(k, S, n, s)Z_{n}^{J}(2s + k, Φ) Φ^{♯}(τ, z).

Here Φ^{♯}(τ, z) = ∑ \overline{c(N, R)} e\{Nτ + R^{t}z\} and μ is a constant depending only on k, S, n and s ([Ar94] (2.4)).

For simplification we put J_{k,S}^{cusp} = J _{k,S}^{1,cusp}, IH = IH_{1}, D = D_{1,1}, Γ = Γ_{1} and Γ_{∞} = P_{1,0}. If n = l = 1, then Z_{1}^{J}(s, Φ) is essentially the Mellin transform of a certain elliptic cusp form. More precisely: Let f be a elliptic cusp form of weight k (i.e f : IH → ℂ holomorphic satisfying f(γ(τ)) = j(γ, τ)^{k} f(τ) for all γ ∈ Γ and vanishing at the cusps). We denote by S_{k} the space of such forms. To a normalized Hecke eigen form f(τ) = ∑_{n=1}^{∞} a(n) e{nτ} ∈ S_{k} one assoziates the L-function L_{f}(s) given by

L_{f}(s) = ∏_{p} L_{f,p}(p^{−s})^{−1} for Re(s) > \frac{k+1}{2}, (27)

where L_{f,p}(X) = 1 − a(p)X + p^{k−1}X^{2}.

Proposition 1.7 Let k be even and Φ ∈ J_{k,S}^{cusp} be a (weak) Hecke-Jacobi eigen form and φ the unique elliptic cusp form of weight 2k − 2 corresponding to Φ by the Shimura correspondance. Then

Z_{1}^{J}(s, Φ) = \frac{L_{φ}(s + k − 3)}{ζ(2s − 2)}. (28)

Proof:

[Ar94] (section 6.1).

In this paper we are mainly interested in a n = 2 and m = l = 1 version of Theorem 1.6.

2 The ’constant’ term of Jacobi Eisenstein series

In this subsection we compute the ’constant’ term of the Eisenstein series E_{n}^{k,S}((τ, z); s) and present a conjecture on the explicit form of the functional equation satisfied by them. We assume that k is an even positive integer throughout this section. It is also convenient to use e(t) = e^{2πi t} for t ∈ ℂ.

2.1 Fourier expansion of Jacobi Eisenstein series

For each μ ∈ ℤ^{l,n}, we define the theta series θ_{μ}(τ, z) to be the sum

∑_{λ∈ℤ^{l,n}} e (tr(S[λ + (2S)^{−1}μ]τ + 2^{t}(λ + (2S)^{−1}μ)Sz)) , ((τ, z) ∈ D_{n,l}).

Then, θ_{μ}(τ, z) depends only on μ mod (2S)ℤ^{l,n}.

We recall the definition of the real analytic Eisenstein series:

E_{n}^{k,S}((τ, z); s) := ∑_{γ∈Γ_{n,l,0}^{J}\Γ_{n,l}^{J}} j_{k,S}(γ, (τ, z))^{−1}δ(M(τ))^{s}.

For each ν (0 ≤ ν ≤ n), the sub series E_{n}^{k,S}((τ, z); s)^{(ν)} of E_{n}^{k,S}((τ, z); s) is given by

E_{n}^{k,S}((τ, z); s)^{(ν)} = ∑_{{γ∈Γ{J \atop n,l,0}\Γ{J \atop n,l} \atop rank c(M)=ν}} j_{k,S}(γ, (τ, z))^{−1}δ(M(τ))^{s},

where M is a Sp_{n}-part of γ and c(M) is a left-lower block of M. In particular if ν = 0, then

E_{n}^{k,S}((τ, z); s)^{(0)} = δ(τ)^{s}θ_{0}(τ, z).

Set Δ_{n} = GL_{n}(ℤ) and for each ν (0 ≤ ν ≤ n) denote by Δ_{ν,n−ν} the subgroup of Δ_{n} consisting of matrices whose right upper ν × n − ν blocks vanish:

Δ_{ν,n−ν} = \{\binom{∗ 0_{ν,n−ν}}{∗} ∈ Δ_{n}\} .

Set Γ_{n}^{∗} = {M ∈ Γ_{n} | c(M) ≠ 0} and P_{n,0}^{J} = P_{n,0}^{J}(ℤ) (see for P_{n,0}^{J}(R) the Notation).

For each T = ( \begin{matrix}N & r^{t}/2 \\ r/2 & S \end{matrix} ) ∈ A_{n+l}, let ζ_{n}(T, s) be a singular series defined by

ζ_{n}(T, s) = ∑_{M∈P_{n,0}\Γ_{n}^{∗}/P_{n,0}^{J}} ∑_{λ∈ℤ^{l,n}/ℤ^{l,n}c^{t}} | det c|^{−s} × e(tr(S[λ]ac^{−1} + Nc^{−1}d − r^{t}λ(c^{−1})^{t})),

where we write M = ( \begin{matrix}a & b \\ c & d \end{matrix} ). We refer to [Ar93] on this singular series.

Let ξ_{n}(η, u; α, β) denote Shimura’s confluent hypergeometric functions ([Sh82]):

ξ_{n}(η, u; α, β) = ∫_{S_{n}(ℝ)} e(−tr(ux)) det(x + iη)^{−α} det(x − iη)^{−β}dx

for u ∈ S_{n}(ℝ), η ∈ S_{n}(ℝ)^{+}, α, β ∈ ℂ. We use the ordinary one for the branch of det(Z)^{−α} (Z ∈ IH_{n}). This integral converges absolutely for Re(α + β) > n.

Proposition 2.1 Let 1 ≤ ν ≤ n. We have

E_{n}^{k,S}((τ, z); s)^{(ν)} = i^{−νl/2}(det 2S)^{−ν/2}δ(τ)^{s} × ∑_{r∈ℤ^{l,ν}/(2S)ℤ^{l,ν}} ∑_{V ∈Δ_{ν,n−ν}\Δ_{n}}

( ∑_{N∈A_{ν}} ζ_{ν}(T, 2s + k)ξ_{ν} (η̂_{1}, T̃, s + k − \frac{l}{2}, s) e(tr(T̃ξ̂_{1}))) × θ_{(r,0)V} (τ, z)

where for each r, N, we put T = ( \begin{matrix}N & r^{t}/2 \\ r/2 & S \end{matrix} ), T̃ = N − \frac{1}{4}r^{t}S^{−1}r, and for each V,

τ̂_{1} = (V τV^{t}) [ \begin{matrix}1_{ν} \\ 0 \end{matrix} ], η̂_{1} = Im(τ̂_{1}), ξ̂_{1} = Re(τ̂_{1}).

Proof: The formula in the proposition corresponds to that of [Ma71], p.306 in the Siegel Eisenstein series case (see also [Mi93]).

First we discuss the case of ν = n. It is easily proved in a manner similar to that of Proposition 4.2 in [Ar93] that

(2.1)  E_{n}^{k,S}((τ, z); s)^{(n)} = ∑_{M∈P_{n,0}\Γ_{n}^{∗}/P_{n,0}^{J}} ∑_{λ∈ℤ^{l,n}/ℤ^{l,n}c^{t}}

(det c)^{−k}| det c|^{−2s} × e(tr(S[λ]ac^{−1})) × F_{k,S,n}(τ + c^{−1}d, z − λ(c^{−1})^{t}; s),

where we set

F_{k,S,n}(τ, z; s) = δ(τ)^{s} ∑_{p∈S_{n}(ℤ)} ∑_{q∈ℤ^{l,n}} det(τ+p)^{−k}| det(τ+p)|^{−2s}×e(−tr(S[z+q](τ+p)^{−1}).

This infinite series is absolutely convergent for k + 2Re(s) > n + l + 1. The Poisson summation formula yields the following identity for F_{k,S,n}(τ, z; s) which will coincide with that of Proposition 4.1, (i) in [Ar93] if s is evaluated at s = 0. If k +2Re(s) > n+l +1, then

(2.2) F_{k,S,n}(τ, z; s) = i^{−nl/2}(det 2S)^{−n/2}δ(τ)^{s} ∑_{x∈A_{n}} ∑_{q∈ℤ^{l,n}} e (tr ((x − \frac{1}{4}S^{−1}[q]) ξ))

×ξ_{n} (η, x − \frac{1}{4}S^{−1}[q], s + k − \frac{l}{2}, s) e (tr (\frac{1}{4}S^{−1}[q]τ + q^{t}z)) ,

where η = Im(τ), ξ = Re(τ). It follows from (2.1), (2.2) that

E_{n}^{k,S}((τ, z); s)^{(n)} = i^{−nl/2}(det 2S)^{−n/2}δ(τ)^{s} ∑_{N∈A_{n}} ∑_{r∈ℤ^{l,n}} ζ_{n}(T; 2s + k)

×ξ_{n} (η, T̃, s + k − \frac{l}{2}, s) e (tr (T̃ξ)) e (tr (\frac{1}{4}S^{−1}[r]τ + r^{t}z)) ,

where for each r, N, we put T = ( \begin{matrix}N & ^{t}r/2 \\ r/2 & S \end{matrix} ) and T̃ = N − \frac{1}{4}S^{−1}[r]. Further

replacing r with r + (2S)λ (r ∈ ℤ^{l,n}/(2S)ℤ^{l,n}, λ ∈ ℤ^{l,n}), we have

(2.3)  E_{n}^{k,S}((τ, z); s)^{(n)} = i^{−nl/2}(det 2S)^{−n/2}δ(τ)^{s}( ∑_{N∈A_{n}} ∑_{r∈ℤ^{l,n}/(2S)ℤ^{l,n}}

ζ_{n}(T; 2s + k)ξ_{n} (η, T̃, s + k − \frac{l}{2}, s) e (tr(T̃ξ))) · θ_{r}(τ, z),

which completes the proof in the case of ν = n.

Next let 1 ≤ ν < n. In a manner similar to the proof in Proposition 4.3 of [Ar93] we infer that

E_{n}^{k,S}((τ, z); s)^{(ν)} = δ(τ)^{s} ∑_{V, λ_{2}} E_{ν}^{k,S}((τ̂_{1}, ẑ_{1}+λ_{2}τ̂_{21}); s)^{(ν)}δ(τ̂_{1})^{−s}e(tr(S[λ_{2}]τ̂_{2}+2λ_{2}^{t}Sẑ_{2})),

where V (resp. λ_{2}) runs over a complete set of representatives of Δ_{ν,n−ν}\Δ_{n} (resp. all elements of ℤ^{l,n−ν}), and where we put, for each V ∈ Δ_{n},

(τ̂, ẑ) = (V τV^{t}, zV^{t})

and write

τ̂ = ( \begin{matrix}τ̂_{1} & ^{t}τ̂_{21} \\ τ̂_{21} & τ̂_{2} \end{matrix} ) , ẑ = (ẑ_{1}, ẑ_{2}) with (τ̂_{1} ∈ IH_{ν}, ẑ_{1} ∈ ℂ^{l,ν}).

By using the result in the case of ν = n, the right hand side of the above identity equals

i^{−νl/2}(det 2S)^{−ν/2}δ(τ)^{s} ∑_{V, λ_{2}} ∑_{r_{1}∈ℤ^{l,ν}/(2S)ℤ^{l,ν}} ∑_{N_{1}∈A_{ν}} ζ_{ν}(T_{1}; 2s + k)

×ξ_{ν} (η̃_{1}, T̃_{1}, s + k − \frac{l}{2}, s) e (tr(T̃_{1}ξ)) · θ_{r_{1}}(τ̂_{1}, ẑ_{1} + λ_{2}τ̂_{21}) · e(tr(S[λ_{2}]τ̂_{2} + 2λ_{2}^{t}Sẑ_{2})),

where T_{1} = ( \begin{matrix}N_{1} & ^{t}r_{1}/2 \\ r_{1}/2 & S \end{matrix} ). Here we note that

∑_{λ_{2}} θ_{r_{1}}(τ̂_{1}, ẑ_{1} + λ_{2}τ̂_{21}) · e(tr(S[λ_{2}]τ̂_{2} + 2λ_{2}^{t}Sẑ_{2}))

= θ_{(r_{1},0)}(τ̂, ẑ)

= θ_{(r_{1},0)V} (τ, z).

Therefore we obtain the identity for 1 ≤ ν < n in the introduction.

2.2 The ’constant’ term

In the above Fourier expansion in Proposition 2.1 the sub series corresponding to those T with T̃ = 0 may be called the ’constant’ term of E_{n}^{k,S}((τ, z); s)^{(ν)}. We write c^{(ν)}((τ, z); s) for this ’constant’ term. Immediately by Proposition 2.1, we have

c^{(ν)}((τ, z); s) = c(ν) δ(τ)^{s}  ∑_{{r∈ℤ^{l,ν} /(2S)ℤ^{l,ν} \atop \frac{1}{4} r^{t}S^{−1}r≡0 modAν}} ∑_{V ∈Δ_{ν,n−ν}^{∞}\Δ_{n}} ζ_{ν}(T, 2s + k) ×

ξ_{ν} (η̂_{1}, 0, s + k − \frac{l}{2}, s) × θ_{(r,0)V} (τ, z),

where c(ν) = i^{− \frac{νl}{2}} (det 2S)^{− \frac{ν}{2}} and T = ( \begin{matrix}\frac{1}{4}r^{t}S^{−1}r & r^{t}/2 \\ r/2 & S \end{matrix} ) for each r. Note that up

to the factors θ_{(r,0)V} (τ, z), the ’constant’ term c^{(ν)}((τ, z); s) is a function of η = Im(τ). In particular if ν = 0,

c^{(0)}((τ, z); s) = δ(τ)^{s}θ_{0}(τ, z).

Set

c_{n}^{k,S}((τ, z), s) = ∑_{ν=0}^{n} c^{(ν)}((τ, z); s),

which is the ’constant’ term of E_{n}^{k,S}((τ, z); s).

For each prime integer p let Γ_{n}^{∗(p)} denote the subset of Γ_{n}^{∗} consisting of all elements M ∈ Γ_{n}^{∗} such that det c(M) = ±p^{m} with some non-negative integers m. We recall from [Ar93] that

ζ_{n}(T, s) = ∑_{M∈P_{n,0}\Γ_{n}^{∗}/P_{n,0}^{J}} ∑_{λ∈ℤ^{l,n}/ℤ^{l,n} c} | det c|^{−s} × e (tr (T [ \begin{matrix}1_{n} \\ λ \end{matrix} ] c^{−1}d))

= ∏_{p<∞} ζ_{n,p}(T; s),

where

ζ_{n,p}(T; s) = ∑_{M∈P_{n,0}\Γ_{n}^{∗(p)}/P_{n,0}^{J}} ∑_{λ∈ℤ^{l,n}/ℤ^{l,n} c} | det c|^{−s}e(tr (T [ \begin{matrix}1_{n} \\ λ \end{matrix} ] c^{−1}d)).

Let | |_{p} denote the normalized p-adic valuation of ℚ_{p}. For each a ∈ ℚ_{p}^{×} denote by ν_{p}(a) the integer ν by which a can be written in the form a = p^{ν}a′, a′ ∈ ℤ_{p}^{×}.

Obviously, |a|_{p} = p^{−ν_{p}(a)}. For each x ∈ S_{n}(ℚ_{p}) choose M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp_{n}(ℤ_{p})

with x = c^{−1}d. Set

ν(x) = | det c|_{p}^{−1} = p^{ν_{p}(det c)},

which is independent of the choice of M ∈ Sp_{n}(ℤ_{p}) satisfying x = c^{−1}d. We define an additive function e_{p} on ℚ_{p} by putting

e_{p}(y) = e(y′)

with y′ ∈ ℚ being chosen so that y − y′ ∈ ℤ_{p}.

Proposition 2.2 Let p > 2 be a prime not dividing det(2S) and T = ( \begin{matrix}N & ^{t}r/2 \\ r/2 & S \end{matrix} ) ∈

A_{n+l} with T̃ = N − \frac{1}{4}^{t}rS^{−1}r = 0.

(i) If l is even, then

ζ_{n,p}(T; s) = \frac{1 − εp^{\frac{l}{2} −s}}{1 − εp^{n+ \frac{l}{2} −s}} ∏_{i=1}^{n} \frac{1 − p^{2i+l−2s}}{1 − p^{n+i+l−2s}} ,

where ε = (\frac{(−1)^{l/2} det(2S)}{p}).

(ii) If l is odd, then

ζ_{n,p}(T; s) = ∏_{i=1}^{n} \frac{1 − p^{2i−1+l−2s}}{1 − p^{n+i+l−2s}} .

Proof: The proof of Proposition 2.2 is based on some results of Shimura [Sh83]. By a little computation, one gets

e (tr (T [ \begin{matrix}1_{n} \\ λ \end{matrix} ] c^{−1}d)) = e (tr (S[λ + (2S)^{−1}r]c^{−1}d) + tr (T̃c^{−1}d)) .

Note that | det c | is a power of p. Since the map λ ⟼ λ + (2S)^{−1}r induces an automorphism of the quotient module ℤ_{p}^{l,n}/ℤ_{p}^{l,n} c, one has

∑_{λ∈ℤ_{p}^{l,n}/ℤ_{p}^{l,n}c} e_{p} (tr (T [ \begin{matrix}1_{n} \\ λ \end{matrix} ] c^{−1}d)) = ∑_{λ∈ℤ_{p}^{l,n}/ℤ_{p}^{l,n}c} e_{p} (tr(S[λ]c^{−1}d)) · e_{p}(tr(T̃c^{−1}d)).

We choose a positive integer m such that p^{m}ℤ_{p}^{l,n} ⊂ ℤ_{p}^{l,n}c. Then we have, for the group indices,

[ℤ_{p}^{l,n}c : p^{m}ℤ_{p}^{l,n}] = \frac{[ℤ_{p}^{l,n} : p^{m}ℤ_{p}^{l,n}]}{[ℤ_{p}^{l,n} : ℤ_{p}^{l,n}c]} = \frac{p^{mln}}{| det c|^{l}} .

Set x = c^{−1}d. Consequently,

∑_{λ∈ℤ_{p}^{l,n}/ℤ_{p}^{l,n}c} e_{p} (tr(S[λ]x)) = \frac{1}{[ℤ_{p}^{l,n}c : p^{m}ℤ_{p}^{l,n}]} ∑_{λ∈ℤ_{p}^{l,n}/p^{m}ℤ_{p}^{l,n}} e_{p} (tr(S[λ]x))

= \frac{| det c|^{l}}{p^{mln}} · p^{mln}ν(x)^{−l/2} · χ_{0,p}(x)
= | det c|^{l/2} · χ_{0,p}(x),

where χ_{0,p}(x) is given as follows:

∑_{λ∈ℤ_{p}^{l,n}/p^{m}ℤ_{p}^{l,n}} e_{p}((tr(S[λ]x)) = ν(x)^{−l/2}p^{mln}χ_{0,p}(x).

It is easy to see that χ_{0,p}(x) depends only on x and is independent of the choice of sufficiently large m. Viewing the above arguments, we have, if p > 2 does not divide det(2S),

ζ_{n,p}(T; s) = ∑_{M∈P_{n,0}\Γ_{n}^{∗(p)}/P_{n,0}^{J}} | det c|^{−(s−l/2)}χ_{0,p}(x)e(tr(T̃x))

= ∑_{x∈S_{n}(ℚ_{p})/S_{n}(ℤ_{p})} ν(x)^{−(s−l/2)}χ_{0,p}(x)e(tr(T̃x)).

If T̃ = 0, then

(2.4)  ζ_{n,p}(T; s) = S_{n,l}(p^{−s}),

where we put

S_{n,l}(t) = ∑_{x∈S_{n}(ℚ_{p})/S_{n}(ℤ_{p})} χ_{0,p}(x)(p^{l/2}t)^{ν_{p}(ν(x))}.

Following Shimura [Sh83], Lemma 4.4, we define a function ω on S_{n}(ℚ_{p}) by putting

∑_{μ∈ℤ_{p}^{n,1}/p^{m}ℤ_{p}^{n,1}} e_{p}((tr(λ^{t}yλ)) = ν(y)^{−1/2}p^{mn}ω(y) (y ∈ S_{n}(ℚ_{p}))

for sufficiently large positive integer m. Then, ω(y) is independent of m. Shimura [Sh83] moreover defined the following Siegel singular series:

α_{k}(y, t) = ∑_{x∈S_{n}(ℚ_{p})/S_{n}(ℤ_{p})} ω(x)^{k}e_{p}(tr(xy))t^{ν_{p}(ν(x))},

where k = 0 or 1, and y ∈ S_{n}(ℚ_{p}) ([Sh83], (4.8a).

Under the assumption that p > 2 does not divide det(2S), it is not difficult to see that

χ_{0,p}(x) = ω(x)^{l} (\frac{(−1)^{l} det(S)}{p})^{ν_{p}(ν(x))} ,

where (\frac{a}{p}) denotes the Legendre quadratic residue symbol. If l is even, then by the

identity ω(x)^{2} = (\frac{−1}{p})^{ν_{p}(ν(x))} in Lemma 4.4 of [Sh83],

χ_{0,p}(x) = (\frac{(−1)^{l/2} det(2S)}{p})^{ν_{p}(ν(x))} .

In this case

S_{n,l}(t) = ∑_{x∈S_{n}(ℚ_{p})/S_{n}(ℤ_{p})} (εp^{l/2}t)^{ν_{p}(ν(x))},

where ε = (\frac{(−1)^{l/2} det(2S)}{p}). Similarly if l is odd, then

χ_{0,p}(x) = ω(x) (\frac{(−1)^{(l+1)/2} det(S)}{p})^{ν_{p}(ν(x))} .

Then,

S_{n,l}(t) = ∑_{x∈S_{n}(ℚ_{p})/S_{n}(ℤ_{p})} ω(x)(ε′p^{l/2}t)^{ν_{p}(ν(x))},

where ε′ = (\frac{(−1)^{(l+1)/2} det(S)}{p}). Summing up with the use of α_{k}(y, t), we conclude that, if l is even,

(2.5)  S_{n,l}(t) = α_{0}(0, εp^{l/2}t)

and that, if l is odd,

(2.6)  S_{n,l}(t) = α_{1}(0, ε′p^{l/2}t).

The exact values of the Siegel singular series α_{k}(y, t) at y = 0 have been computed by Shimura [Sh83], Proposition 5.1. If l is even, then

(2.7)  α_{0}(0, t) = \frac{1 − t}{1 − p^{n}t} · ∏_{i=1}^{[n/2]} \frac{1 − p^{2i}t^{2}}{1 − p^{2n+1−2i}t^{2}} = \frac{1 − t}{1 − p^{n}t} · ∏_{i=1}^{n} \frac{1 − p^{2i}t^{2}}{1 − p^{n+i}t^{2}} .

If l is odd, then

(2.8)  α_{1}(0, t) = ∏_{i=1}^{[(n−1)/2]} \frac{1 − p^{2i+1}t^{2}}{1 − p^{2n−2i}t^{2}} = ∏_{i=1}^{n} \frac{1 − p^{2i−1}t^{2}}{1 − p^{n+i}t^{2}} .

Taking the identities from (2.4) to (2.8) into account of, we obtain the assertions (i), (ii) of the proposition.

To compute c^{(ν)}((τ, z); s) we need an explicit value of ξ_{m}(η, 0; α, β). Set

Γ_{m}(s) = π^{m(m−1)/4} ∏_{j=0}^{m−1} Γ (s − \frac{j}{2}) .

By Shimura [Sh82], p.275, (1.31), we get

ξ_{m}(η, 0; α, β) = i^{m(β−α)}2^{− \frac{m(m−1)}{2}} · (2π)^{\frac{m(m+1)}{2}} · \frac{Γ_{m} (α + β − \frac{m+1}{2})}{Γ_{m}(α)Γ_{m}(β)} · det(2η)^{\frac{m+1}{2} −α−β}.

Thus,

ξ_{ν} (η̂_{1}, 0, s + k − \frac{l}{2}, s) = i^{ν( \frac{l}{2} −k)}2^{ν}π^{\frac{ν(ν+1)}{2}} · \frac{Γ_{ν} (2s + k − \frac{l}{2} − \frac{ν+1}{2})}{Γ_{ν}(s + k − \frac{l}{2})Γ_{ν}(s)}

× det(2η̂_{1})^{\frac{l}{2} + \frac{ν+1}{2} −k−2s}.

Now we assume a maximality condition on S:

(2.9) If S[x] ∈ ℤ for x ∈ (2S)^{−1}M_{l,1}(ℤ), then necessarily, x ∈ M_{l,1}(ℤ).

This condition ensures the uniqueness of the Eisenstein series. Under this condition the ’constant’ term c^{(n)}((τ, z); s) is expressed as

c^{(n)}((τ, z); s) = i^{−nk}(det 2S)^{− \frac{n}{2}} · 2^{n}π^{\frac{n(n+1)}{2}} · 2^{n( \frac{l+n+1}{2} −k−2s)} ×

ζ_{n}(T_{0}, 2s + k) · \frac{Γ_{n} (2s + k − \frac{l+n+1}{2})}{Γ_{n}(s + k − \frac{l}{2})Γ_{n}(s)}(det η)^{\frac{l+n+1}{2} −k−s} · θ_{0}(τ, z),

T_{0} being ( \begin{matrix}0 & 0 \\ 0 & S \end{matrix} ) ∈ A_{n+l}.

Set κ = \frac{k−l/2}{2}. Then the above formula changes into the following Proposition 2.3 We have

c^{(n)}((τ, z); s − κ) = Φ^{(n)}(s)δ(τ)^{\frac{n+1}{2} −s−κ}θ_{0}(τ, z),

where we set

Φ^{(n)}(s) = i^{−nk} det(2S)^{− \frac{n}{2}} 2^{n}π^{\frac{n(n+1)}{2}} · 2^{n( \frac{n+1}{2} −2s)} \frac{Γ_{n} (2s − \frac{n+1}{2})}{Γ_{n}(s + κ)Γ_{n}(s − κ)} · ζ_{n} (T_{0}, 2s + \frac{l}{2}) .

2.3 Some conjectures on the functional equation

It has been proved in [Ar90], [Ar92] that, if n = 1, then the Eisenstein series E_{1}^{k,S}((τ, z); s) can be continued to a meromorphic function of s in the whole complex plane satisfying a certain functional equation. For instance if l = 1 and S is a square free positive integer, then the explicit functional equation is given by

E_{1}^{k,S}((τ, z); s − κ) = Φ(s)E_{1}^{k,S}((τ, z); 1 − s − κ),
Φ(s)Φ(1 − s) = 1

where κ = (k − 1/2)/2 and

(2.10)  Φ(s) = \frac{e^{−πik/2}}{\sqrt{2S}} · \frac{2^{2−2s}πΓ(2s − 1)}{Γ(s + κ)Γ(s − κ)} · \frac{ζ(4s − 2)}{ζ(4s − 1)} ∏_{p | S} \frac{1 + p^{3/2−2s}}{1 + p^{1/2−2s}}

(see [Ar90], Proposition 3.5). In this case the ’constant’ term c_{1}^{k,S}((τ, z); s − κ) of E_{1}^{k,S}((τ, z); s − κ) is given by

c_{1}^{k,S}((τ, z); s − κ) = (η^{s−κ} + η^{1−s−κ}Φ(s))θ_{0}(τ, z)

([Ar90], Proposition 3.3, (3.15)). We are able to know the exact functional equations of E_{1}^{k,S}((τ, z); s − κ) and Φ(s) from the explicit expression of the ’constant’ term.

Now we go gack to the case of degree n. Assume that S satisfies the condition (2.9). Set

κ = (k − l/2)/2.

We guess that among the ’constant’ terms c^{(ν)}((τ, z); s − κ) of E_{n}^{k,S}((τ, z); s − κ)^{(ν)} (0 ≤ ν ≤ n) the first term

c^{(0)}((τ, z); s − κ) = δ(τ)^{s−κ}θ_{0}(τ, z)

and the last term

c^{(n)}((τ, z); s − κ) = Φ^{(n)}(s)δ(τ)^{(n+1)/2−s−κ}θ_{0}(τ, z)

will be replaced with each other up to some constant factor. Namely we guess that

c^{(0)}((τ, z); s − κ) = Φ^{(n)}(s)c^{(n)}((τ, z); (n + 1)/2 − s − κ).

Our precise conjecture is as follows.

Conjecture 2.4 (Main Conjecture) Under the maximality condition (2.9) on S the function Φ^{(n)}(s) will satisfy the functional equation

Φ^{(n)}(s)Φ^{(n)} (\frac{n + 1}{2} − s) = 1.

Moreover the Eisenstein series E_{n}^{k,S}((τ, z); s) will satisfy the functional equation

E_{n}^{k,S}((τ, z); s − κ) = Φ^{(n)}(s)E_{n}^{k,S} ((τ, z); \frac{n + 1}{2} − s − κ) .

Remark 2.5 (i) If n = 1, Conjecture 2.4 holds true. The case of l = 1 has been discussed in [Ar90], while in [Ar92] the conjecture for n = 1 and arbitrary l has been proved.

(ii) If the general theory of Langlands for the Eisenstein series of semi-simple Lie groups is available to our Jacobi Eisenstein series, then we have only to determine the explicit form of Φ^{(n)}(s).

Finally we consider the case of l = 1 and S = 1 (the simplest but the most important case). In view of Proposition 2.2 we present a conjecture on ζ_{n,2}(T_{0}; s) which will lead us to the explicit computation of Φ^{(n)}(s).

Conjecture 2.6 Let l = 1 and S = 1. The assertion (ii) in Proposition 2.2 holds true also for p = 2. Namely, if T̃ = 0 (namely T = T_{0}), then

ζ_{n,2}(T; s) = ∏_{i=1}^{n} \frac{1 − 2^{2i−2s}}{1 − 2^{n+i+1−2s}} .

Therefore,

ζ_{n}(T; s) = ∏_{i=1}^{n} \frac{ζ(2s − n − 1 − i)}{ζ(2s − 2i)}.

Remark 2.7 (i) If n = 1, then Conjecture 2.6 can be easily shown from the definition of ζ_{1,2}(T_{0}, s) directly.

(ii) Let l = 1 and S = 1. Then under Conjecture 2.6 we have, for arbitrary n,

Φ^{(n)}(s) = i^{−nk}π^{\frac{n(n+1)}{2}} × 2^{n( \frac{n+2}{2} −2s)} \frac{Γ_{n} (2s − \frac{n+1}{2})}{Γ_{n}(s + κ)Γ_{n}(s − κ)} × ∏_{i=1}^{n} \frac{ζ(4s − n − 1 − i)}{ζ(4s + 1 − 2i)},

which is consistent with (2.10) if n = 1. From this expression we can prove directly the functional equation

Φ^{(n)}(s)Φ^{(n)} (\frac{n + 1}{2} − s) = 1.

3 A Pullback formula

In this section we prove a pullback formula of a Jacobi Eisenstein series of Klingen type of genus two. Let us assume l = 1. Let U_{⊥} = \{\binom{∗ ∗}{ε ν} ∈ P_{1,0}^{J}\ Γ | ε, ν > 0\}, Φ ∈ J_{k,t}^{cusp} and (τ_{1}, z_{1}), (τ_{2}, z_{2}) ∈ D. Then we put

ψ_{Φ}^{ε,ν}((τ_{1}, z_{1}), (τ_{2}, z_{2}); s) = Im (τ_{1}τ_{2}(ε^{2}τ_{1} + ν^{2}τ_{2})^{−1})^{s} Φ(ε^{2}τ_{1} + ν^{2}τ_{2}, εz_{1} + νz_{2}). (29)

Theorem 3.1 Let k, t ∈ ℕ and k even. If Φ ∈ J_{k,t}^{cusp} and k + 2Re(s) > 4, then

E_{2,1}^{k,t}(Φ, (τ_{1}, z_{1}) × (τ_{2}, z_{2})), s) (30)

(31) = E_{1}^{k,t}((τ_{1}, z_{1}); s) Φ(τ_{2}, z_{2}) + E_{1}^{k,t}((τ_{2}, z_{2}); s) Φ(τ_{1}, z_{1})

+ ∑_{γ_{1},γ_{2}∈Γ_{∞}\Γ} ∑_{\binom{ab}{εν}∈ U_{⊥}} ∑_{λ∈ℤ}

(ψ_{Φ}^{ε,ν}|_{k,t,1} ((λa, 0, 0), γ_{1}) |_{k,t,2}((λb, 0, 0), γ_{2})) ((τ_{1}, z_{1}), (τ_{2}, z_{2}); s).

Here |_{k,t,j} is the operator |_{k,t} corresponding to (τ_{j}, z_{j}).

Proof: First we have to choose an appropriate representative system for Γ_{2,1,1}^{J}\Γ_{2}^{J}. Let us take the set {(λ0, 0, 0), M ∈ Γ_{2}^{J} | λ ∈ ℤ and M ∈ P_{2,1}\Γ_{2}}, with the decomposition R = R_{1} ⊔ R_{2} ⊔ R_{3} of P_{2,1}\Γ_{2}, given by

(32) (Γ_{∞}\ Γ × 1_{2}) ⊔ π(1_{2} × Γ_{∞}\ Γ) ⊔ T (U_{⊥})(Γ_{∞}\ Γ × Γ_{∞}\ Γ).

Here T (U_{⊥}) = \{T_{⊥}(u) = \binom{u 0}{0 ∗} ∈ Γ_{2} | u ∈ U_{⊥}\} and

π = ( \begin{matrix}0 1 1 0 & 0 \\ 0 & 0 1 1 0 \end{matrix} ) .

The Theorem can be proven in the same style as Theorem 2.3 in [He96], with only minor changes. Thus we only give some useful formulas. Let g be an element of R_{3}:

g = ((λ0, 0, 0), T \binom{a b}{ε ν} (γ_{1} × γ_{2})).

Then we have:

Φ(g((τ_{1}, z_{1}) × (τ_{2}, z_{2}))_{∗_{J}} ) = Φ(ε^{2}γ_{1}(τ_{1}) + ν^{2}γ_{2}(τ_{2}), εz_{1}J(γ_{1}, τ_{1})^{−1}
+ νz_{2}J(γ_{2}, τ_{2})^{−1} + λ(aεγ_{1}(τ_{1}) + bνγ_{2}(τ_{2}))),

(\frac{δ (T \binom{a b}{ε ν}(γ_{1} × γ_{2})(τ_{1} × τ_{2}))}{δ (T \binom{a b}{ε ν}(γ_{1} × γ_{2})(τ_{1} × τ_{2}))_{∗}} )^{s} = (∏_{j=1}^{2} δ(γ_{j}(τ_{j}))^{s}) δ(ε^{2}γ_{1}(τ_{1}) + ν^{2}γ_{2}(τ_{2}))^{−s}, (33)

j_{k,t}(g, (τ_{1}, z_{1}) × (τ_{2}, z_{2}))^{−1} (34)

= j_{k,t}((λa, 0, 0)γ_{1}, (τ_{1}, z_{1}))^{−1}j_{k,t}((λb, 0, 0)γ_{2}, (τ_{2}, z_{2}))^{−1}.

The formula (31) in Theorem 3.1 and the results of Theorem 1.6 make it transparent, that pullbacks of Jacobi Eisenstein series and Eisenstein series attached to the Siegel modular group have similar properties. The main point is here, that this is true in the non-holomorphic case !.

4 Certain Dirichlet series attached to three Jacobi forms

In this section we calculate the iterated scalar product of a diagonal-restricted Jacobi Eisenstein series against three Jacobi cusp forms. The main ingredient is the pullback formula (31) developed in the last section.

Let us first simplify and specialize our notation a little more. Moreover we fix the weight k even and the index t ∈ A_{1}^{+} = ℕ of our forms. We denote by τ = τ_{x} + i τ_{y} and z = z_{x} + i z_{y} the decomposition of (τ, z) ∈ D in real- and imaginary part. Hence the G_{1,1}^{J}(ℝ)− invariant volume element on D has the form

dω_{1,1}(τ, z) = τ_{y}^{−3}dτ_{x} dτ_{y} dz_{x} dz_{y}

and the function C_{k,t} is equal to τ_{y}^{k} e{2i t z_{y}^{2}τ_{y}^{−1}}.

Let Φ_{1}, Φ_{2}, Φ_{3} ∈ J_{k,t}^{cusp} and s ∈ ℂ with Re(s) > 5. We are mainly interested in calculating the expression

(35) D_{1,2;3}(s) := ⟨⟨E_{2,1}^{k,t}(Φ_{3}^{♯}, ((τ_{1}, z_{1}) × (τ_{2}, z_{2})), s) , Φ_{1}(τ_{1}, z_{1}) Φ_{2}(τ_{2}, z_{2})⟩_{J}⟩_{J}.

A fundamental domain F of (Γ_{∞} × H_{1,1}) \ IH × ℂ is given by

(36) F = {(τ, z) ∈ D | 0 ≤ τ_{x} < 1; 0 < τ_{y} < ∞ and 0 ≤ z_{x} < 1; 0 ≤ z_{y} < τ_{y}}.

After some elementary calculations we get the

Proposition 4.1 Let Φ_{1}, Φ_{2}, Φ_{3} ∈ J_{k,t}^{cusp}. Then

D_{1,2;3}(s) = ∫_{0}^{1}dτ_{1,x} ∫_{0}^{1}dτ_{2,x} ∫_{0}^{∞} dτ_{1,y} ∫_{0}^{∞}dτ_{2,y} ∫_{0}^{1} dz_{1,x} ∫_{0}^{1}dz_{2,x}

∫_{0}^{τ_{1,y}} dz_{1,y} ∫_{0}^{τ_{2,y}}dz_{2,y} ∑_{{n, r \atop 4nt>r^{2}}} ∑_{{n_{1}, r_{1} \atop 4n_{1}t>r^{2}_{1}}} ∑_{{n_{2}, r_{2} \atop 4n_{2}t>r^{2}_{2}}} ∑_{λ∈ℤ} ∑_{\binom{ab}{εν}∈U_{⊥}}

\overline{c_{1}(n_{1}, r_{1})}\overline{c_{2}(n_{2}, r_{2})}\overline{c_{3}(n, r)} τ_{1,y}^{s+k−3}τ_{2,y}^{s+k−3}(ε^{2}τ_{1,y} + ν^{2}τ_{2,y})^{−s} × e{−n_{1}τ_{1,x} + in_{1}τ_{1,y} − r_{1}z_{1,x} + ir_{1}z_{1,y}} × e{−n_{2}τ_{2,x} + in_{1}τ_{2,y} − r_{1}z_{2,x} + ir_{2}z_{2,y}} × e{nε^{2}τ_{1,x} + inε^{2}τ_{1,y} + nν^{2}τ_{2,x} + inν^{2}τ_{2,y}} × e{rεz_{1,x} + irεz_{1,y} + rνz_{2,x} + iνrz_{2,y}

+rλaετ_{1,x} + irλaετ_{1,y} + rλbντ_{2,x} + irλbντ_{2,y}} × e{λ^{2}a^{2}tτ_{1,x} + iλ^{2}a^{2}tτ_{1,y} + 2λatz_{1,x} + 2iλatz_{1,y}} × e{λ^{2}b^{2}tτ_{2,x} + iλ^{2}b^{2}tτ_{2,y} + 2λbtz_{2,x} + 2iλbtz_{2,y} × e{2iπtz_{1,y}^{2}/τ_{1,y}} e{2iπtz_{2,y}^{2}/τ_{2,y}}.

Before we state the main theorem of this section we have to make several remarks concerning the notation and properties of Fourier coefficients of Jacobi cusp forms. Let

Φ(τ, z) = ∑_{( \begin{matrix}n & r/2 \\ r/2 & t \end{matrix} )∈A_{2}^{+}} c^{Φ}(n, r) e\{nτ + rz\} ∈ J_{k,t}^{cusp}. (37)

As a simple consequence of the invariance of Φ with respect to the Heisenberg group, one has

(38) c^{Φ}(n, r) = c^{Φ}(n + rλ + tλ^{2}, r + 2tλ) for all λ ∈ ℤ.

Hence c^{Φ}(n, r) = c^{Φ}(n′, r′), whenever r ≡ r′(mod 2t) and 4n t−r^{2} = 4n′ t−(r′)^{2}. Thus the Fourier coefficients depend only on the ‘discriminant’ D = 4n t − r^{2} and r (mod 2t). Moreover, because we have choosen k even, c^{Φ}(n, r) = c^{Φ}(n, −r) and if t = 1 or equal to a prime p, then the Fourier coefficients depend only on the discriminant ([E-Z85], Theorem 2.2). Let c_{r}^{Φ}(D) = c^{Φ}(\frac{r^{2}+D}{4t}, r), then

Φ(τ, z) = ∑_{{r∈ℤ, D∈ℕ \atop −D≡r^{2} mod(4t)}} c_{r}^{Φ}(D) e\{\frac{r^{2}+D}{4t} τ + rz\}. (39)

For Φ_{j} ∈ J_{k,t}^{cusp} we put c^{j}(n, r) = c^{Φ_{j}}(n, r) and c_{r}^{j}(D) = c_{r}^{Φ_{j}}(D).

Theorem 4.2 Let k, t be natural integers with k even and k + 2Re(s) > 4 for s ∈ ℂ. Let Φ_{1}, Φ_{2}, Φ_{3} ∈ J_{k,t}^{cusp}, then

D_{1,2;3}(s) = a(k, s, t) Γ_{∞}^{∗}(s) D_{1,2;3}^{∗}(s), (40)

here a(k, s, t) = 4^{−1}π^{−(s+2k−3)}t^{s+2k−4},

Γ_{∞}^{∗}(s) = \frac{Γ(2k + s − 3)Γ(s + k − 3/2)^{2}}{Γ(2k + 2s − 3)}

D_{1,2;3}^{∗}(s) = ∑_{r_{0}=0}^{2t−1} ∑_{{D∈ℕ \atop −D≡r^{2}_{0} mod 4t}} ∑_{{ε,ν=1 \atop (ε,ν)=1}}^{∞}

\overline{c_{r_{0}}^{3}(D_{0})} \overline{c_{r_{0}ε}^{1}(D_{0}ε^{2})} \overline{c_{r_{0}ν}^{2}(D_{0}ν^{2})}(εν)^{−(2s+2k−3)}D_{0}^{−(s+2k−3)}.

Proof: First we employ the integration with respect to dτ_{1,x}; dτ_{2,x}; dz_{1,x} and dz_{2,x}. This leads to

D_{1,2;3}(s) = ∑_{{n,r∈ℤ \atop 4nt>r^{2}}} ∑_{λ=−∞}^{∞} ∑_{\binom{ab}{εν}∈U_{⊥}} \overline{c^{1}(n_{1}, r_{1})} \overline{c^{2}(n_{2}, r_{2})} \overline{c^{3}(n, r)}
∫_{0}^{∞} ∫_{0}^{∞}dτ_{1,y} dτ_{2,y} ∫_{0}^{τ_{1,y}} dz_{1,y} ∫_{0}^{τ_{2,y}} dz_{2,y} τ_{1,y}^{s+k−3}τ_{2,y}^{s+k−3}(ε^{2}τ_{1,y} + ν^{2}τ_{2,y})^{−s}
× e{2it(z_{1,y}^{2}/τ_{1,y} + r_{1}/t z_{1,y} + n_{1}/t)}
× e{2it(z_{2,y}^{2}/τ_{2,y} + r_{2}/t z_{2,y} + n_{2}/t)}.

Here n_{1}, n_{2}, r_{1} and r_{2} are parameterized by (\begin{matrix}a & b \\ ε & ν \end{matrix}) and n, r, λ in the following way:

n_{1} = (λa)^{2}t + nε^{2} + rλaε,
n_{2} = (λb)^{2}t + nν^{2} + rλbν

and

r_{1} = rε + 2λat and r_{2} = rν + 2λbt.

The invariance properties of the Fourier coefficients lead to the useful identity

c^{1}(n_{1}, r_{1}) c^{2}(n_{2}, r_{2}) c^{3}(n, r) = c^{1}(nε^{2}, rε) c^{2}(nν^{2}, rν) c^{3}(n, r)
= c_{rε}^{1}(Dε^{2}) c_{rν}^{2}(Dν^{2}) c_{r}^{3}(D),

here D = 4nt − r^{2}. Substituting z_{j,y} by p_{j}τ_{j,y} and p_{j} by p̃_{j} − \frac{r_{j}}{2t} leads to

D_{1,2;3}(s) = ∑_{{n,r∈ℤ \atop 4nt>r^{2}}} ∑_{\binom{ab}{εν}∈U_{⊥}} \overline{c_{rε}^{1}(Dε^{2})} \overline{c_{rν}^{2}(Dν^{2})} \overline{c_{r}^{3}(D)}

∫_{0}^{∞} ∫_{0}^{∞}dτ_{1,y} d_{τ_{2},y} τ_{1,y}^{s+k−2} τ_{2,y}^{s+k−2} (ε^{2}τ_{1,y} + ν^{2}τ_{2,y})^{−s}e^{−4πDt^{−1}(ε^{2}τ_{1,y}+ν^{2}τ_{2,y})}

∑_{λ=−∞}^{∞} ∫_{r_{1}/2t}^{(r_{1}+2t)/2t}∫_{r_{2}/2t}^{(r_{2}+2t)/2t}dp̃_{1} dp̃_{2} e^{−4π t(τ_{1,y}p̃_{1}^{2}+τ_{2,y}p̃_{2}^{2})}.

Let D = {(D, r) ∈ ℕ × ℤ | − D ≡ r^{2} mod 4t} and R_{0} = {0, 1, …, 2t − 1}. Then we have

D = ⨆_{r_{0}∈R_{0}} D_{r_{0}}, here D_{r_{0}} = {(D, r) ∈ D | r ≡ r_{0} mod 2t}.

Hence we have

D_{1,2;3}(s) = ∑_{r_{0}∈R_{0}} ∑_{{D∈ℕ \atop −D≡r^{2}_{0} mod 4t}} \overline{c_{r_{0}ε}^{1}(Dε^{2})} \overline{c_{r_{0}ν}^{2}(Dν^{2})} \overline{c_{r_{0}}^{3}(D)}

∫_{0}^{∞} ∫_{0}^{∞}dτ_{1,y} d_{τ_{2},y} τ_{1,y}^{s+k−2} τ_{2,y}^{s+k−2} (ε^{2}τ_{1,y} + ν^{2}τ_{2,y})^{−s}e^{−4πDt^{−1}(ε^{2}τ_{1,y}+ν^{2}τ_{2,y})}

∑_{λ,u∈ℤ}∫_{\frac{r_{0}ε}{2t} +uε+λ a}^{\frac{r_{0}ε}{2t} +uε+λ a+1} ∫_{\frac{r_{0}ν}{2t} +uν+λ b}^{\frac{r_{0}ν}{2t} +uν+λ b+1} dp̃_{1} dp̃_{2} e^{−4π t(τ_{1,y}p̃_{1}^{2}+τ_{2,y}p̃_{2}^{2})}.

But this can easily be simplified to

D_{1,2;3}(s) = ∑_{r_{0}∈R_{0}} ∑_{{D∈ℕ \atop −D≡r^{2}_{0} mod 4t}} \overline{c_{r_{0}ε}^{1}(Dε^{2})} \overline{c_{r_{0}ν}^{2}(Dν^{2})} \overline{c_{r_{0}}^{3}(D)} ∫_{0}^{∞}∫_{0}^{∞}dτ_{1,y} d_{τ_{2},y} ∫_{−∞}^{∞} ∫_{−∞}^{∞}dx dy

τ_{1,y}^{s+k−2} τ_{2,y}^{s+k−2} (ε^{2}τ_{1,y} + ν^{2}τ_{2,y})^{−s}e^{−4πDt^{−1}(ε^{2}τ_{1,y}+ν^{2}τ_{2,y})}e^{−4πt(τ_{1,y}x^{2}++τ_{2,y}y^{2})}

Using ∫_{−∞}^{∞} e^{−ab^{2}}db = (\frac{π}{a})^{1/2} and obvious substitutions give the desired result.

Theorem 4.3 Let k, t be natural integers with k even and Φ_{1}, Φ_{2}, Φ_{3} ∈ J_{k,t}^{cusp}, then the Dirichlet series D_{1,2;3}^{∗}(s) has a meromorphic continuation to the whole complex plane.

Proof: It follows from the work of Arakawa [Ar90], [Ar92], [Ar94] and Heim [He97], that all Jacobi Eisenstein series have a meromorphic continuation to the whole complex plane.

Remark 4.4 The results given in [Ar94] and [He97] (e.g. Theorem 4.1) can be used to determine the location of the possible poles of real analytic Jacobi Eisenstein series of arbitary degree and hence the analytical behaviour of D_{1,2;3}^{∗}(s). And moreover, the program started in §2 of this paper, will finally give us a functional equation of this series.

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Tsuneo Arakawa Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Tokyo 171, Japan arakawa@rkmath.rikkyo.ac.jp Bernhard Heim Max-Planck Institut für Mathematik Gottfried-Claren Strasse 26 53225 Bonn, Germany heim@mpim-bonn.mpg.de