Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic L-functions

By Bernhard E. Heim

Abstract In this paper a new Rankin-Selberg integral is presented, which is a Petersson scalar product of a diagonal-restricted Eisenstein series against an elliptic modular form and two Siegel modular forms of degree n. This generalizes Paul Garrett’s work on the Rankin triple product L-function. Surprisingly the case n = 2 leads, as n = 1, to an Euler product. We prove, by choosing one of the Siegel forms as a Saito-Kurokawa lift, that our approach gives a new integral representation of a degree eight L-function for GSp_{2} × GL_{2}, the twisted spinor L-function. ^{1}

This is a extended version of a talk given by the author at the NSF-CBMS conference: Euler Products and Eisenstein Series held at the Texas Christian University, Fort Worth, Texas, May 20-24, 1996 in honor of Professor Goro Shimura.

Introduction

The purpose of this paper is to present new insights into the connection between pullbacks of Eisenstein series and automorphic L-functions. Surprisingly, special properties of a certain Hecke-Jacobi algebra will play a decisive role. The paper has its origin in a question asked by S. Böcherer to the author.

Let us consider the following situation. Let (n_{i})_{i=1}^{l} be a partition of a natural number n and let F_{i} ∈ S_{n_{i}}^{k} be Siegel cusp forms of degree n_{i} and weight k. Let further E_{n}^{k}(z; s) be the analytic Eisenstein series of Siegel type, here z is in the Siegel half space IH_{n} of degree n and s ∈ ℂ with 2 Re(s) + k > n + 1. Let us further denote by ⟨; ⟩ the Petersson scalar product and let us assume that all the F_{i}’s are Hecke eigenforms. Then P. Garrett [Ga84], S. Böcherer [Boe85b] and I. Piatetski-Shapiro and S. Rallis [P-R84], [G-P-R87] have shown, that the iterated scalar product

Φ^{n}(F_{1}, …, F_{l}; s) := ⟨ … ⟨E_{n}^{k} (( \begin{matrix}z_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & z_{l} \end{matrix} ); s); F_{1}(z_{1})⟩ … F_{l}(z_{l})⟩ (1)

for l = 2 is essentially the standard L-function D_{F_{1}}^{n_{1}} of F_{1} multiplied with ⟨F_{1}, F_{2}⟩, if n_{1} = n_{2} and zero otherwise. By considering the case l = 3 with n_{1} = n_{2} = n_{3} = 1, P. Garrett [Ga87] succeeded in refining his previous work to get an integral representation of the triple product L-function for the involved elliptic modular forms. This was generalized by I. Piatetski-Shapiro and S. Rallis [P-R87] to 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]). (Of course, to get these integral representations a profound study of the pullbacks of the Eisenstein series via the embeddings IH_{n_{1}} × … × IH_{n_{l}} ↪ IH_{n} is needed.) These results were the starting point and a basis for a lot of important mathematical work, e.g. [Or87],[Sa87],[H-K91],[G-H93], [G-K95],[B-S95],[Sh95]. A good overview over the results concerning the triple product L-function can be found in the introduction to the paper [G-H93] by P. Garrett and M. Harris. Since the approach in (1) has been so successful it is certainly natural to ask:

11991 Mathematics Subject Classification. Primary 11F66, 11F67; Secondary 11F46, 11F60.

Key words and phrases. Pullbacks of Eisenstein series, Hecke-Jacobi theory, automorphic L-functions, Rankin triple-product L-function, special values of L-functions.

Are there any other partitions of a natural number n, which lead to integral representations of an automorphic L-functions in the just described way?

Already in 1985, at the conference of modular functions in Oberwolfach, Garrett mentioned at the end of his talk, that it seems to be unlikely. Nevertheless, this paper gives a positive answer. We produce a family of new pullbacks of Eisenstein series attached to partitions 1 + n + n of 2n + 1, calculate Φ(F_{1}, F_{2}, F_{3}; s) explicitly as a Dirichlet series and show that in a special case we have established a new integral representation of an automorphic L-series.

To state our main results we have to introduce the following notation. Let Γ_{n} be the Siegel modular group. Writing an element of Γ_{n} in the form γ = \binom{a b}{c d}, with matrices a, b, c, d of size n, we can define the action of Γ_{n} on IH_{n} by γ < z >= (az + b)(cz + d)^{−1}. Let P_{n,0} the subgroup of Γ_{n} consisting of elements γ for which c = 0. Then we can introduce the Eisenstein series

E_{n}^{k}(z; s) = ∑_{γ∈P_{n,0}\Γ_{n}} det (cz + d)^{−k} det (Im γ < z >)^{s}.

Here z ∈ IH_{n}, s ∈ ℂ and k ∈ 2ℤ. Let IH_{1} × IH_{n} × IH_{n} be diagonally embedded in IH_{2n+1}. Let ⟨; ⟩, ⟨; ⟩_{J} be the Petersson scalar products of Siegel modular and Jacobi forms. In the following, we assume for convenience the condition totally real for the Fourier coefficients of Siegel cusp forms, which can be easily dropped ([Sh76],[Boe85a]).

Theorem 1 Let F, G ∈ S_{n}^{k} and h ∈ S_{1}^{k} be Hecke eigenforms with totally real Fourier coefficients and k even. Then for s ∈ ℂ with 2 Re (s) + k > 2n + 2

Φ^{2n+1}(h, G, F; s) = a(n, s, k) L_{h,G,F}^{n} (s + 2k − n − 1).

Here a(n, s, k) is essentially a product of gamma functions and

L_{h,G,F}^{n} (s + 2k − n − 1) = {ζ(2s + k) ∏_{j=1}^{n} ζ(4s + 2k − 2j)}^{−1}
D_{F}^{n} (2s + k − n) D_{h,G;F} (s),

where D_{F}^{n} is the standard zeta function of F and D_{h,G;F} (s) denotes a certain Dirichlet series (see section 2.4) involving the Fourier coefficients of h and the Fourier-Jacobi coefficients of F and G amalgamated together by the scalar product ⟨; ⟩_{J}.

Has this anything to do with Euler products? If n = 2 (the case n = 1 corresponds to the triple product L-function [Ga87]) we can prove the following Main Theorem Let F, G ∈ S_{2}^{k} and h ∈ S_{1}^{k} be Hecke eigenforms with totally real Fourier coefficients. Let k be even and h normalized. Further we assume that F is in the Maass Spezialschar with corresponding f ∈ S_{1}^{2k−2}. Then for s ∈ ℂ with 2 Re (s) + k > 6

Φ^{5}(h, G, F; s) = a(2, s, k)⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J}
×ζ(2s + k)^{−1}ζ(4s + 2k − 2)^{−1}ζ(4s + 2k − 4)^{−1}
×L_{f}(2s + 2k − 4) Z_{G⊗h}(s + 2k − 3).

Here L_{f} denotes the Mellin transform of f and Z_{G⊗h} the convolution of the spinor zeta functions attached to G and h, in other words the L-functions to GSp_{2} × GSp_{1}. And with Φ_{N}^{H} we denote the N-th Fourier-Jacobi coefficient of a cusp form H.

If we assume that the first Fourier-Jacobi coefficient of G is not identically zero, we can conclude that Z_{G⊗h}(s) has a meromorphic continuation to the whole s-plane and get a functional equation (see Corollary 5.2). Taking G with totally real algebraic Fourier coefficients and of weight k > 6, and σ ∈ Gal(\overline{ℚ}/ℚ) we get

(π^{8−5k} \frac{Z_{G⊗h}(2k − 3)}{⟨G; G⟩⟨h; h⟩})^{σ} = π^{8−5k} \frac{Z_{G^{σ}⊗h^{σ}}(2k − 3)}{⟨G^{σ}; G^{σ}⟩⟨h^{σ}; h^{σ}⟩} ∈ \overline{ℚ}.

Related work: In [K-S89], W. Kohnen and N.-P Skoruppa studied analytic and arithmetic properties of D_{F,G}(s) = ∑_{N=1}^{∞}⟨Φ_{N}^{F}; Φ_{N}^{G}⟩_{J} N^{−s} with F, G ∈ S_{2}^{k}. They proved that D_{F,G}(s) has a meromorphic continuation to the whole s-plane. This was generalized in [Ya90] with respect to F, G ∈ S_{n}^{k}. Let F be in the Maass Spezialschar and G a Hecke eigenform of degree 2, then

D_{F,G}(s) = ⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J} Z_{G}^{2}(s).  (2)

Here, Z_{G}^{2}(s) denotes the spinor zeta function of G. If Φ_{1}^{G} is identically zero, then one replaces F by a certain paramodular cusp form such that one can still use the approach of Kohnen and Skoruppa to study Z_{G}^{2}(s) for an arbitrary eigenform G. This refinement is due to V. Gritsenko [Gr93]. A different approach to study the L-function Z_{G⊗h}(s) was given in 1993 by M. Furusawa in [Fu93]. He showed that the L-function is essentially a scalar product of a Hermitian Eisenstein series attached to h and G, if G satisfies certain conditions. His results are compatible with our results concerning the meromorphic continuation, functional equation and the special value property of Z_{G⊗h}(s). For certain reasons it was impossible for him with his approach to state the Galois behaviour of the special value.

Brief outline of the Article: After reviewing some known results on Siegel modular forms, analytic Eisenstein series and a Hecke theory for the symplectic group, we state the Theorem of Garrett and Böcherer, which gives a first insight in the correspondence of pullbacks of Eisenstein series of Siegel and Klingen type. This leads us to the problem of understanding P_{n+1,n}(ℤ)\Sp_{n+1}(ℤ) with respect to the action of Sp_{1}(ℤ) × Sp_{n}(ℤ) ↪

Sp_{n+1}(ℤ) from the right, here P_{n+1,n}(ℚ) = \{\binom{a ∗}{0 ∗} ∈ Sp_{n+1}(ℚ)|a ∈ ℚ^{×}\}.

In section 2, we present briefly a viewpoint, which reduces this problem to the calculation of the double coset space

P_{n+1,n}(ℚ)\Sp_{n+1}(ℚ)/Sp_{1}(ℚ) × Sp_{n}(ℚ)

and indicate how one can do that in a systematic manner. Behind all that stands a new method, which allows us to understand orbit structures of much more general shape. This will be the main subject of a forthcoming paper. After stating a pullback formula of the relevant Eisenstein series of Klingen type, we prove Theorem 1 and give some first applications.

In the next two sections we show, that L_{h,G,F}^{n} (s) has an Euler product for n = 2 (the case n = 1 is due to Garrett [Ga87]) and that Fourier-Jacobi coefficients of eigenfunctions of Siegel modular forms have certain multiplicative properties. For that, we have chosen the following strategy:

In section 3, we construct a modified Hecke-Jacobi algebra H̃^{J} in which one can embed the usual symplectic Hecke algebra H^{2} of degree 2. Highlights of our considerations are factorizations of the Hecke and Rankin polynomials of degree 2 (cf. [Gr82],[Gr93] and [An87]). We interpret Jacobi forms as automorphic forms (P-forms) on a higher dimensional space to maximal parabolic subgroups of the symplectic group (Klingen [Kl89]), which explains many properties of Jacobi forms in a transparent way. With the results developed in section 2 and 3 we prove in section 4, that L_{h,G,F}^{2} (s) has an Euler product and calculate the local factors explicitly.

In section 5, we summarize our investigations and deduce important properties of the L-function Z_{G⊗h}(s), assuming that the first Fourier-Jacobi coefficient of G is not identically zero. The reader should not be surprised about that condition, because already in the case n = 1 ([Ga87]), the Fourier(-Jacobi) coefficients play a normalizing role. This paper is rounded out by a unifying presentation of some properties of the triple product L-function and the L-function Z_{G⊗h}(s). This makes it possible to translate and apply techniques, that have been developed to analyse the triple product L-function, to Z_{G⊗h}(s) and get a deeper understanding of it in the future.

T his paper contains the main parts of my thesis (Mannheim 97) which was written under the supervision of Prof. Dr. S. Böcherer. I would like to thank him for introducing me into the subject and for his advice. I am grateful to the Max-Planck-Institut für Mathematik in Bonn for its hospitality and support. Finally I would like to take the opportunity to thank the organizers of the conference, Robert S. Doran, Ze-Li Dou and George T . Gilbert for their excellent job and their warm hospitality.

————————

Notation: For an associative ring R with identity element, we denote by R^{×} the group of all invertible elements of R and by M(m, n; R) 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) = M(n, n; R), 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)\} ,
C_{n,r}(R) = \{( \begin{matrix}∗ & ∗ \\ 0_{n−r,n+r} & ∗ \end{matrix} ) ∈ Sp_{n}(R)\} ,
C_{n,r}^{+}(R) = \{( \begin{matrix}∗ & ∗ \\ 0_{n−r,n+r} & 1_{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 a subring R of ℝ we put R^{+} = {r ∈ R | r > 0} and define

G^{+}Sp_{n}(R) = {M ∈ Gl_{2n}(R)|M^{t}J_{n}M = μ(M)J_{n} with μ(M) ∈ R^{+}}.

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. Further we write [A_{1}, …, A_{m};A_{m+1} …, A_{2m}] for numbers A_{j}, if n = 2m. The disjoint union of sets is denoted by ⨆. Let T, Z ∈ M_{n}(ℂ), then we put e{T Z} = e^{2πi (tr T Z)} and Re(Z), Im(Z) for the real and imaginary part of Z and for M ∈ M(m, n; ℂ) we put occasionally x_{M} = Re(M) and y_{M} = Im(M). Further δ(Z) = det (Im (Z)). By \overline{ℚ} we denote the algebraic closure of ℚ, which we assume to be embedded into ℂ and \overline{a} denotes the complex conjugation of an element a ∈ ℂ. Let us always assume, that p is a prime number.

1 Preliminaries

1.1 Siegel modular forms

The group of positive symplectic similitudes G^{+}Sp_{n}(ℝ) acts on Siegel’s half space IH_{n} = {Z = Z^{t} ∈ M_{n}(ℂ)| Im(Z) > 0} of degree n as a group of biholomorphic automorphisms by

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

where M = \binom{A B}{C D} with A, B, C, D ∈ M_{n}(ℝ). We denote the factor of automorphy by j(M, Z) = det (CZ + D). For F : IH_{n} → ℂ and k ∈ ℤ we define the Petersson operator

(F |_{k}M)(Z) := μ(M)^{nk− \frac{n(n+1)}{2}} j(M, Z)^{−k}F (M < Z >).  (3)

The real positive number μ(M) is called factor of similitude. Let 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}. Every F ∈ M_{n}^{k} satisfies the transformation law

F |_{k}M = F, for all M ∈ Γ_{n} = Sp_{n}(ℤ)  (4)

and has a Fourier expansion of the form

F (Z) = ∑_{T ∈A_{n}} a(T)e{T Z},

with a(T) ∈ ℂ and A_{n} denotes the set of semi-integral positive semidefinite matrices of size n. For cusp forms we have only to sum over A_{n}^{+} = {T ∈ A_{n}|T > 0}. It is well known ([Sh75]) that, if σ is an automorphism of ℂ and F ∈ S_{n}^{k}, then we have

F^{σ}(Z) = ∑_{T ∈A_{n}^{+}} a(T)^{σ}e{T Z} ∈ S_{n}^{k}.  (5)

For arbitrary complex valued functions F and G on IH_{n}, which satisfy (4), one can define the Petersson integral, convergence assumed,

⟨F, G⟩ = ∫_{Γ_{n}\IH_{n}} F (Z) \overline{G(Z)} det (Im Z)^{k}d^{∗}Z.  (6)

Here d^{∗}Z = det (Y)^{−(n+1)}dXdY denotes the symplectic volume element. For more details the reader is referred to Andrianov [An87] or Klingen [Kl90].

1.2 Classical Hecke theory for the symplectic group

We start by recalling the basic facts on abstract Hecke rings ([An87], §3.1; [Sh71] Chapter 3). Let (R, S) be a Hecke pair, this means R is a subgroup of the group S and for s ∈ S the coset space R\RsR is finite. Let A be a principal ideal domain, then R acts from the right on the A−module

L_{A}(R, S) = {X = ∑_{i=1}^{n} a_{i} Rs_{i} | a_{i} ∈ A, s_{i} ∈ S and n ∈ IN}.

The subset of elements H_{A}(R, S) invariant under this action forms a ring with

(∑_{i} a_{i}Rg_{i}) ∘ (∑_{j} b_{j}Rh_{j}) = ∑_{i,j} a_{i}b_{j}Rg_{i}h_{j}.

It is called Hecke ring or Hecke algebra to stress either the ring or algebra properties. Identifying double cosets RsR = ⨆_{i=1}^{n} Rs_{i} with ∑_{i=1}^{n} Rs_{i} ∈ H_{A}(R, S) makes it possible to see them as a basis of the A-module H_{A}(R, S).

Lemma 1.1 Let (Γ_{1}, S_{1}) and (Γ_{2}, S_{2}) be two Hecke pairs included in a bigger group. If we assume that

Γ_{2} ⊂ Γ_{1}, S_{1} ⊂ Γ_{1}S_{2}, and Γ_{1} ∩ S_{2}S_{2}^{−1} ⊂ Γ_{2},

then we one can define an embedding i of H_{A}(Γ_{1}, S_{1}) into H_{A}(Γ_{1}, S_{1}) in the following way: write X ∈ L_{A}(Γ_{1}, S_{1}) in the form ∑_{i} a_{i}Γ_{1}s_{i,2} with s_{i,2} ∈ S_{2} and a_{i} ∈ A, then i(X) := ∑_{i} a_{i}Γ_{2}s_{i,2}.

Lemma 1.2 If (R, S) is a Hecke pair, then (R, S) is also a Hecke pair and

j : \{ \begin{matrix}H_{A}(R, S) & ⟶ & H_{A}(R, S^{−1}) \\ RsR & ↦ & Rs^{−1}R \end{matrix}  (7)

is an anti-isomorphism which commutes with the embedding i defined in the last lemma.

1.2.1 Symplectic Hecke algebra H^{n}

By applying the elementary divisor theorem, it is not difficult to see that the symplectic Hecke algebra H^{n} := H_{ℚ}(Γ_{n}, G^{+}Sp_{n}(ℚ)) decomposes into an infinite tensor product

H^{n} = ⊗_{p}H_{p}^{n}.

The local Hecke algebra H_{p}^{n} = H_{ℚ}(Γ_{n}, GSp_{n}^{+}(ℤ[p^{−1}])) is isomorphic by the Satake- isomorphism Φ_{p}^{n} to ℚ[x_{0}^{±1}, …, x_{n}^{±1}]^{W_{n}}. Here W_{n} denotes the Weyl group of Sp_{n} (for more details see Freitag [Fr83] Kapitel 4). The Hecke algebra H_{p}^{n} is generated by

T^{(n)}(p) = Γ_{n}[1, …, 1; p, …, p]Γ_{n}, T_{n}^{(n)}(p)^{−1}
T_{i}^{(n)}(p^{2}) = Γ_{n}[\underbrace{1, …, 1,}_{n−i} \underbrace{p, …, p}_{i}; p^{2}, …, p^{2}, p, …, p]Γ_{n}

for i = 0, …, n. Furthermore we define T^{(n)}(a) = ∑_{M∈SD_{n}(a)} Γ_{n}MΓ_{n} with

SD_{n}(a) = {[e_{1}, …, e_{n}; f_{1}, …, f_{n}] | e_{i}, f_{j} ∈ ℕ; e_{i}|e_{i+1}; e_{n}|f_{n} and e_{i}f_{i} = a}.

Definition 1.3 The Hecke polynomial Q_{p}^{n}(X) and Rankin polynomial D_{p}^{n}(X) of degree n are defined by

Q_{p}^{n}(X) := (1 − x_{0,p}X) ∏_{r=1}^{n} ∏_{1≤i_{1}<···<i_{r}≤n}(1 − x_{0,p}x_{i_{1},p} · . . . · x_{i_{r},p}X)

D_{p}^{n}(X) := (1 − X) ∏_{i=1}^{n}(1 − x_{i,p}X)(1 − x_{i,p}^{−1}X).

Taking the pre-images of the coefficients of these polynomials under Φ_{p}^{n}, we get polynomials in H_{p}^{n}[X] which we denote by q_{p}^{n}(X) and d_{p}^{n}(X). We denote the formal spinor- and standard zeta function of degree n by Z^{n}(s) = ∏_{p} (Q_{p}^{n}(p^{−s}))^{−1} and D^{n}(s) = ∏_{p} (D_{p}^{n}(p^{−s}))^{−1} for s ∈ ℂ.

By modifying the map j slightly we get an (anti-)automorphism

∗ : H^{n} ⟶ H^{n}, Γ_{n}MΓ_{n} ↦ (Γ_{n}MΓ_{n})^{∗} = Γ_{n}μ(M) M^{−1}Γ_{n}.  (8)

To simplify our notation in the case n = 1, we put T (a, d) = Γ\binom{a 0}{0 d}Γ and T (n) = T^{(1)}(n). Then

T (p) = ∑_{a mod p} Γ ( \begin{matrix}1 & a \\ 0 & p \end{matrix} ) + Γ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) = Γ ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) U_{[1,p]},

where U_{[a,b]} denotes a representative system of (( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )^{−1} Γ ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ∩ Γ) \ Γ.

In H_{p}^{n}[X], we have the following formal identity

∑_{n=0}^{∞} T (p^{n})X^{n} = (1 − T (p)X + pT (p, p)X^{2})^{−1} = q_{p}^{1}(X)^{−1}.  (9)

It is not difficult to see that

pT_{1}^{(2)}(p^{2}) + p(p^{2} + 1)Δ_{p}^{(2)} = T^{(2)}(p)^{2} − T^{(2)}(p^{2}) − p^{2}Δ_{p}^{(2)}  (10)

holds in H_{p}^{2}. Important for our purpose is the explicit form of q_{p}^{2}(X) and d_{p}^{2}(X).

Lemma 1.4 In H_{p}^{2}[X], we have

q_{p}^{2}(X) = 1 − A_{1}X + A_{2}X^{2} − p^{3}Δ_{p}A_{1} + p^{6}(Δ_{p})^{2}X^{4}  (11)
d_{p}^{2}(X) = (1 − X)(1 − B_{1}X + B_{2}X^{2} − B_{1}X^{3} + X^{4}),  (12)

with A_{1} = T^{(2)}(p), A_{2} = p(T_{1}^{(2)}(p^{2}) + (p^{2} + 1)Δ_{p}) and

B_{1} = Δ_{p}^{−1}p^{−3}(T^{(2)}(p)^{2} − T^{(2)}(p^{2}) − p^{2}Δ_{p} − 2Δ_{p}p^{3}),
B_{2} = Δ_{p}^{−1}p^{−3}(2 T^{(2)}(p^{2}) − T^{(2)}(p)^{2} + 2p^{3}Δ_{p} + 2p^{2}Δ_{p}).

The space S_{n}^{k} can be seen as an H^{n}-module. Let X = ∑ a_{i}Γ_{n}g_{i} ∈ H^{n}, then

F ↦ F |_{k}D(X) := ∑_{i} a_{i}F |_{k}g_{i}.  (13)

It is well known that there exists a basis of S_{n}^{k} consisting of eigenforms. An eigenform F induces for every local algebra H_{p}^{n} an algebra homomorphism to ℂ, which can be characterized by (α_{0,p}, α_{1,p}, …, α_{n,p}) ∈ (ℂ^{×})^{n+1}, the Satake-parameters of F. The eigenvalue p^{nk−n(n+1)} of Δ_{p}^{n} gives α_{0,p}^{2}α_{1,p} … α_{n,p} = p^{nk− \frac{n(n+1)}{2}} . (The parameters are unique up to the action of the Weyl group). To such a F we can define Q_{p,F}^{n} (X), D_{p,F}^{n} (X), Z_{F}^{n}(s), D_{F}^{n} (s). It is known that the last two functions are absolutely convergent for Re(s) large. In the case n = 1, it is convenient to use a different notation. One defines α_{p} = α_{0,p} and β_{p} = α_{0,p}α_{1,p} for Satake-parameters α_{0,p}, α_{1,p} of a normalized (a_{1}^{f} = 1) Hecke eigenform f ∈ S_{1}^{k} where a_{m}^{f} denotes the m^{th} Fourier coefficient of f. We usually want to assume that an eigenform in S_{1}^{k} is normalized. Then one has

α_{p}β_{p} = p^{k−1} and α_{p} + β_{p} = a_{p}^{f}.

We write L_{f}(s) for Z_{f}^{1}(s) and L_{p,f}(X) for Q_{p,f}^{1}(X). With this notation we introduce the L-function Z_{G⊗h} corresponding to two eigenforms G ∈ S_{2}^{k} and h ∈ S_{1}^{k} with parameters α_{0,p}, α_{1,p}, α_{2,p} and α_{p}, β_{p} for every prime number p:

Z_{G⊗h}(s) = ∏_{p} {Q_{p,G}^{2}(α_{p}p^{−s}) Q_{p,G}^{2}(β_{p}p^{−s})}^{−1} .

The triple product L-function of eigenforms f_{1}, f_{2}, f_{3} ∈ S_{1}^{k} is

L_{f_{1}⊗f_{2}⊗f_{3}}(s) = ∏_{p} det (1_{8} − ( \begin{matrix}α_{p}^{f_{1}} & 0 \\ 0 & β_{p}^{f_{1}} \end{matrix} ) ⊗ ( \begin{matrix}α_{p}^{f_{2}} & 0 \\ 0 & β_{p}^{f_{2}} \end{matrix} ) ⊗ ( \begin{matrix}α_{p}^{f_{3}} & 0 \\ 0 & β_{p}^{f_{3}} \end{matrix} ) p^{−s})^{−1} .

Remark 1.5 For each subring A ⊂ ℂ, let S_{n}^{k}(A) be the A−module of all F ∈ S_{n}^{k} for which all the Fourier coefficients are in A. Let F be an eigenform, especially we have F |_{k}T^{(n)}(m) = λ_{m}^{F}F for all m ∈ ℕ and define a subfield ℚ(F) of \overline{ℚ} generated by ℚ and the set of all λ_{m}^{F}. Then ℚ(F) is a totally real finite extension of ℚ and for k ≥ 2n + 2 there exists an orthogonal eigenbasis {F_{j}}_{j=1}^{d} of S_{n}^{k} such that F_{j} ∈ S_{n}^{k}(ℚ(F_{j})) and the F_{j} are permuted by the action of Gal(\overline{ℚ}/ℚ), the absolute Galois group of ℚ. Such a basis is called a \underline{distinguished} basis. Every eigenform F ∈ S_{n}^{k}(ℚ(F)) can be extended to such a basis ([Mi91], Appendix A).

1.3 Eisenstein series

In this section, we recall some basic facts on Eisenstein series of Klingen type and study some abstract properties of their pullbacks.

Definition 1.6 Let r, k be rational integers and n a natural number. Further let k be even and 0 ≤ r ≤ n. If s ∈ ℂ with k + 2 Re(s) > n + r + 1 and F ∈ S_{r}^{k}, then we denote by

E_{n,r}^{k}(Z, F; s) = ∑_{γ∈P_{n,r}\Γ_{n}} F (γ < Z >_{∗}) j(γ, Z)^{−k} (\frac{det Imγ < Z >}{det Imγ < Z >_{∗}} )^{s} ,

the Eisenstein series of Klingen type on IH_{n}.

The series is absolutely convergent and admits a meromorphic continuation to the whole s−plane ([La76]). With regard to k and Γ_{n}, it satisfies the transformation law (4) and it is orthogonal to S_{n}^{k} for r < n. When s = 0 and k > n+r+1, we get the holomorphic Klingen Eisenstein series ([Kl90], chapter 5). Siegel ([Si39]) has already proved that the Fourier coefficients of E_{n,0}^{k}(Z, 1; 0) are rational numbers. For convenience we put E_{n}^{k}(Z; s) = E_{n,0}^{k}(Z, 1; s), Γ_{m}(s) = ∏_{j=1}^{m} Γ(s − \frac{j−1}{2}) and ξ(s) = π^{− \frac{s}{2}} Γ(\frac{s}{2})ζ(s).

Lemma 1.7 If we put IE_{n}^{k}(Z; s) := ϵ(n, s) E_{n}^{k}(Z; s − \frac{k}{2}) with

ϵ(n, s) = ξ(2s)\frac{Γ_{n}(s+ \frac{k}{2} )}{Γ_{n}(s)} ∏_{j=1}^{[ \frac{n}{2} ]} ξ(4s − 2j). Then we have

IE_{n}^{k}(Z; s) = IE_{n}^{k}(Z; \frac{n + 1}{2} − s).

Now we associate to a given partition (n_{i})_{i=1}^{l} of n (i.e. ∑_{i=1}^{l} n_{i} = n with n_{i} ∈ ℕ) the embeddings IH_{n_{1}} × … × IH_{n_{l}} ↪ IH_{n} and Γ_{n_{1}} × … × Γ_{n_{l}} ↪ Γ_{n} given by

(z_{1}, …, z_{l}) ↦ [z_{1}, …, z_{l}]

( \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ) × … × ( \begin{matrix}a_{l} & b_{l} \\ c_{l} & d_{l} \end{matrix} ) ↦ ( \begin{matrix}a_{1} & & 0 & b_{1} & & 0 \\ & ... & & & ... \\ 0 & & a_{l} & 0 & & b_{l} \\ c_{1} & & 0 & d_{1} & & 0 \\ & ... & & & ... \\ 0 & & c_{l} & 0 & & d_{l} \end{matrix} ) .

It is convenient to use the same notation for the images and pre-images.

Definition 1.8 Let n, r, k be as before and (n_{i})_{i=1}^{l} a partition of n. Assume further that F ∈ S_{r}^{k} and s ∈ ℂ with Re(s) big enough. Then we call

ϕ_{F} (z_{1}, …, z_{l}; s) = E_{n,r}^{k} (( \begin{matrix}z_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & z_{l} \end{matrix} ) , F; s)

a diagonal-restricted Eisenstein series attached to F of type ((n_{i})_{i=1}^{l}; r).

It is easy to see that such functions satisfy (4) in each variable. If we can describe ϕ_{F} (z_{1}, …, z_{l}; s) explicitly as a function in the variables z_{1}, …, z_{l}, then we will call such an expression a pullback formula ([Ga84], Main formula). Let us put (n_{1}, …, n_{l}) = (n_{1}, …, n_{l}; 0) and ϕ := ϕ_{1}. The following theorem has served as the main motivation for us to study pullback formulas.

Theorem 1.9 Let k be an even rational integer and s ∈ ℂ with k + 2 Re(s) > n + 1. Further let (n_{i})_{i=1}^{l} be a partition of n and F_{i} ∈ S_{n_{i}}^{k}. Then

Φ(F_{1}, …, F_{l}; s) := ⟨...⟨ϕ(z_{1}, …, z_{l}; s); F_{1}(z_{1})⟩...F (z_{l})⟩

satisfies

(a) Φ(F_{1}, …, F_{l}; s) has an analytic continuation to a meromorphic function on ℂ and inherits a functional equation from Lemma 1.7.

(b) If we assume that the F_{i}’s are Hecke eigenforms with F_{i} ∈ S_{n_{i}}^{k}(ℚ(F_{i})) and k > max{n + 1, max{n_{i}} + 1}, then

\frac{Φ(F_{1}^{σ}, …, F_{l}^{σ}; 0)}{∏_{i=1}^{l}⟨F_{i}^{σ}; F_{i}^{σ}⟩} = (\frac{Φ(F_{1}, …, F_{l}; 0)}{∏_{i=1}^{l}⟨F_{i}; F_{i}⟩})^{σ} ∈ \overline{ℚ} with σ ∈ Gal(ℚ̄/ℚ).

Proof: Part (a) is clear. To prove (b), we assume for convenience that l = 2, for arbitrary l one proceeds in the same way. We can complete f_{1} = F_{1} and g_{1} = F_{2} to a distinguished basis of S_{n_{1}}^{k} and S_{n_{2}}^{k} respectively and then to an orthogonal basis of M_{n_{1}}^{k} and M_{n_{2}}^{k}. This can be done in such a way that the orthogonal complements of S_{n_{1}}^{k} and S_{n_{2}}^{k} are invariant under all automorphisms of ℂ (cf. [Boe85a], section 6). From ϕ(z_{1}, z_{2}; 0) ∈ M_{n_{1}}^{k} ⊗ M_{n_{2}}^{k} we can deduce

ϕ(z_{1}, z_{2}; 0) = ∑_{i,j} c_{i,j}f_{i}(z_{1})g_{j}(z_{2}) (c_{ij} ∈ ℂ).

The rationality of the Fourier coefficients of ϕ(z_{1}, z_{2}; 0) leads to

(\frac{Φ(F_{1}, F_{2}; 0)}{⟨F_{1}; F_{1}⟩ ⟨F_{2}; F_{2}⟩})^{σ} = c_{11}^{σ} = \frac{⟨⟨ϕ(z_{1}, z_{2}; 0); F_{1}^{σ}F_{2}^{σ}⟩⟩}{⟨F_{1}^{σ}; F_{1}^{σ}⟩ ⟨F_{2}^{σ}; F_{2}^{σ}⟩},

for all automorphisms σ of ℂ. The assumptions in (b) imply that c_{11} is algebraic.

Remark 1.10 If we denote by ℚ(F_{1}, …, F_{l}) the totally real number field which is generated by ℚ(F_{1}), …, ℚ(F_{l}), then

\frac{Φ(F_{1}, …, F_{l}; 0)}{∏⟨F_{i}; F_{i}⟩} ∈ ℚ(F_{1}, …, F_{l}).

It is of course possible to state the theorem under weaker assumptions. For example, the purely technical condition totally real can easily be dropped (e.g. [Boe85a], Satz 14).

A first insight into the connection between Φ(F_{1}, …, F_{l}; s) and automorphic L-functions is given by the following

Theorem 1.11 (Garrett, Böcherer) Let m, n ∈ ℕ, k ∈ 2ℤ and s ∈ ℂ with s ∈ ℂ with k + 2Re(s) > m + n + 1. If F ∈ S_{n}^{k} is an eigenform then the following assertion holds

⟨E_{m+n}^{k} (( \begin{matrix}z & 0 \\ 0 & ∗ \end{matrix} ); s); F ⟩ = \{ \begin{matrix}μ(n, k, s) \{ζ(2s + k) ∏_{j=1}^{n} ζ(4s + 2k − 2j)\}^{−1} \\ × D_{F}^{n} (2s + k − n)E_{m,n}^{k}(z, F^{#}; s)  if m ≥ n, \\ 0  otherwise. \end{matrix}

Here # denotes complex-conjugation and

μ(n, k, s) = 2^{\frac{n^{2}+3n}{2} −2ns−nk+1}(−1)^{\frac{nk}{2}} π^{\frac{n(n+1)}{2}} \frac{Γ_{n}(k + s − \frac{n+1}{2})}{Γ_{n}(k + s)}.  (14)

See [Ga84] and [Boe85b] for more details. In the special case n = m, s = 0 we have

Corollary 1.12 Let F ∈ S_{n}^{k}(ℚ(F)) be a Hecke eigenform and k > 2n even. Then

(\frac{π^{\frac{3n(n+1)}{2} −2nk−k}D_{F}^{n} (k − n)}{⟨F; F ⟩})^{σ} = \frac{π^{\frac{3n(n+1)}{2} −2nk−k}D_{F^{σ}}^{n}(k − n)}{⟨F^{σ}; F^{σ}⟩} ∈ ℚ(F)  (15)

for all σ ∈ Gal(\overline{ℚ}/ℚ).

2 A family of Dirichlet series

This section centers around the explicit calculation of the integral of an Eisenstein series of type (1, n, n) against three cusp forms. To do this, we start with some Lineare Zahlentheorie (cf. Siegel [Si68] 3. Abschnitt).

2.1 Representatives for P_{n+1,n}\Γ_{n+1}

The fact, that for every g ∈ Sp_{m}(ℚ) there exists p_{g} ∈ ⋂_{r=0}^{m−1} P_{m,r}(ℚ) with

p_{g} · g ∈ Γ_{m}  (16)

allows us to find a suitable system of representatives for RS^{ℤ} of P_{n+1,n}\Γ_{n+1} in a rather simple way, that suits our purposes.

Proposition 2.1 The left cosets space P_{n+1,n}(ℚ)\Sp_{n+1}(ℚ) has representatives

RS^{ℚ} = R_{1}^{ℚ} ⊔ R_{2}^{ℚ} ⊔ R_{3}^{ℚ}.

R_{1}^{ℚ} = P_{1,0}(ℚ)\Sp_{1}(ℚ) × 1_{2n},  (17)
R_{2}^{ℚ} = π_{n} · (1_{2} × P_{n,n−1}(ℚ)\Sp_{n}(ℚ)),  (18)
R_{3}^{ℚ} = ξ_{n} · (P_{1,0}(ℚ)\Sp_{1}(ℚ) × ((T × 1_{2(n−1)}) · P_{n,n−1}(ℚ)\Sp_{n}(ℚ)))  (19)

with π_{n} = ( \begin{matrix}\begin{matrix}0 1 1 0 & 0 \\ 0 & 1_{n−1} \end{matrix} & 0 \\ 0 & \begin{matrix}0 1 1 0 & 0 \\ 0 & 1_{n−1} \end{matrix} \end{matrix} )

and ξ_{n} = ( \begin{matrix}\begin{matrix}1 0 1 1 & 0 \\ 0 & 1_{n−1} \end{matrix} & 0 \\ 0 & \begin{matrix}\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} & 0 \\ 0 & 1_{n−1} \end{matrix} \end{matrix} ) .

Furthermore T = T (ℚ) = \{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) | a ∈ ℚ^{×}\}.

Proof: By Witt’s Theorem ([Bou59] §4, n^{0}3; Corollaire 2), we can identify P_{n+1,n}(ℚ)\Sp_{n+1}(ℚ) with the set X of one-dimensional (isotropic) subspaces of ℚ^{2(n+1)}. To prove the assertion of the proposition, we have to determine the orbits of the action of Sp_{1}(ℚ) × Sp_{n}(ℚ) ≤ Gl_{2(n+1)}(ℚ) on X. We decompose ℚ^{2(n+1)} = ℚ^{2} ⊕ ℚ^{2n} and denote by p_{1}(U), p_{2}(U) the projection of U ∈ X on ℚ^{2} and ℚ^{2n} respectively. Then it is clear, that the orbits are characterized by the ranks (rank(p_{1}(U)), rank(p_{2}(U))) of the projection: (1, 0), (0, 1) (1, 1). So we get three orbits. Then we fix an element in each orbit and determine its isotropy group. Translating these observations into the language of matrices proves the proposition.

To get the desired representatives over ℤ, we have to alter RS^{ℚ} by suitable elements of P_{n+1,n}(ℚ) from the left.

Corollary 2.2 The left cosets space P_{n+1,n}\Γ_{n+1} has representatives

RS^{ℤ} = R_{1}^{ℤ} ⊔ R_{2}^{ℤ} ⊔ R_{3}^{ℤ}.
R_{1}^{ℤ} = P_{1,0}\Γ_{1} × 1_{2n},  (20)
R_{2}^{ℤ} = π_{n} · (1_{2} × P_{n,n−1}\Γ_{n}),  (21)
R_{3}^{ℤ} = ⨆_{{ε∈ℕ,ν∈ℤ\\{0\} \atop (ε,ν)=1}} ξ_{n}^{ε,ν} · (P_{1,0}\Γ × P_{n,n−1}\Γ_{n})  (22)

with ξ_{n}^{ε,ν} = ( \begin{matrix}∗ & ∗ & 0 & 0 \\ ε & ν & 0 & 0 \\ 0 & 0 & ν & −ε \\ 0 & 0 & ∗ & ∗ \end{matrix} ) × 1_{2(n−1)} ∈ Γ_{n+1}.

Proof: The property (16) implies that there is an 1-1 correspondence between P_{n,r}(ℚ)\Sp_{n,r}(ℚ) and P_{n,r}\Γ_{n}. So it remains only to translate R_{3}^{ℚ} back to R_{3}^{ℤ}. That means we are looking for a p_{a} ∈ P_{n,r}(ℚ) with

p_{a} · (1_{2} × ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) × 1_{2(n−1)}) ∈ Γ_{n+1}.

If we parametrize a ∈ ℚ^{×} with \frac{ν}{ε} (ν ∈ ℤ\\{0\}, ε ∈ ℕ), then

p_{ε,ν} = ( \begin{matrix}\begin{matrix}ν^{−1} & μ′ \\ 0 & ε \end{matrix} & 0 \\ 0 & \begin{matrix}ν & 0 \\ μ & ε^{−1} \end{matrix} \end{matrix} ) × 1_{2(n−1)}

does the job, if we choose μ, μ′ ∈ ℤ in such a way that μ ε ≡ 1 mod ν.

The assertion of the Proposition 2.1 still holds for arbitrary fields. The corollary is of course not true for every integral domain.

2.2 Pullback formulas

This subsection is devoted to explain how one can use Corollary 2.2 to produce pullback formulas. We achieve this by giving two examples.

Theorem 2.3 Let k ∈ 2ℤ and s ∈ ℂ with 2 Re(s) + k > 2n + 2. If F ∈ S_{n}^{k} and τ ∈ IH, τ′ ∈ IH_{n}, then we get

E_{n+1,n}^{k} (( \begin{matrix}τ & 0 \\ 0 & τ′ \end{matrix} ) , F; s) = E_{1}^{k}(τ, s)F (τ′) + \{ \begin{matrix}E_{1}^{k}(τ′; s)F (τ) & n = 1 \\ E_{n,n−1}^{k}(τ′, F_{τ}; s) & n > 1 \end{matrix}

+ 2 δ(τ)^{s} δ(τ′)^{s} ∑_{{ϵ,ν=1 \atop (ϵ,ν)=1}}^{∞} ∑_{{γ_{1}∈Γ∞\Γ \atop γ_{2}∈P_{n,n−1}\Γn}} χ_{j}^{k,s}(γ_{1}, τ) χ_{j}^{k,s}(γ_{2}, τ′)

× F (( \begin{matrix}ε^{2}γ_{1} < τ > & 0 \\ 0 & 0 \end{matrix} ) + (γ_{2} < τ′ >) [( \begin{matrix}ν & 0 \\ 0 & 1_{n−1} \end{matrix} )])

×δ (( \begin{matrix}ε^{2}γ_{1} < τ > & 0 \\ 0 & 0 \end{matrix} ) + (γ_{2} < τ′ >) [( \begin{matrix}ν & 0 \\ 0 & 1_{n−1} \end{matrix} )])^{−s} .

Here F_{τ}(z) = F \binom{τ 0}{0 z} ∈ S_{n−1}^{k} and χ_{j}^{k,s}(γ_{j}, τ) = j(γ_{j}, τ)^{−k}|j(γ_{j}, τ)|^{−2s}.

Proof: It is convenient to rewrite the Eisenstein series in the following manner

E_{n+1,n}^{k} (( \begin{matrix}τ & 0 \\ 0 & τ′ \end{matrix} ) , F; s) = δ(τ)^{s} δ(τ′)^{s} ∑_{g∈P_{n+1,n}\Γ_{n+1}} χ_{j}^{k,s}(g, ( \begin{matrix}τ & 0 \\ 0 & τ′ \end{matrix} ))

× F (g < ( \begin{matrix}τ & 0 \\ 0 & τ′ \end{matrix} ) >_{∗}) δ (g < ( \begin{matrix}τ & 0 \\ 0 & τ′ \end{matrix} ) >_{∗})^{−s} ,

because χ_{j}^{k,s} satisfies the same cocyle relation as the factor of automorphy j. Then it is easy to see that the subseries E_{1}^{k}(τ, s) F (τ′) corresponds to R_{1}^{ℤ} . If n = 1, then R_{2}^{ℤ} leads to E_{1}^{k}(τ′)F (τ) and to E_{n,n−1}^{k}(τ′, F_{τ}; s) otherwise, because the \binom{0 1}{1 0} block of π_{n} interchanges τ and τ′. To interpret R_{3}^{ℤ} it suffices to look at

(( \begin{matrix}\begin{matrix}∗ & ∗ \\ ε & ν \end{matrix} & 0 \\ 0 & 1_{n−1} \end{matrix} ) ( \begin{matrix}τ̃ & 0 \\ 0 & \begin{matrix}z̃_{1} & z̃_{2} \\ z̃_{2}^{t} & z̃_{4} \end{matrix} \end{matrix} ) ( \begin{matrix}\begin{matrix}∗ & ε \\ ∗ & ν \end{matrix} & 0 \\ 0 & 1_{n−1} \end{matrix} ))_{∗} = ( \begin{matrix}ε^{2}τ + ν^{2}z̃_{1} & νz̃_{2} \\ νz̃_{2}^{t} & z̃_{4} \end{matrix} ) .

Remark 2.4 (1) The case n = 1 is essentially due to Garrett [Ga87].

(2) For n > 1 a new phenomenon shows up, namely the occurrence of diagonal-restricted cusp forms. This will play no role in this paper.

(3) If we put U_{+} =\{\binom{∗ ∗}{c d} ∈ Γ_{∞}\Γ | c, d > 0\}, then we can essentially describe R_{3}^{ℤ} by

⨆_{u∈U_{+}} (( \begin{matrix}u & 0 \\ 0 & u^{τ} \end{matrix} ) × 1_{2(n−1)}) (P_{1,0}\Γ × P_{n,n−1}\Γ_{n}) .

We are now in a good position. The pullback formula in Theorem 2.3 can easily be generalized to get pullback formulas of type (n_{1}, …, n_{l}, 1; n). It is further possible to calculate Φ_{F} (F_{1}, …, F_{l+1}; s) in an explicit way (This vein will be developed in a forthcoming paper). To convince the reader, we present here the case (1, 1, 1; 2).

Corollary 2.5 Let k ∈ 2ℤ and F ∈ S_{2}^{k}. Then for s ∈ ℂ with Re(s) > 6 we have

E_{3,2}^{k} ([z_{1}, z_{2}, z_{3}], F; s)

= ∑_{(a,b,c)∈P} E_{1}^{k}(z_{a}; s)F ( \begin{matrix}z_{b} & 0 \\ 0 & z_{c} \end{matrix} )

+2 δ(z_{a})^{s}δ(z_{b})^{s} ∑_{u∈U_{+}} ∑_{γ_{1},γ_{2}∈Γ_{∞}\Γ} χ_{j}^{k,s}(γ_{1}, z_{a}) χ_{j}^{k,s}(γ_{1}, z_{b})

×F_{z_{c}} (( \begin{matrix}γ_{1} < z_{a} > & 0 \\ 0 & γ_{2} < z_{b} > \end{matrix} ) [u^{t}]_{∗}) δ (( \begin{matrix}γ_{1} < z_{a} > & 0 \\ 0 & γ_{2} < z_{b} > \end{matrix} ) [u^{t}]_{∗})^{−s}

+2 δ(z_{1})^{s}δ(z_{2})^{s}δ(z_{3})^{s} ∑_{u_{1},u_{2}∈U_{+}} ∑_{γ_{1},γ_{2},γ_{3}∈Γ_{∞}\Γ} ∏_{i=1}^{3} χ_{j}^{k,s}(γ_{i}, z_{i})

×F (( \begin{matrix}γ_{1} < z_{1} > & 0 & 0 \\ 0 & γ_{2} < z_{2} > & 0 \\ 0 & 0 & γ_{3} < z_{3} > \end{matrix} ) [( \begin{matrix}1 & 0 \\ 0 & u_{1}^{t} \end{matrix} ) ( \begin{matrix}u_{2}^{t} & 0 \\ 0 & 1 \end{matrix} )]_{∗})

×δ (( \begin{matrix}γ_{1} < z_{1} > & 0 & 0 \\ 0 & γ_{2} < z_{2} > & 0 \\ 0 & 0 & γ_{3} < z_{3} > \end{matrix} ) [( \begin{matrix}1 & 0 \\ 0 & u_{1}^{t} \end{matrix} ) ( \begin{matrix}u_{2}^{t} & 0 \\ 0 & 1 \end{matrix} )]_{∗})^{−s} ,

with P = {(1, 2, 3), (3, 1, 2), (2, 3, 1)}.

2.3 The integral representation

We calculate the integral of a diagonal-restricted Eisenstein series of type (1, n, n) against three cusp forms. To sharpen our result, we recall some fundamental principles on Jacobi forms (see [E-Z85],[Kl89] or section 3.4). Let F ∈ S_{n}^{k}, then one denotes by

F ( \begin{matrix}τ & z \\ z^{t} & τ′ \end{matrix} ) = ∑_{N=1}^{∞} Φ_{N}^{F}(τ, z) e\{Nτ′\} (τ′ ∈ IH)  (23)

the Fourier-Jacobi expansion of F with respect to τ′. The Fourier-Jacobi coefficients Φ_{N}^{F}(τ, z) are Jacobi forms J_{n−1,1}^{k}(N) of degree (n − 1, 1), weight k and index N. By ⟨; ⟩_{J} = ⟨; ⟩_{J,N} we denote the Petersson scalar product for such spaces. Further we put

U_{l} : \{ \begin{matrix}J_{m,1}^{k}(N) & ⟶ & J_{m,1}(Nl^{2}) \\ Φ_{N}^{F}(τ, z) & ↦ & (Φ_{N}^{F}|U_{l})(τ, z) = Φ_{N}^{F}(τ, lz). \end{matrix}  (24)

In the case n = 1, we interpret the Fourier-Jacobi coefficients as the Fourier coefficients, the map U_{l} as the identity map and ⟨; ⟩_{J} as the standard scalar product in ℂ^{2}.

Theorem 2.6 Let k be an even number, F, G ∈ S_{n}^{k} and h ∈ S_{1}^{k} with totally real Fourier coefficients. Then for s ∈ ℂ with k + 2 Re(s) > 2n + 2

⟨⟨E_{n+1,n}^{k} (( \begin{matrix}W & 0 \\ 0 & Z \end{matrix} ) , F; s); G(Z)⟩; h(W)⟩  (25)

= 2 (4π)^{−(2k+s−n−1)} \frac{Γ(s + 2k − n − 1)Γ(s + k − n)Γ(s + k − 1)}{Γ(2k − n − 1 + 2s)}

× ∑_{{N,ε,l=1 \atop (ε,l)=1}}^{∞}⟨Φ_{N}^{F}|U_{l}; Φ_{Nl^{2}}^{G}⟩_{J} a_{Nε^{2}}^{h} ε^{−2(k+s−1)}l^{−2(k+s−n)}N^{−(2k+s−n−1)}.

Proof: To calculate (25), we insert the pullback formula developed in Theorem 2.3. After interchanging the integration and summation, we get by “unwinding ”

⟨⟨E_{n+1,n}^{k} (( \begin{matrix}W & 0 \\ 0 & Z \end{matrix} ) , F; s); G(Z)⟩; h(W)⟩

= 2 ∫_{Γ_{∞}\IH} d^{∗}W ∫_{P_{n,n−1}\IH_{n}} d^{∗}Z ∑_{{ε,l=1 \atop (ε,l)=1}}^{∞} F (( \begin{matrix}ε^{2}W & 0 \\ 0 & 0 \end{matrix} ) + Z [( \begin{matrix}l & 0 \\ 0 & 1_{n−1} \end{matrix} )]) (26)

× δ (( \begin{matrix}ε^{2}W & 0 \\ 0 & 0 \end{matrix} ) + Z [( \begin{matrix}l & 0 \\ 0 & 1_{n−1} \end{matrix} )])^{−s} δ(W)^{k+s}δ(Z)^{k+s}\overline{G(Z)h(W)}.

To use the notation in [Kl89] we rewrite this as

2 ∫_{Γ_{∞}\IH} d^{∗}W ∫_{C_{n,n−1}\IH_{n}}d^{∗}Z ∑_{{ε,l=1 \atop (ε,l)=1}}^{∞} F (( \begin{matrix}0 & 0 \\ 0 & ε^{2}W \end{matrix} ) + Z [( \begin{matrix}1_{n−1} & 0 \\ 0 & l \end{matrix} )])

×δ (( \begin{matrix}0 & 0 \\ 0 & ε^{2}W \end{matrix} ) + Z [( \begin{matrix}1_{n−1} & 0 \\ 0 & l \end{matrix} )])^{−s} δ(W)^{k+s}δ(Z)^{k+s}\overline{G(Z)h(W)}.

If we denote by B_{n−1,1} a fundamental domain of the action of the group C_{n,n−1}^{+}/Z^{n,n−1} on IH_{n−1} × ℂ^{n−1}, here we denote by Z^{n,n−1} the center of C_{n,n−1}^{+}, then

Q_{n,1} = \{Z = ( \begin{matrix}τ & z \\ z^{t} & τ′ \end{matrix} ) ∈ IH_{n} | (τ, z) ∈ B_{n−1,1} and 0 ≤ x_{τ′} < 1\}  (27)

is a fundamental domain of C_{n,n−1}^{+}\IH_{n} ([Kl89]). This leads to

⟨⟨E_{n+1,n}^{k} (( \begin{matrix}W & 0 \\ 0 & Z \end{matrix} ) , F; s); G(Z)⟩; h(W)⟩

= 2 ∑_{{ε,l=1 \atop (ε,l)=1}}^{∞} ∫_{B_{n−1,1}}dτdz ∫_{y_{τ′}>y_{τ}^{−1}[y_{z}]} dy_{τ′} ∫_{0}^{1}dx_{τ′} ∫_{0}^{1}dx_{W} ∫_{0}^{∞}dy_{W} δ(Z)^{k+s−(n+1)}y_{W}^{k+s−2}

× ∑_{N=1}^{∞} Φ_{N}^{F}(τ, lz)e\{N(ε^{2}W + l^{2}τ′)\}\overline{∑_{M=1}^{∞} a_{M}^{h}e\{MW \}}\overline{∑_{L=1}^{∞} Φ_{L}^{G}(τ, z)e\{Lτ′\}}

× δ (( \begin{matrix}0 & 0 \\ 0 & ε^{2}W \end{matrix} ) + Z[( \begin{matrix}1 & 0 \\ 0 & l \end{matrix} )])^{−s}

= 2 ∫_{B_{n−1,1}} dτdz ∑_{{N,ε,l=1 \atop (ε,l)=1}}^{∞} Φ_{N}^{F}(τ, lz)\overline{Φ_{Nl^{2}}^{G}(τ, z)}\overline{a_{Nε^{2}}^{h}} ∫_{y_{τ′}>y_{τ}^{−1}[y_{z}]}dy_{τ′} ∫_{0}^{∞}dy_{W}

δ (( \begin{matrix}0 & 0 \\ 0 & ε^{2}W \end{matrix} ) + Z[( \begin{matrix}1 & 0 \\ 0 & l \end{matrix} )])^{−s} δ(Z)^{k+s−(n+1)} y_{W}^{k+s−2} e^{−4πN(ε^{2}y_{W} +l^{2}y_{τ′})}.

2 A FAMILY OF DIRICHLET SERIES 19 In the last step, we have integrated over the real parts of τ′ and W. It remains to calculate the integration over dy_{W} and dy_{τ′}. We have

∫_{y_{τ′}>y_{τ}^{−1}[y_{z}]}dy_{τ′} ∫_{0}^{∞}dy_{W} δ( \begin{matrix}y_{τ} & ly_{z} \\ ly_{z}^{t} & ε^{2}W + l^{2}y_{τ′} \end{matrix} )^{−s} δ ( \begin{matrix}y_{τ} & y_{z} \\ y_{z}^{t} & y_{τ′} \end{matrix} )^{k+s−(n+1)}

y_{W}^{k+s−2} e^{−4πN(ε^{2}y_{W} +l^{2}y_{τ′})}

=^{(1)} ∫_{0}^{∞}dt ∫_{0}^{∞}dy_{W} δ ( \begin{matrix}y_{τ} & ly_{z} \\ ly_{z}^{t} & ε^{2}y_{W} + l^{2}(t + y_{τ}^{−1}[y_{z}]) \end{matrix} )^{−s} δ ( \begin{matrix}y_{τ} & y_{z} \\ y_{z}^{t} & t + y_{τ}^{−1}[y_{z}] \end{matrix} )^{k+s−n−1}

y_{W}^{k+s−2} e^{−4πNε^{2}y_{W}} e^{−4πNl^{2}t}e^{−4πNl^{2}y_{τ}^{−1}[y_{z}]}

= ∫_{0}^{∞}dt ∫_{0}^{∞}dy_{W} δ ( \begin{matrix}y_{τ} & ly_{z} \\ ly_{z}^{t} & ε^{2}y_{W} + l^{2}(t + y_{τ}^{−1}[y_{z}]) \end{matrix} )^{−s} δ ( \begin{matrix}y_{τ} & y_{z} \\ y_{z}^{t} & l^{2}t + l^{2}y_{τ}^{−1}[y_{z}] \end{matrix} )^{k+s−n−1}

l^{−2(k+s−(n+1))} y_{W}^{k+s−2} e^{−4πNε^{2}y_{W}} e^{−4πNl^{2}t}e^{−4πNl^{2}y_{τ}^{−1}[y_{z}]}

=^{(2)} ∫_{0}^{∞}dt ∫_{0}^{∞}dy_{W} t^{k+s−(n+1)}(ε^{2}y_{W} + l^{2}t)^{−s}δ(y_{τ})^{k−(n+1)} y_{W}^{k+s−2}

e^{−4πNε^{2}y_{W}} e^{−4πNl^{2}t}e^{−4πNl^{2}y_{τ}^{−1}[y_{z}]}. In (1) we have substituted y_{τ′} by t + y_{τ}^{−1}[y_{z}] and in (2) we have used

δ ( \begin{matrix}y_{τ} & ly_{z} \\ ly_{z}^{t} & ε^{2}y_{W} + l^{2}(t + y_{τ}^{−1}[y_{z}]) \end{matrix} )^{−s} δ ( \begin{matrix}y_{τ} & ly_{z} \\ ly_{z}^{t} & l^{2}t + l^{2}y_{τ}^{−1}[y_{z}] \end{matrix} )^{k+s−n−1}

= (l^{2}t)^{k+s−(n+1)}(ε^{2}y_{W} + l^{2}t)^{−s}δ(y_{τ})^{k−(n+1)}

Recall that the invariant measure for the Petersson scalar product in J_{n−1,1}^{k}(m) is

δ(y_{τ})^{k−((n−1)+2)}e^{−4πmy_{τ}^{−1}[y_{z}]}dτdz.  (28)

So it remains only to calculate

∫_{0}^{∞} \frac{dt}{t} ∫_{0}^{∞} \frac{dy_{W}}{y_{W}} t^{k+s−n}y_{W}^{k+s−1}(ε^{2}y_{W} + l^{2}t)^{−s}e^{−4πNε^{2}y_{W}} e^{−4πNl^{2}t},

which is an Eulerian integral of first kind. Applying the formula

∫_{0}^{∞} ∫_{0}^{∞} \frac{dx}{x} \frac{dy}{y} x^{α}y^{β} (\frac{xy}{x + y} )^{γ} e^{−(x+y)} = \frac{Γ(α + β + γ) Γ(α + γ) Γ(β + γ)}{Γ(α + β + 2γ)}

gives the desired result.

2.4 First results

The Dirichlet series we obtained in the last subsection we now denote by

D_{h,G;F} (s) = ∑_{{N,ε,l=1 \atop (ε,l)=1}}^{∞}⟨Φ_{N}^{F}|U_{l}; Φ_{Nl^{2}}^{G}⟩_{J} a_{Nε^{2}}^{h} ε^{−2(k+s−1)}l^{−2(k+s−n)}N^{−(2k+s−n−1)}. (29)

Combining results of the Theorem 2.6 with the Theorem of Garrett and Böcherer and Theorem 1.9 gives the following first résumé.

Theorem 2.7 Let F, G ∈ S_{n}^{k}, h ∈ S_{1}^{k} Hecke eigenforms with totally real Fourier coefficients. Let s ∈ ℂ with 2 Re(s) + k > 2n + 2 and k even. Then we have

Φ(h, G, F; s) = a(n, s, k) L_{h,G,F}^{n} (s + 2k − n − 1) with  (30)

L_{h,G,F}^{n} (s + 2k − n − 1) = {ζ(2s + k) ∏_{j=1}^{n} ζ(4s + 2k − 2j)}^{−1}
× D_{F}^{n} (2s + k − n) D_{h,G;F} (s)

and a(n, s, k) = a^{∗}(n, s, k) Γ^{∗}(n, s, k) with

a^{∗}(n, s, k) = 2^{−2s(n+1)−nk−4k+3+ \frac{n^{2}+7n}{2}} (−1)^{\frac{nk}{2}} 2π^{\frac{n^{2}+3n}{2} −s−2k+1},
Γ^{∗}(n, s, k) = \frac{Γ(s + 2k − 1 − n)Γ(s + k − n)Γ(s + k − 1)}{Γ(2k − n − 1 + 2s)} \frac{Γ_{n}(k + s − \frac{n+1}{2})}{Γ_{n}(k + s)}.

Corollary 2.8 With the same assumptions as above, we have:

(i) L_{h,G,F}^{n} has meromorphic continuation to ℂ and satisfies the functional equation

L_{h,G,F}^{n,∗} (s) = L_{h,G,F}^{n,∗} (3k − n − 1 − s),  (31)

where

L_{h,G,F}^{n,∗} (s) = ϵ(2n + 1, s + n + 1 − 3/2 k) a(n, n + 1 + s − 2k, k) L_{h,G,F}^{n} (s).

(ii) Let k > 2n, h ∈ S_{1}^{k}(ℚ(h)), G ∈ S_{2}^{k}(ℚ(G)) and F ∈ S_{2}^{k}(ℚ(F)). Then we have for all σ ∈ Gal(\overline{ℚ}/ℚ)

(\frac{a(n, 0, k)L_{h,G,F}^{n} (2k − n − 1)}{⟨h; h⟩⟨G; G⟩⟨F; F ⟩})^{σ} = (\frac{a(n, 0, k)L_{h^{σ},G^{σ},F^{σ}}^{(n)}(2k − n − 1)}{⟨h^{σ}; h^{σ}⟩⟨G^{σ}; G^{σ}⟩⟨F^{σ}; F^{σ}⟩}) ∈ \overline{ℚ}.

The restriction k > 2n can be weakened. The assumption totally real is only of technical nature. It can easily be removed (see [Sh76], [Boe85a]). All of the results stated so far have only been known for n = 1. In that case, P. Garrett [Ga87] has shown that L_{h,G,F}^{1} is essentially the triple product L-function for three elliptic eigenforms h, G, F ∈ S_{1}^{k}. So there is some hope that L_{h,G,F}^{n} has something to do with Euler products in general. The rest of this paper is devoted to show that one really gets an Euler product in the case of n = 2, if one chooses the cusp forms in an appropriate way.

3 Hecke-Jacobi theory

In this section, a certain Hecke-Jacobi theory ([An77b],[An87],[Gr82],[Gr93]) is developed, which serves mainly as an aid for studying the arithmetic properties of the Dirichlet series D_{h,G;F} (s) introduced in (29). We proceed in a self-contained way to make the underlying structure as transparent as possible and to fix a good notation. This enables us to prove new results which are essential in getting an Euler product for the Dirichlet series.

3.1 The Jacobi group G_{R}^{J}

For R = ℤ, ℤ[\frac{1}{p}] or ℚ, we define the Heisenberg group H_{R}, the group Γ_{R} and the Jacobi group G_{R}^{J}. Let

H_{R} = {(X; κ)| X ∈ R × R, κ ∈ R}

with group law (X_{1}; κ_{1}) ∘ (X_{2}; κ_{2}) = (X_{1} + X_{2}; κ_{1} + κ_{2} + det \binom{X_{1}}{X_{2}} ), further let Γ_{R} be the direct product of the groups G^{+}Sp_{1}(R) and R^{+}, and

G_{R}^{J} = \{M = ( \begin{matrix}a & 0 & b & μ′ \\ λ & α & μ & κ \\ c & 0 & d & −λ′ \\ 0 & 0 & 0 & β \end{matrix} ) | \begin{matrix}\binom{a b}{c d} ∈ G^{+}Sp_{1}(R), β > 0 \\ αβ = μ(M) = μ\binom{a b}{c d} and \\ (λ, μ; κ) ∈ R^{3} with β^{−1}(λ′, μ′)\binom{a b}{c d} = (λ, μ) \end{matrix} \}

a subgroup of G^{+}Sp_{2}(R). The exact sequence

1 ⟶ H_{R} ⟶^{i} G_{R}^{J} ⟶^{p} Γ_{R} ⟶ 1  (32)

with i(λ, μ; κ) = ( \begin{matrix}1 & 0 & 0 & μ \\ λ & 1 & μ & κ \\ 0 & 0 & 1 & −λ \\ 0 & 0 & 0 & 1 \end{matrix} ) and p ( \begin{matrix}a & 0 & b & μ′ \\ λ & α & μ & κ \\ c & 0 & d & −λ′ \\ 0 & 0 & 0 & β \end{matrix} ) = (\binom{a b}{c d}, β)

splits by

q : \{ \begin{matrix}Γ_{R} & ⟶ & G_{R}^{J} \\ (( \begin{matrix}a & b \\ c & d \end{matrix} ) , β) & ↦ & ( \begin{matrix}a & b \\ c & d \end{matrix} ) × ( \begin{matrix}∗ & 0 \\ 0 & β \end{matrix} ) \end{matrix} .

That is the same as to say that G_{R}^{J} can be seen as a semi-direct product of H_{R} and Γ_{R}. If γ = (M, β), then ρ_{γ} given by ρ_{γ}(X; κ) = (αXM^{−1}; \frac{α}{β}κ) is an automorphism of H_{R}. The semi-direct product H_{R} ⋊ Γ_{R} is the set H_{R} × Γ_{R} with the group law

(h, γ) (h̃, γ̃) := (hρ_{γ}(h̃), γγ̃).  (33)

Proposition 3.1 The groups H_{R} ⋊ Γ_{R} and G_{R}^{J} are isomorphic by

ϕ : \{ \begin{matrix}H_{R} ⋊ Γ_{R} & ⟶ & G_{R}^{J} \\ ((x_{1}, x_{2}; κ); (( \begin{matrix}a & b \\ c & d \end{matrix} ) , β)) & ↦ & ( \begin{matrix}a & 0 & b & βx_{2} \\ y_{1} & α & y_{2} & βκ \\ c & 0 & d & −βx_{1} \\ 0 & 0 & 0 & β \end{matrix} ) \end{matrix} .  (34)

It is convenient not to distinguish between image and pre-image of the embeddings i, q and ϕ. Then the group law in G_{R}^{J} becomes

((X; κ); (M; β)) ∘ ((X̃; κ̃); (M̃, β̃)
= ((X + αX̃M^{−1}; κ + \frac{α}{β}κ̃ + det ( \begin{matrix}X \\ αX̃M^{−1} \end{matrix} ) ); (MM̃, ββ̃))

and for h = (X; κ) ∈ H_{R} and γ = (M, β) ∈ Γ_{R} we have

(h; γ) = h γ = γ (α^{−1}XM; \frac{β}{α}κ).

3.2 Hecke-Jacobi algebra

In the seventies, Andrianov (e.g. [An77b]) recognized that it is possible to factorize the Hecke and Rankin polynomials in a convenient way, if one embeds the Hecke algebra H^{n} in a Hecke algebra belonging to the Siegel parabolic C_{n,0} of Sp_{n}. This embedding has the advantage that one can see the action of H^{n} on M_{n}^{k} directly on the Fourier coefficients of the underlying modular form. So it is rather obvious that one should exchange, in the proceeding of Andrianov, the Siegel parabolic by another maximal parabolic, if one is interested in analyzing the arithmetic properties of the Fourier-Jacobi coefficients of a Siegel eigenform. This has been done by Gritsenko ([Gr82],[Gr88],[Gr95]) in some special cases. We want to recall some of his results by stressing the underlying general structures already developed by Andrianov and modify and generalize them for our purposes.

Let us now introduce a Hecke theory to the Hecke pair

(Γ^{J} := C_{2,1}^{+}(ℤ), G^{J} := G_{ℚ}^{J}) .

That this is really a Hecke pair can be proven by a slight modification of the proof of Prop 1.1 in [S-Z89]. We call the corresponding Hecke algebra H^{J} the Hecke-Jacobi algebra and the local Hecke algebra H_{p}^{J} associated to the pair (Γ^{J}, G_{p}^{J} = G_{ℤ[p^{−1}]}^{J}) the local Hecke-Jacobi algebra. It is an important question how a double coset decomposes into left cosets. The following result gives a first answer.

Lemma 3.2 a) Let r be a positive rational number and g ∈ G^{J}. Then

Γ^{J}([r, r; r, r]) Γ^{J} = Γ^{J}([r, r; r, r]) and
Γ^{J}g Γ^{J} ∘ Γ^{J}([r, r; r, r]) Γ^{J} = Γ^{J}g([r, r; r, r]) Γ^{J}.

b) Let a, d, α, β be natural numbers with a | d and ad = αβ. Then

⨆_{j} (( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) , β) ⨆_{λ mod a/(a, α) μ mod d/(d, α) κ mod β/(β, α)} (λ, μ; κ) (U_{j} × 1_{2})  (35)

is a representative system of Γ^{J}\Γ^{J}(\binom{a 0}{0 d}, β)Γ^{J}. Here U_{j} runs through a representative system of (\binom{a 0}{0 d}^{−1}Γ\binom{a 0}{0 d} ⋂ Γ)\Γ.

The first condition is clear and the second can easily be proven (cf. [Gr82], Lemma 3.1).

By Lemma 1.1, it is easy to see that

i : H^{2} ↪ H^{J}; ∑_{j} a_{j}Γ_{2} g_{j} ↦ ∑_{j} a_{j}Γ^{J}g_{j}

gives an embedding if one chooses g_{j} ∈ Γ^{J}. This map commutes with the anti-automorphism ∗ of Γ^{J} obtained by

Γ^{J} (h; (M, β)) Γ^{J} ↦ (Γ^{J} (h; (M, β)) Γ^{J})^{∗} = Γ^{J}μ(M) (h; (M, β))^{−1} Γ^{J}.  (36)

With (35) in Lemma 3.2, one sees that

j_{−} : \{ \begin{matrix}H^{1} & ⟶ & H^{J} \\ Γ ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) Γ & ↦ & Γ^{J} (( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ); 1) Γ^{J} \end{matrix}  (37)

j_{+} : \{ \begin{matrix}H^{1} & ⟶ & H^{J} \\ Γ ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) Γ & ↦ & Γ^{J} (( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ); ad) Γ^{J} \end{matrix}  (38)

lead to two algebra monomorphisms and that j_{+}(X) = (j_{−}(X))^{∗}, if X ∈ H^{1}. We denote the images of these maps by H_{−}^{J} and H_{+}^{J}. One has (H_{−}^{J})^{∗} = H_{+}^{J} and all elements of H^{2} are fixed by ∗. By these constructions we have distinguished certain subalgebras H_{−}^{J}, H_{+}^{J} and i(H^{2}) of H^{J}, which are commutative, have no zero divisors and decompose in their local components.

Next we want to introduce some important elements of H^{J}. Let r ∈ ℚ and n ∈ ℕ:

T_{−}^{g}(n) := j_{−}(T (n)),  T_{+}^{g}(n) := j_{+}(T (n)),
Λ_{−}^{g}(n) := j_{−}(T (n, n)),  Λ_{+}^{g}(n) := j_{+}(T (n, n)),
∇(r) := Γ^{J} ((0; r); (1_{2}, 1)) Γ^{J}, Δ_{n} := Γ^{J}[n, n; n, n]Γ^{J},
∇_{n}^{g} := Δ_{n} ∑_{b mod n} ∇(\frac{b}{n}),  Ξ_{n}^{g} := Δ_{n} ∑_{λ,μ,κ mod n} Γ^{J}(\frac{λ}{n}, \frac{μ}{n}; \frac{κ}{n}),

T^{J,g}(n) := Γ^{J}[1, n; n^{2}, n]Γ^{J}.

For T (a, b) ∈ H^{1} with a, b ∈ ℕ, we put T_{−}^{g}(a, b) = Γ^{J}[a, ab; b, 1]Γ^{J} and T_{+}^{g}(a, b) = Γ^{J}[a, 1; b, ab]Γ^{J}. If ab = n^{2} with n ∈ ℕ, we put T_{0}^{g}(a, b) = Γ^{J}[a, n; b, n]Γ^{J}. The double cosets ∇(a) and Δ_{n} decompose into only one left coset and Ξ_{n}^{g}, ∇_{n}^{g}, Δ_{n} and T^{J,g}(n) are fixed by ∗. Furthermore, we have (T_{−}^{g}(n))^{∗}= T_{+}^{g}(n) and (Λ_{−}^{g}(n))^{∗} = Λ_{+}^{g}(n). Further we put ∇_{n}^{r} = (Δ_{n})^{−1}∇_{n}^{g}, Ξ_{n}^{r} = (Δ_{n})^{−1}Ξ_{n}^{g} and T^{J,r}(n) = (Δ_{n})^{−1}T^{J,g}(n) (as a rule, we do this only when it is very convenient). By abuse of notation we put Δ_{n} for Γ_{2}[n, n; n, n]Γ_{2} and Γ^{J}[n, n; n, n]Γ^{J}.

Remark 3.3 (1) The Hecke-Jacobi algebra H^{J} is not commutative and has zero divisors, because Λ_{−}^{g}(p) · (∇_{p}^{r} − p) = 0 and Λ_{−}^{g}(p)T_{+}^{g}(p) ≠ T_{+}^{g}(p)Λ_{−}^{g}(p).

(2) It is not clear how one can decompose H^{J} into its local components. Let for example p, q be prime numbers, then

∇(\frac{1}{q})∇(\frac{1}{p}) = ∇(\frac{p + q}{pq}),

and if we assume that p^{2} ≢ 1(q) we have Λ_{−}^{g}(p) ∇(\frac{1}{q}) ≠ ∇(\frac{1}{q}) Λ_{−}^{g}(p).

The following proposition indicates how one can get a local-global result.

Proposition 3.4 The elements T_{−}^{g}, T_{+}^{g}, T^{J,g}, ∇^{g}, Ξ^{g}, Λ_{−}^{g}, Λ_{+}^{g} and Δ in H^{J} commute with each other when we only allow coprime arguments. For Λ_{−}^{g}, Λ_{+}^{g} and Δ, the assumption on the coprimeness can be dropped.

Proof: Case by case. To indicate how one can do that, we prove that

T_{−}^{g}(n)T_{+}^{g}(m) = T_{+}^{g}(m)T_{−}^{g}(n) for (n, m) = 1.

Since H^{1} = ⊗_{p}H_{p}^{1} and H_{p}^{1} is generated by T (p), T (p, p) and T (p, p)^{−1}, it is enough to show that T_{−}^{g}(p)T_{+}^{g}(q) = T_{+}^{g}(q)T_{−}^{g}(p) for two different prime numbers p and q. But this can be deduced directly from Lemma 3.2.

Corollary 3.5 If we denote by H̃^{J} the subalgebra of H^{J} generated by T_{−}^{g}, T_{+}^{g}, T^{Jg}, ∇^{g} Ξ^{g}, Λ_{−}^{g}, Λ_{+}^{g}, Δ and if we put H̃_{p}^{J} = H_{p}^{J} ⋂ H̃^{J}, then we have

H̃^{J} = ⊗_{p}H̃_{p}^{J}.  (39)

3.3 The Hecke algebra H̃_{p}^{J}

First we want to show that H^{2} can be seen as a subalgebra of H̃^{J}. To do that, it is enough to determine the image of the generators of H_{p}^{2} under the embedding i. A slight modification of Lemma 3.3.32 in [An87] gives us

i(T^{(2)}(p)) = T_{−}^{g}(p) + T_{+}^{g}(p)
i(T_{1}^{(2)}(p^{2})) = Λ_{−}^{g}(p) + Λ_{+}^{g}(p) + T^{J,g}(p) + ∇_{p}^{g} − Δ_{p}
i(T_{0}^{(2)}(p^{2})) = T_{−}^{g}(1, p^{2}) + T_{+}^{g}(1, p^{2}) + T^{J,g}(p)(∇_{p}^{r} − 1) + Ξ_{p}^{g} − ∇_{p}^{g}
i(Δ_{p}^{(2)}) = Δ_{p},

which proves our claim. To simplify our notation we put T_{−}^{g}, T_{+}^{g}, Λ_{−}^{g}, Λ_{+}^{g}, T^{J,g} for T_{−}^{g}(p), T_{+}^{g}(p), Λ_{−}^{g}(p), Λ_{+}^{g}(p), T^{J,g}(p), if we are in a fixed local situation. Gritsenko ([Gr95], Abschnitt 5.1) introduced the following very convenient invariants. Let X = Γ^{J}(h; (M, β))Γ^{J} ∈ H̃_{p}^{J}, then we denote by Sig(X) = \frac{β^{2}}{μ(M)} the signature of X and for

r ∈ ℚ^{×} and Y = ∑_{i} a_{i}X_{i} ∈ H̃_{p}^{J} we put Sig_{r}(Y) = ∑_{i,Sig_{(}X_{i})=r} a_{i}X_{i}. For M_{1}, M_{2} ∈ G^{J} we have

Sig (Γ^{J}M_{1}Γ^{J} ∘ Γ^{J}M_{2}Γ^{J}) = Sig (Γ^{J}M_{1}Γ^{J}) · Sig (Γ^{J}M_{2}Γ^{J})  (40)

and for X = ∑_{u} Sig_{u}(X) and X = ∑_{v} Sig_{v}(X) we have

Sig_{r}(XY) = ∑_{uv=r} Sig_{u}(X)Sig_{v}(Y)  (41)
= ∑_{uv=r} Sig_{v}(Y)Sig_{u}(X).  (42)

Applying this for example to the case X = T^{(2)}(p) and Y = T_{1}^{(2)}(p^{2}), leads us to

T_{+}^{g}Λ_{+}^{g} = Λ_{+}^{g}T_{+}^{g}
T_{−}^{g}Λ_{+}^{g} + T_{+}^{g}T^{J,g} + T_{+}^{g}(∇_{p}^{g} + Δ_{p}) = Λ_{+}^{g}T_{−}^{g} + T^{J,g}T_{+}^{g} + (∇_{p}^{g} + Δ_{p})T_{+}^{g}
T_{+}^{g}Λ_{−}^{g} + T_{−}^{g}T^{J,g} + T_{−}^{g}(∇_{p}^{g} + Δ_{p}) = Λ_{−}^{g}T_{+}^{g} + T^{J,g}T_{−}^{g} + (∇_{p}^{g} + Δ_{p})T_{−}^{g}
T_{−}^{g}Λ_{−}^{g} = Λ_{−}^{g}T_{−}^{g}.

Such identities can be used to determine commutators in H̃_{p}^{J}. Next, we want to simplify terms of the form T_{−}^{g}Λ_{+}^{g}, Λ_{−}^{g}Λ_{+}^{g} and so on.

Lemma 3.6 The following relations in H̃_{p}^{J} hold:

T_{−}^{g}Λ_{+}^{g} = p^{2}Δ_{p}T_{+}^{g}  (43)
Λ_{−}^{g}T_{+}^{g} = p^{2}Δ_{p}T_{−}^{g}  (44)
Λ_{−}^{g}Λ_{+}^{g} = p^{4}(Δ_{p})^{2}  (45)
T_{−}^{g}T_{+}^{g} = pT^{J,g} + (p^{2} + p^{3})Δ_{p}.  (46)

Proof: By Lemma 3.2, we have

T_{−}^{g} = Γ^{J}[1, p; p, 1] (U_{[1,p]} × 1_{2})
Λ_{−}^{g} = Γ^{J}[p, p^{2}; p, 1]
T_{+}^{g} = Γ^{J}[1, 1; p, p] ∑_{μ,κ mod p}(0, μ; κ) (U_{[1,p]} × 1_{2})
Λ_{+}^{g} = Γ^{J}[p, 1; p, p^{2}] ∑_{{λ,μ mod p \atop κ mod p^{2}}}(λ, μ; κ).

This leads to

T_{−}^{g}Λ_{+}^{g} = p^{−2}Γ^{J}[p, p; p^{2}, p^{2}] (U_{[1,p]} × 1_{2}) ∑_{λ,μ,κ mod p^{2}}(λ, μ; κ)  (47)
= p^{−2}Δ_{p}Γ^{J}[1, 1; p, p] ∑_{λ,μ,κ mod p^{2}}(λ, μ; κ) (U_{[1,p]} × 1_{2})  (48)

which proves (43). Here we used that GL_{2}(ℤ) normalizes the Heisenberg group H_{ℤ/p^{2}ℤ}. If we apply the anti-automorphism ∗ to equation (43) we get (44).

It is easy to see that

Λ_{−}^{g}Λ_{+}^{g} = Γ^{J} [p, p^{2}; p, 1][p, 1; p, p^{2}] ∑_{{λ,μ mod p \atop κ mod p^{2}}}(λ, μ; κ)

= Γ^{J}[p^{2}, p^{2}; p^{2}, p^{2}] ∑_{{λ,μ mod p \atop κ mod p^{2}}}(λ, μ; κ)

= p^{4}(Δ_{p})^{2}.

One can prove the last assumption of the lemma as follows:

T_{−}^{g}T_{+}^{g} = (Γ^{J}[1, p; p, 1](U_{[1,p]} × 1_{2}))(Γ^{J}[1, 1; p, p] ∑_{μ,κ mod p}(0, μ; κ)(U_{[1,p]} × 1_{2}))
= p^{−1}Γ^{J} ((T^{(1)}(p) · T^{(1)}(p)) × [p; p]) ∑_{λ,μ,κ mod p}(0, μ; κ)
= p^{−1}Γ^{J} ((T^{(1)}(1, p^{2}) + T^{(1)}(p, p) + pT^{(1)}(p, p)) × [p; p]) ∑_{μ,κ mod p}(0, μ; κ)

= p^{−1}Γ^{J} (([1, p^{2}]U_{[1,p^{2}]}) × [p; p]) ( ∑_{μ,κ mod p}(0, μ; κ))
+(p^{2} + p^{3})Δ_{p}

which gives the desired result.

We hope to have given the reader an impression of how one can prove such statements. The following list contains relations between the generators of H̃_{p}^{J}. At the (i,j) position we have put the product of the corresponding row and column elements. With the subscript we want to indicate, which kind of relation is listed in the second chart. In the left column the position, in the middle one the relation and in the right eventually a reference is given. The number in the reference indicates another equation, which is the same as the given one. If the number comes with ∗, one has to apply ∗ to the corresponding equation. Finally, 15^{m} indicates a connection with equation 15.

\begin{matrix}∘ & T_{−}^{g} & Λ_{−}^{g} & T_{+}^{g} & Λ_{+}^{g} & T^{J,g} & ∇_{p}^{g} & Ξ_{p}^{g} \\ T_{−}^{g} & 11_{F} & 12_{F} & 13^{√} & 14^{√} & 15_{[ ]} & 16^{√} & 17_{U} \\ Λ_{−}^{g} & 21_{F} & 22_{F} & 23^{√} & 24^{√} & 25^{√} & 26^{√} & 27^{√} \\ T_{+}^{g} & 31_{U} & 32_{U} & 33_{F} & 34_{F} & 35_{U} & 36_{?} & 37_{?} \\ Λ_{+}^{g} & 41_{U} & 42^{√} & 43_{F} & 44_{F} & 45_{?} & 46_{?} & 47_{?} \\ T^{J,g} & 51_{[]} & 52_{?} & 53_{[]} & 54^{√} & 55_{F} & 56_{?} & 57_{?} \\ ∇_{p}^{g} & 61_{?} & 62_{?} & 63^{√} & 64^{√} & 65_{?} & 66_{F} & 67^{√} \\ Ξ_{p}^{g} & 71_{?} & 72_{?} & 73_{U} & 74^{√} & 75_{?} & 76^{√} & 77_{U} \end{matrix} √ simplification F formula U easy transformation [ ] essentially the commutator ? no real simplification (49)

11 T_{−}^{g}T_{−}^{g} = T_{−}(p^{2}) + pΛ_{−}^{g}
12 T_{−}^{g}Λ_{−}^{g} = T_{−}^{g}(p, p^{2})
13 T_{−}^{g}T_{+}^{g} = pT^{J,g} + (p^{2} + p^{3})Δ_{p}
14 T_{−}^{g}Λ_{+}^{g} = p^{2}Δ_{p}T_{+}^{g}
15 T_{−}^{g}T^{J,g} = p(p − 1)Δ_{p}T_{−}^{g} + ∇_{p}^{g}T_{−}^{g} − T_{+}^{g}Λ_{−}^{g} + T^{J,g}T_{−}^{g}
16 T_{−}^{g}∇_{p}^{g} = pT_{−}^{g}Δ_{p}^{g}
17 T_{−}^{g}Ξ_{p}^{g} = p^{2}(Δ_{p} − p^{−1}∇_{p}^{g})T_{−}^{g} + pT_{+}^{g}Λ_{−}^{g}
21 Λ_{−}^{g}T_{−}^{g} = T_{−}^{g}(p, p^{2})  12
22 Λ_{−}^{g}Λ_{−}^{g} = Λ_{−}^{g}(p^{2})
23 Λ_{−}^{g}T_{+}^{g} = p^{2}Δ_{p}T_{−}^{g}  14^{∗}
24 Λ_{−}^{g}Λ_{+}^{g} = p^{4}(Δ_{p})^{2}
25 Λ_{−}^{g}T^{J,g} = pΔ_{p}T_{−}^{g}(1, p^{2})
26 Λ_{−}^{g}∇_{p}^{g} = pΔ_{p}Λ_{−}^{g}
27 Λ_{−}^{g}Ξ_{p}^{g} = p^{3}Δ_{p}Λ_{−}^{g}
31 T_{+}^{g}T_{−}^{g} = Δ_{p}^{−1}T^{J,g}∇_{p}^{g} + p∇^{g} + Ξ_{p}^{g}
32 T_{+}^{g}Λ_{−}^{g} = p^{−1}T_{−}^{g}Ξ_{p}^{g} − (pΔ_{p} − ∇_{p}^{g})T_{−}^{g}
33 T_{+}^{g}T_{+}^{g} = T_{+}^{g}(p^{2}) + pΛ_{+}^{g}  11^{∗}
34 T_{+}^{g}Λ_{+}^{g} = T_{+}^{g}(p, p^{2})  12^{∗}
35 T_{+}^{g}T^{J,g} = p(1 − p)Δ_{p}T_{+}^{g} − T_{+}^{g}∇_{p}^{g} + Λ_{+}^{g}T_{−}^{g} + T^{J,g}T_{+}^{g} 15^{m}
36 T_{+}^{g}∇_{p}^{g} = − − −
37 T_{+}^{g}Ξ_{p}^{g} = − − −
41 Λ_{+}^{g}T_{−}^{g} = p^{−1}Ξ_{p}^{g}T_{+}^{g} − T_{+}^{g}(pΔ_{p} − ∇_{p}^{g})  32^{∗}
42 Λ_{+}^{g}Λ_{−}^{g} = p^{−1}Δ_{p}^{−1}∇_{p^{2}}^{g}Ξ_{p}^{g}
43 Λ_{+}^{g}T_{+}^{g} = T_{+}^{g}(p, p^{2})  34
44 Λ_{+}^{g}Λ_{+}^{g} = Λ_{+}^{g}(p^{2})  22^{∗}
45 Λ_{+}^{g}T^{J,g} = − − −
46 Λ_{+}^{g}∇_{p}^{g} = − − −
47 Λ_{+}^{g}Ξ_{p}^{g} = − − −

51 T^{J,g}T_{−}^{g} = T_{+}^{g}Λ_{−}^{g} − p(p − 1)Δ_{p}T_{−}^{g} − ∇_{p}^{g}T_{−}^{g} + T_{−}^{g}T^{J,g}  35^{∗}
52 T^{J,g}Λ_{−}^{g} = − − −  45^{∗}
53 T^{J,g}T_{+}^{g} = p(p − 1)Δ_{p}T_{+}^{g}∇_{p}^{g} − Λ_{+}^{g}T_{−}^{G} + T_{+}^{g}T^{J,g}  15^{∗}
54 T^{J,g}Λ_{+}^{g} = pΔ_{p}T_{+}^{g}(1, p^{2})  25^{∗}
55 T^{J,g}T^{J,g} = T^{J,g}(p^{2}) + (p − 1)∇_{p}^{g}T^{J,g}(p) + pΔ_{p}(Ξ_{p}^{g} − ∇_{p}^{g} + p^{2} + p)
56 T^{J,g}∇_{p}^{g} = − − −
57 T^{J,g}Ξ_{p}^{g} = − − −
61 ∇_{p}^{g}T_{−}^{g} = − − − 36^{∗}
62 ∇_{p}^{g}Λ_{−}^{g} = − − −  46^{∗}
63 ∇_{p}^{g}T_{+}^{g} = pΔ_{p}T_{+}^{g} 16^{∗}
64 ∇_{p}^{g}Λ_{+}^{g} = pΔ_{p}Λ_{+}^{g}  26^{∗}
65 ∇_{p}^{g}T^{J,g} = − − −  56
66  ∇_{p}^{g}∇_{p}^{g} = ∇_{p^{2}}^{g}
67  ∇_{p}^{g}Ξ_{p}^{g} = pΔ_{p}Ξ_{p}^{g}
71  Ξ_{p}^{g}T_{−}^{g} = − − −  37^{∗}
72  Ξ_{p}^{g}Λ_{−}^{g} = − − −  47^{∗}
73  Ξ_{p}^{g}T_{+}^{g} = p^{2}T_{+}^{g}(Δ_{p} − p^{−1}∇_{p}^{g}) + pΛ_{+}^{g}T_{−}^{g}  17^{∗}
74  Ξ_{p}^{g}Λ_{+}^{g} = p^{3}Δ_{p}Λ_{+}^{g}  27^{∗}
75 Ξ_{p}^{g}T^{J,g} = − − −  57^{∗}
76 Ξ_{p}^{g}∇_{p}^{g} = pΔ_{p}Ξ_{p}^{g} 67^{∗}
77  Ξ_{p}^{g}Ξ_{p}^{g} = pΔ_{p}^{−1}Ξ_{p}^{g}∇_{p^{2}}^{g} + p^{3}Δ_{p}∇_{p}^{g} − p^{2}∇_{p^{2}}^{g}

Now we are able to state and prove the main results of this section, the announced factorization of the Hecke and Rankin polynomials q_{p}^{2}(X) and d_{p}^{2}(X). It is tricky to construct such decompositions, in contrast it is easy to prove the correctness of them. A good overview over this topic is given in [An87], §3.5. For a proof of the next theorem the reader is referred to [Gr93].

Theorem 3.7 (Gritsenko) The Hecke polynomial q_{p}^{2}(X) has the following factorization in H̃_{p}^{J}[X]:

q_{p}^{2}(X) = j_{−} (q_{p}^{1}(X)) (1 + pΔ_{p}(∇_{p}^{r} − p)X^{2}) j_{+} (q_{p}^{1}(X))

with q_{p}^{1}(X) = 1 − T^{(1)}(p)X + pΔ_{p}^{(1)}X^{2}.

Next, we want to present a new factorization of the Rankin polynomial. It is an unpleasant fact, that the suggested formula in [Gr88], Theorem 3, is not correct.

Theorem 3.8 The Rankin polynomial d_{p}^{2}(X) has the following factorization in H̃_{p}^{J}[X]:

d_{p}^{2}(X) = (1 − X) (1 − p^{−2}Λ_{−}^{r}X) S^{(2)}(X) (1 − p^{−2}Λ_{+}^{r}X)  (50)

with S^{(2)}(X) = ∑_{j=0}^{3}(−1)^{j}S_{j}^{(2)}X^{j}. Here

S_{0}^{(2)} = 1
S_{1}^{(2)} = p^{−2} (T^{J,r} + ∇_{p}^{r} − p^{2})
S_{2}^{(2)} = p^{−3} (T^{J,r}(∇_{p}^{r} − p) + Ξ^{r} − p∇_{p}^{r} + p^{2})
S_{3}^{(2)} = p^{−2} (∇_{p}^{r} − p) .

Proof: Most of the work has already been done. We multiply the right side of the equation (50) out and get a polynomial of degree 5. With the aid of our charts, the stated formulas given in section 1.2 and at the beginning of section 3.3, we see that this polynomial is nothing else but the Rankin polynomial of degree 4.

3.4 Jacobi forms as P-forms

We begin with a very convenient definition.

Definition 3.9 Let n, s and k be natural numbers. A P-Jacobi form of degree (n, s) and weight k is a holomorphic function F on IH_{n+s}, which satisfies

(i) F |_{k}γ = F for all γ ∈ C_{n+s,n}^{+}(ℤ)

(ii) There exists t ∈ A_{s} such that F has the Fourier expansion

F (Z) = ∑_{T =\binom{∗ ∗}{∗ t}∈A_{n+s}} a(T) e\{T Z\} .  (51)

We call such t the index of F. A P-Jacobi form is called P-Jacobi cusp form, (if the index is positive definite and) if the summation in (51) may be restricted to positive definite matrices. The space which is generated by all P-Jacobi forms (P-Jacobi cusp forms) of degree (n, s) and weight k is denoted by the space of P-forms PJ_{n,s}^{k} (space of P-cusp forms PJ_{n,s}^{k,cusp}). Their elements are called P-forms and P-cusp forms respectively. Finally we denote by PJ_{n,s}^{k}(t) the subspace of P-Jacobi forms and by PJ_{n,s}^{k,cusp}(t) of P-Jacobi cusp forms for fixed index t. We use the convention PJ_{n,s}^{k,cusp}(t) = {0}, if t is not half-integral.

To introduce the Petersson scalar product for P-forms, we put ℂ^{n,s} = Mat(n, s; ℂ), IH_{n,s} = IH_{n} × ℂ^{n,s}, Γ_{n,s}^{J} = C_{n+s,n}^{+}(ℤ) and Γ_{n,s}^{J,red} = Γ_{n,s}^{J}/Z^{n,s}, where Z^{n,s} denotes the center of Γ_{n,s}^{J}. Let γ ∈ Γ_{n,s}^{J}, then

γ < ( \begin{matrix}z_{1} & z_{2} \\ ∗ & ∗ \end{matrix} ) >= ( \begin{matrix}z′_{1} & z′_{2} \\ ∗ & ∗ \end{matrix} )

induces the operation (z_{1}, z_{2}) ⟼ γ < z_{1}, z_{2} >= (z′_{1}, z′_{2}) of Γ_{n,s}^{J} on IH_{n,s}, which is in fact an action of Γ_{n,s}^{J,red}. Then

Q_{n,s} = \{Z ∈ IH_{n+s}| (z_{1}, z_{2}) ∈ B_{n,s} and |Re(z_{4})_{ij}| ≤ \frac{1}{2}\}

is a fundamental domain of the action of Γ_{n,s}^{J} on IH_{n+s}, if B_{n,s} is a fundamental domain of the action of Γ_{n,s}^{J,red} on IH_{n,s}. Every F ∈ PJ_{n,s}^{k}(t) has a decomposition of the form F (Z) = ϕ^{F} (z_{1}, z_{2}) e{tz_{4}}.

Definition 3.10 Let F, G ∈ PJ_{n,s}^{k}. Then we denote by the value of

⟨F, G⟩_{A} = ∫_{Q_{n,s}} F (Z)\overline{G(Z)} (detY)^{k} d^{∗}Z  (52)

the Petersson scalar product of F and G for d^{∗}Z = dXdY/( dety)^{n+s+1}, if it exists.

Remark 3.11

(i) The space of Jacobi forms of degree (n, s), weight k and index t is given by

J_{n,s}^{k}(t) = {ϕ : IH_{n} × ℂ^{n,s} ⟶ ℂ | ϕ̃_{t}(Z) := ϕ(z_{1}, z_{2}) e{tz_{4}} ∈ PJ _{n,s}^{k}}. (53)

In the same way one defines the space of Jacobi cusp forms J_{n,s}^{k,cusp}(t) of index t, and more generally the spaces J_{n,s}^{k} and J_{n,s}^{k,cusp}.

(ii) Let F, G ∈ PJ_{n,s}^{k,cusp}(t) with k > n + s and y_{i} =Im(z_{i}). Then

⟨F, G⟩_{A} = β_{s,k−n} (dett)^{\frac{s+1}{2} +n−k}
∫_{B_{n,s}} ϕ^{F} (z_{1}, z_{2}) \overline{ϕ^{G}(z_{1}, z_{2})}e^{−4πtr(ty_{1}^{−1}[y_{2}])}( dety_{1})^{k} d^{∗}Z_{n,s} .

Here β_{s,k−n} = π^{\frac{s(s−1)}{4}} (4π)^{\frac{s(s+1)}{2} −s(k−n)} ∏_{ν=1}^{s} Γ(k − n − \frac{s+ν}{2} ) and

d^{∗}Z_{n,s} = dz_{1} dz_{2}/( dety_{1})^{n+s+1} (cf. Klingen [Kl89] Satz 3, Dulinski [Du95]).

Let ϕ, ψ ∈ J_{n,s}^{k,cusp}(t) and k > n + s, then we get the Petersson scalar product ⟨ϕ, ψ⟩_{J,t} for Jacobi forms by

⟨ϕ̃; ψ̃⟩_{A} = β_{s,k−n} (det t)^{\frac{s+1}{2} +n−k}⟨ϕ, ψ⟩_{J,t} .

Let us fix in the case s = n = 1 the following notation.

For Z = ( \begin{matrix}τ & z \\ z & τ′ \end{matrix} ) ∈ IH_{2}, we put ( \begin{matrix}τ_{x} & z_{x} \\ z_{x} & τ′_{x} \end{matrix} ) = Re(Z) and ( \begin{matrix}τ_{y} & z_{y} \\ z_{y} & τ′_{y} \end{matrix} ) = Im(Z).

Let F, G ∈ PJ_{1,1}^{k,cusp}(t), k > 2 and β_{k} = (4π)^{k−2}Γ(k − 2)^{−1}, then

⟨ϕ^{F} , ϕ^{G}⟩_{J,t} = β_{k}t^{k−2}⟨F, G⟩_{A}.  (54)

If F : IH_{2} ⟶ ℂ is invariant under the |_{k}-action of Γ^{J}, then for X = ∑ a_{j}Γ^{J}g_{j} ∈ H̃^{J}

F |_{k}D(X) := ∑_{j} a_{j} F |_{k}g_{j}  (55)

has the same property. Let F (Z) = ∑_{n=1}^{∞} f_{n}(τ, z) e{nτ′} ∈ S_{2}^{k} and X ∈ H^{2}, then (55) is compatible with the action introduced in (13) and the embedding i. More generally, let X ∈ H̃^{J}, then we put f_{m}∥_{k}X for the m-th Fourier-Jacobi coefficient of F |_{k}D(X) resp., τ′. Especially the action in (55) can be seen as an action of H̃^{J} on PJ_{1,1}^{k,cusp} (J_{1,1}^{k,cusp}). To be more precise, if X ∈ H̃^{J} is a double coset with signature r ∈ ℚ^{+} and F ∈ PJ_{1,1}^{k,cusp}(t), then F |_{k}D(X) ∈ PJ_{1,1}^{k,cusp}(tr). By Proposition 4.1.7, page 218 [An87], it is clear that

⟨F |_{k}D(X); G⟩_{A} = ⟨F; G|_{k}D(X^{∗})⟩_{A},  (56)

with X ∈ H̃^{J} and F, G ∈ PJ_{1,1}^{k,cusp}. Further, if ϕ̃ ∈ PJ_{1,1}^{k,cusp}(t) and (p, t) = 1, then ϕ̃|_{k}D(X) = 0, if X = ∇_{n}^{g}, Ξ_{n}^{g}, T_{+}^{g}(n) or Λ_{+}^{g}(n).

Finally, we are able to compute the adjoint operators of the classical Hecke-Jacobi operators (e.g. U_{l}, V_{l}, T_{l}) introduced in [E-Z85], §4 in a rather simple way. For T_{p}, we put T_{p}^{EZ}. Let A be a linear operator on J_{1,1}^{k,cusp}, then we denote by Ã the corresponding operator on PJ_{1,1}^{k,cusp}, further by A^{ad} resp., (Ã)^{ad} the adjoint operators. Assuming that Ã has signature r, we have Ã^{ad} = r^{k−2}(Ã)^{ad}. Then the dictionary

\begin{aligned}D(Λ_{−}^{g}(l)) & = & l^{3(k−2)}Ũ_{l} \\ D(T_{−}^{g}(l)) & = & l^{k−2}Ṽ_{l} \\ D(T^{J,g}(p)) & = & p^{3−k}T̃_{p}^{EZ} \end{aligned} leads to \begin{aligned}Ũ_{l}^{ad} & = & l^{2−k}D(Λ_{+}^{g}(l)) \\ Ṽ_{l}^{ad} & = & D(T_{+}^{g}(l)) \\ { \atop (T_{p}^{EZ})^{ad}} & = & T̃_{p}^{EZ} \end{aligned} .  (57)

The following notation and observation will turn out to be convenient later on. If we put

S^{(2)}(X)^{factor} = 1 − p^{−2}T^{J,r}(p)X + p^{−1}X^{2}  (58)
S^{(2)}(X)^{prim} = 1 − (p^{−2}T^{J,r}(p) − 1)X + (p^{−1} − p^{−2})T^{J,r}(p)X^{2} + p^{−1}X^{3} (59)
= S^{(2)}(X)^{factor} (1 + X)
S^{(2)}(X)^{n.prim} = 1 − (p^{−2}T^{J,r}(p) + p^{−1} − 1)X + p^{−3}Ξ_{p}^{r}X^{2},  (60)

then D(S^{(2)}(X)) = D(S^{(2)}(X)^{prim}) on PJ_{1,1}^{k,cusp}(t), if (p, t) = 1 and D(S^{(2)}(X)) = D(S^{(2)}(X)^{n.prim}) otherwise. Further we have for all F ∈ PJ_{1,1}^{k,cusp}(p):

F |_{k}D(S^{(2)}(X)T_{+}^{g}(p)) = F |_{k}D(T_{+}^{g}(p) (1 − p^{−2}T^{J,r}(p)X + p^{−1}X^{2})).  (61)

3.5 Weak rationality theorems

Let F ∈ S_{2}^{k} be a Hecke eigenform and (f_{n})_{n∈ℕ} the Fourier-Jacobi coefficients of F. Further let Q_{p,F}^{2} and D_{p,F}^{2} the corresponding Hecke and Rankin polynomials. Then the following identities hold in the ring of formal power series.

Proposition 3.12 (Gritsenko) Let F ∈ S_{2}^{k} be a Hecke eigenform and m ∈ ℕ, then

Q_{p,F}^{2} (X) ∑_{δ=0}^{∞} f_{mp^{δ}}|_{k}D (T_{+}^{g}(p^{δ})X^{δ})

= (f_{m} − f\frac{m}{p} |_{k}D(T_{−}^{g}(p)X) + pf_{\frac{m}{p^{2}}} |_{k}D(Λ_{−}^{g}(p)X^{2}))|_{k}D(1 + p(∇_{p}^{r} − p)Δ_{p}X^{2})

with f_{m}|_{k}D(1 + p(∇_{p}^{r} − 1)Δ_{p}X^{2}) = \{{f_{m} if m≡0(p) \atop (1−p^{2k−4}X^{2})f_{m} otherwise} .

Proposition 3.13 Let F ∈ S_{2}^{k} be a Hecke eigenform and m ∈ ℕ, then

D_{p,F}^{2} (X) ∑_{δ=0}^{∞} f_{mp^{2δ}}|_{k}D (Λ_{+}^{r}(p^{δ}) (p^{−2}X)^{δ})

= (1 − X) (f_{m}|_{k}D(S^{(2)}(X)) − f_{\frac{m}{p^{2}}} |_{k}D(Λ_{−}^{r}(p) S^{(2)}(X)p^{−2}X)) .

Theorem 3.7, Theorem 3.8 and results from our charts give the desired results. Also a proof of the Proposition 3.12 is given in [Gr93], the proof of the second one proceeds in the same way. It may seem that the two propositions are of the same type, but that is not the case. Let for example (m, p) = 1: then one can describe f_{mp^{δ}}|_{k}D(T_{+}^{g}(p^{δ})) in terms of eigenvalues and the function f_{m}, but this is not the case for f_{mp^{2δ}}|_{k}D(Λ_{+}^{g}(p^{δ})), where one needs to know more about the nature of f_{m}|_{k}D(T^{J,g}(p)).

3.6 The Maass Spezialschar

There is a correspondence between certain elliptic modular forms, Jacobi forms and Siegel modular forms of degree 2, which can be described in the language of L-functions. To do that, we recall the definition of a certain subspace of S_{2}^{k}.

Definition 3.14 The Maass Spezialschar S_{2}^{k,M} of weight k is the space

{F (Z) = ∑_{l=1}^{∞} l^{2−k} (ϕ̃_{1}|_{k}D(T_{−}^{g}(l)) (Z) | ϕ̃_{1} ∈ PJ_{1,1}^{k,cusp}(1)} .  (62)

It is well known that S_{2}^{k,M} is a subspace of S_{2}^{k} and that it is stable under the action of H^{2}. For simplicity we have omitted reference to Eisenstein series. But much more is known ([E-Z85] §6): Theorem 3.15 (Saito-Kurokawa Conjecture) The space S_{2}^{k,M} is spanned by Hecke eigenforms. These are in 1-1 correspondence with normalized Hecke eigenforms f ∈ S_{1}^{2k−2}, the correspondence being such that

Z_{F}^{2} (s) = ζ(s − k + 1)ζ(s − k + 2)L_{f}(s).  (63)

The degeneration of the spinor zeta function leads to the degeneration of the standard zeta function. Theorem 3.16 Let us choose F and f as in Theorem 3.15. Then

D_{F}^{2} (s) = ζ(s)L_{f}(s + k − 1)L_{f}(s + k − 2).  (64)

Proof: To prove the latter theorem one has to determine the Satake-parameters α_{0,p}, α_{1,p}, α_{2,p}, that is to say, one has to determine the tuple (α_{1,p}, α_{1,p}^{−1}, α_{2,p}, α_{2,p}^{−1}), which is unique up to permutations of the entries. If α_{p} and β_{p} denote the p-parameters of f, then it turns out, that one can choose α_{1,p} = α_{p}p^{1−k} and α_{2,p} = β_{p}p^{1−k}, which leads to the desired result.

Remark 3.17 There exists a distinguished basis (F_{j})_{j} of S_{2}^{k,M} with corresponding f_{j} ∈ S_{1}^{2k−2}(ℚ(F_{j})). Let σ ∈ Gal(\overline{ℚ}/ℚ), then f_{j}^{σ} corresponds to F_{j}^{σ}. This remark, the explicit form of D_{F}^{2} and Corollary 1.12 lead to Proposition 3.18 Let F ∈ S_{2}^{k,M}(ℚ(F)) be a Hecke eigenform with corresponding f ∈ S_{1}^{2k−2} and k > 4 even, then for all σ ∈ Gal(\overline{ℚ}/ℚ) we have

(\frac{π^{7−4k}L_{f}(2k − 3) L_{f}(2k − 4)}{⟨F; F ⟩})^{σ} = \frac{π^{7−4k}L_{f^{σ}}(2k − 3) L_{f^{σ}}(2k − 4)}{⟨F^{σ}; F^{σ}⟩} ∈ ℚ(F). (65)

4 Euler product of a Dirichlet series of type (1,2,2)

From now on let F, G ∈ S_{2}^{k}, h ∈ S_{1}^{k} Hecke eigenforms with h normalized and F in the Maass Spezialschar with corresponding f ∈ S_{1}^{2k−2}. We show that

D_{h,G;F} (s) = ∑_{{n,ε,l=1 \atop (ε,l)=1}}^{∞}⟨Φ_{n}^{F} |_{k}U_{l}; Φ_{nl^{2}}^{G}⟩_{J} a_{nε^{2}}^{h} ε^{−2(k+s−1)}l^{−2(k+s−2)}n^{−(2k+s−3)}  (66)

has an Euler product by using the language of P-forms and the Hecke-Jacobi theory developed in the last section.

4.1 Preparation of the Dirichlet series

First interpreting the Fourier-Jacobi coefficients of F and G as P-forms, then using our definition of the Maass Spezialschar and the translation rules in (57), leads to the formula

D_{h,G;F} (s) = β_{k} ∑_{{n,ε,l=1 \atop (ε,l)=1}}^{∞}⟨Φ̃_{1}^{F} |_{k}D(T_{−}^{g}(n) Λ_{−}^{g}(l)); Φ̃_{nl^{2}}^{G}⟩_{A} a_{nε^{2}}^{h}  (67)

ε^{−2(k+s−1)}l^{−3k−2s+6}n^{−(2k+s−3)}.

Next we put X = p^{−(s+k−1)}, N = p^{k−1} and define

D_{h,G;F}^{(p)} (s) = ∑_{{n,ε,l=0 \atop min(ε,l)=0}}^{∞}⟨Φ̃_{1}^{F} |_{k}D (T_{−}^{g}(p^{n}) Λ_{−}^{g}(p^{l})); Φ̃_{p^{n+2l}}^{G}⟩_{A} a_{p^{n+2ε}}^{h}

X^{2ε} (X^{2}N^{−1}p^{3})^{l} (pXN^{−1})^{n},  (68)

which will later lead to D_{h,G;F} (s) = β_{k}⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A} ∏_{p}(D_{h,G;F}^{(p)} (s)/⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}). Our first goal is to calculate D^{(p)} as explicitly as possible. To do that, we decompose D^{(p)} by the condition min(ε, l) = 0 into three sums:

D_{h,G;F}^{(p)} (s) = D_{(ε)}(s) + D_{(l)}(s) − D_{(ε,l)}(s).  (69)

Let α and α′ be the p-parameters of h determined by α + α′ = a_{p}^{h} and αα′ = N. Then

D_{(ε)}(s) (α − α′) = α ∑_{n,l=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{n+2l}}^{G}|_{k}D(T_{+}^{g}(p^{n})Λ_{+}^{g}(p^{l}))⟩_{A}(X^{2}N^{−1}p^{3})^{l} (αpXN^{−1})^{n}

−α′ ∑_{n,l=0}^{∞}⟨Φ̃_{1}^{F};Φ̃_{p^{n+2l}}^{G}|_{k}D(T_{+}^{g}(p^{n})Λ_{+}^{g}(p^{l}))⟩_{A}(X^{2}N^{−1}p^{3})^{l} (α′pXN^{−1})^{n}

D_{(l)}(s) (α − α′) = α ∑_{n,ε=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{n}}^{G}|_{k}D(T_{+}^{g}(p^{n}))⟩_{A} (αX)^{2ε} (αpXN^{−1})^{n}

−α′ ∑_{n,ε=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{n}}^{G}|_{k}D(T_{+}^{g}(p^{n}))⟩_{A} (α′X)^{2ε} (α′pXN^{−1})^{n}

D_{(ε,l)}(s) (α − α′) = α ∑_{n=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{n}}^{G}|_{k}D(T_{+}^{g}(p^{n}))⟩_{A} (αpXN^{−1})^{n}

−α′ ∑_{n=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{n}}^{G}|_{k}D(T_{+}^{g}(p^{n}))⟩_{A} (α′pXN^{−1})^{n}.

4.2 Local calculations

Using Proposition 3.12, we get the following two lemmas.

Lemma 4.1 Let X_{1} = αpXN^{−1} and X′_{1} = α′pXN^{−1}, then

D_{(l)} − D_{(ε,l)} = \frac{⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}}{α − α′} (\frac{α^{3}X^{2}}{Q_{p,G}^{2}(X_{1})} − \frac{α′^{3}X^{2}}{Q_{p,G}^{2}(X′_{1})}) .  (70)

Lemma 4.2 Let Y = pNX^{2} and l a non-negative rational integer, then

∑_{n=0}^{∞} Φ̃_{p^{n+2l}}^{G}|_{k}D(T_{+}^{g}(p^{n})Λ_{+}^{g}(p^{l})) X_{1}^{n}(X^{2}N^{−1}p^{3})^{l}  (71)
= Q_{p,G}^{2}(X_{1})^{−1}( {Φ̃_{p^{2l}}^{G} − Φ̃_{p^{2l−1}}^{G}|_{k}D(T_{−}^{g}(p)X_{1}) + pΦ̃_{p^{2l−2}}^{G}|_{k}D(Λ_{−}^{g}(p)X_{1}^{2})}
|_{k}D((1 + p(∇_{p}^{r} − p)Δ_{p}X_{1}^{2})Λ_{+}^{r}(p^{l}) (p^{−2}Y)^{l}).  (72)

Notice that the eigenvalue of Δ_{p}p^{5}N^{−1}X^{2} is Y and that X, Y are independent of h, in contrast to X_{1} and X′_{1}. For simplicity, let us use T_{−}, T_{+}, Λ_{−} and Λ_{+} for T_{−}(p), T_{+}(p), Λ_{−}(p) and Λ_{+}(p). To calculate D_{(ε)}, we have to analyze the infinite sums over l corresponding to Φ̃_{p^{2l}}^{G}, Φ̃_{p^{2l−1}}^{G}, Φ̃_{p^{2l−2}}^{G} as suggested in the last lemma.

Proposition 4.3 We have

α Q_{p,G}^{2}(X_{1})^{−1} ∑_{l=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{2l}}^{G}|_{k}D ((1 + p(∇_{p}^{r} − p)Δ_{p}X_{1}^{2})Λ_{+}^{r}(p^{l}))⟩_{A} (p^{−2}Y)^{l}

−α′ Q_{p,G}^{2}(X′_{1})^{−1} ∑_{l=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{2l}}^{G}|_{k}D ((1 + p(∇_{p}^{r} − p)Δ_{p}(X′_{1})^{2})Λ_{+}^{r}(p^{l}))⟩_{A} (p^{−2}Y)^{l}

= (αQ_{p,G}^{2}(X_{1})^{−1} − α′Q_{p,G}^{2}(X′_{1})^{−1}) \frac{S_{F}^{(2)}(Y)^{prim}⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}}{D_{p,G}^{2,∗}(Y)}

+ (\frac{α′^{3}X^{2}}{Q_{p,G}^{2}(X′_{1})} − \frac{α^{3}X^{2}}{Q_{p,G}^{2}(X_{1})}) ⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}  (73)

with S_{F}^{(2)}(Y)^{prim}Φ̃_{1}^{F} = Φ̃_{1}^{F} |_{k}D(S^{(2)}(Y)^{prim}) and D_{p,G}^{2,∗}(Y) = \frac{D_{p,G}^{2}(Y)}{1−Y}.

Proof: Let X_{∗} be X_{1} or X′_{1}, then

∑_{1}^{G}(X_{∗}) := ∑_{l=0}^{∞} Φ̃_{p^{2l}}^{G}|_{k}D ((1 + p(∇_{p}^{r} − p)Δ_{p}X_{∗}^{2})Λ_{+}^{r}(p^{l})) (p^{−2}Y)^{l}

= ∑_{l=0}^{∞} Φ̃_{p^{2l}}^{G}|_{k}D(Λ_{+}^{r}(p^{l})) (p^{−2}Y)^{l} − p^{2}Φ̃_{1}^{G}|_{k}D(Δ_{p}X_{∗}^{2}).

Here we used the peculiarity of l = 0. Applying Proposition 3.13 leads to

∑_{1}^{G}(X_{∗}) = (Φ̃_{1}^{G}|_{k}D(S^{(2)}(Y)) D_{p,G}^{2,∗}(Y)^{−1} − (α_{∗})^{2}X^{2}Φ̃_{1}^{G},

where α_{∗} is adapted to the meaning of X_{∗}, that means α_{∗} = α, if X_{∗} = X_{1} and α′ otherwise. The property that Φ̃_{1}^{F} is an eigenform of H̃_{0}^{J} (the subalgebra of all elements of H̃^{J} with signature one) and that S^{(2)}(Y)^{prim} is hermitian proves the proposition.

We want to point out the cancellations caused by Lemma 4.1 and Proposition 4.3. By using again Proposition 3.13 and the already calculated relations between the generators of H̃_{p}^{J}, we get

Proposition 4.4 We have

αQ_{p,G}^{2}(X_{1})^{−1} ∑_{l=0}^{∞} p⟨Φ̃_{1}^{F}; Φ̃_{p^{2l−2}}^{G}|_{k}D ((Λ_{−}^{g}X_{1}^{2})(1 + p(∇_{p}^{r} − p)Δ_{p}X_{1}^{2})(Λ_{+}^{r}p^{−2}Y)^{l})⟩_{A}

−α′Q_{p,G}^{2}(X′_{1})^{−1} ∑_{l=0}^{∞} p⟨Φ̃_{1}^{F}; Φ̃_{p^{2l−2}}^{G}|_{k}D ((Λ_{−}^{g}X′_{1}^{2})(1 + p(∇_{p}^{r} − p)Δ_{p}X′_{1}^{2})(Λ_{+}^{r}p^{−2}Y)^{l})⟩_{A}

= (\frac{α^{3}Np^{2}X^{4}}{Q_{p,G}^{2}(X_{1})} − \frac{α′^{3}Np^{2}X^{4}}{Q_{p,G}^{2}(X′_{1})} ) (\frac{S_{F}^{(2)}(Y)^{prim}⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}}{D_{G}^{2,∗}(Y)} ) .

To calculate the sum corresponding to Φ̃_{p^{2l−1}}^{G} is the main point here. Of course the specialness of F is essential.

Proposition 4.5 Let λ_{p}^{F,J} be defined by Φ̃_{1}^{F} |_{k} D(T^{J,r}(p)) = λ_{p}^{F,J} Φ̃_{1}^{F}. Then

α Q_{p,G}^{2}(X_{1})^{−1} ∑_{l=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{2l−1}}^{G}|_{k}D ((T_{−}^{g}X_{1})(1 + p(∇_{p}^{r} − p)Δ_{p}X_{1}^{2})Λ_{+}^{r}(p^{l}) (p^{−2}Y)^{l})⟩_{A}

−α′ Q_{p,G}^{2}(X′_{1})^{−1} ∑_{l=0}^{∞}⟨Φ̃_{1}^{F}; Φ̃_{p^{2l−1}}^{G}|_{k}D ((T_{−}^{g}X′_{1})(1 + p(∇_{p}^{r} − p)Δ_{p}(X′_{1})^{2})(Λ_{+}^{r}p^{−2}Y)^{l})⟩_{A}

= (α^{2}p^{2}X^{3}Q_{p,G}^{2}(X_{1})^{−1} − (α′)^{2}p^{2}X^{3}Q_{p,G}^{2}(X′_{1})^{−1})
λ_{p}^{G}(1 − p^{−2}λ_{p}^{F,J}Y + p^{−1}Y^{2})⟨Φ̃_{1}^{F} , Φ̃_{1}^{G}⟩_{A} (D_{p,G}^{2,∗}(Y))^{−1} .

Proof: \underline{First step} We start with the calculation of

∑_{3}^{G}(X_{∗}) := ∑_{l=0}^{∞} Φ̃_{p^{2l−1}}^{G}|_{k}D ((T_{−}^{g}X_{∗})(1 + p(∇_{p}^{r} − p)Δ_{p}X_{∗}^{2})Λ_{+}^{r}(p^{l}) (p^{−2}Y)^{l}) . (74)

By T_{−}^{g}Λ_{+}^{r} = p^{2}T_{+}^{g}, we have

∑_{3}^{G}(X_{∗}) = ∑_{l=1}^{∞} Φ̃_{p^{2l−1}}^{G}|_{k}D (Λ_{+}^{r}(p^{l−1}) (p^{−2}Y)^{l−1}Y T_{+}X_{∗})

= ∑_{l=0}^{∞} Φ̃_{p^{2l+1}}^{G}|_{k}D (Λ_{+}^{r}(p^{l}) (p^{−2}Y)^{l}Y X_{∗}T_{+}^{g})
= D_{p,G}^{2,∗}(Y)^{−1}(Φ̃_{p}^{G}|_{k}D(S^{(2)}(Y)T_{+}^{g}) Y X_{∗}.

Here we used Proposition 3.13 and that the terms corresponding to Λ_{−}^{r} provide no contribution. With the identity given in (61), we have

∑_{3}^{G}(X_{∗}) = α_{∗}p^{2}X^{3} D_{p,G}^{2,∗}(Y)^{−1} Φ̃_{p}^{G}|_{k}D(T_{+}^{g}(p)S^{(2)}(Y)^{prim}(1 + Y)^{−1}).  (75)

\underline{Second step}

The P-form Φ̃_{1}^{F} is an eigenfunction of H̃_{0}^{J} and S^{(2)}(Y)^{prim} is self adjoint. This gives us

⟨Φ̃_{1}^{F}; Φ̃_{p}^{G}|_{k}D (T_{+}^{g}S^{(2)}(Y)^{prim})⟩_{A}/(1 + Y)
= ⟨Φ̃_{1}^{F} |_{k}D(1 − p^{−2}T^{J,r}(p)Y + p^{−1}Y^{2}); Φ̃_{p}^{G}|_{k}D(T_{+}^{g})⟩_{A} (1 + Y)^{−1}
= (1 − p^{−2}λ_{p}^{F,J}Y + p^{−1}Y^{2}) ⟨Φ̃_{1}^{F}; Φ̃_{p}^{G}|_{k}D(T_{+}^{g})⟩_{A} (1 + Y)^{−1}
= λ_{p}^{G} (1 − p^{−2}λ_{p}^{F,J}Y + p^{−1}Y^{2}) ⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}.

Here Φ̃_{1}^{F} |_{k}D(T^{J,r}(p)) = λ_{p}^{F,J} Φ̃_{1}^{F} and Φ̃_{p}^{G}|_{k}D(T_{+}^{g}) = λ_{p}^{G}Φ̃_{1}^{G}. Thus the Proposition is proven.

We have proven the last lemmas and propositions in such a way that one can easily extract the following crucial fact.

Corollary 4.6 Let n′, l′, ε′ be coprime to p. Then we have

∑_{{n,ε,l=0 \atop min(ε,l)=0}}^{∞} ⟨Φ̃_{1}^{F} |_{k}D (T_{−}^{g}(n′p^{n}) Λ_{−}^{g}(l′p^{l}));Φ̃_{n′l′^{2}p^{n+2l}}^{G}⟩_{A} a_{n′(ε′)^{2}p^{n+2ε}}^{h}X^{2ε} (X^{2}N^{−1}p^{3})^{l}(pXN^{−1})^{n}

= ⟨Φ̃_{1}^{F} |_{k}D (T_{−}^{g}(n′) Λ_{−}^{g}(l′)) ; Φ̃_{n′l′^{2}}^{G}⟩_{A} a_{n′(ε′)^{2}}^{h} (D_{h,G;F}^{(p)} (s)⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}^{−1}) .

Thus we have proven the following Theorem 4.7 Let F, G ∈ S_{2}^{k} and h ∈ S_{1}^{k} Hecke eigenforms, with h normalized. Let further F be in the Maass Spezialschar. Then we have for k even and s big enough:

D_{h,G;F} (s) = β_{k}⟨Φ̃_{1}^{F} , Φ̃_{1}^{G}⟩_{A} ∏_{p}(D_{h,G;F}^{(p)} (s)/⟨Φ̃_{1}^{F} , Φ̃_{1}^{G}⟩_{A})  (76)

with D_{h,G;F}^{(p)} (s) = \frac{⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}S_{F}^{(2)}(Y)^{prim}}{(α − α′)D_{p,G}^{2,∗}(Y)}

× (\frac{α}{Q_{p,G}^{2}(X_{1})} − \frac{α′}{Q_{p,G}^{2}(X′_{1})} + \frac{α^{3}Np^{2}X^{4}}{Q_{p,G}^{2}(X_{1})}  (77)

−\frac{α′^{3}Np^{2}X^{4}}{Q_{p,G}^{2}(X′_{1})} + \frac{α′^{2}p^{2}X^{3}λ_{p}^{G}}{Q_{p,G}^{2}(X′_{1})(1 + Y)} − \frac{α^{2}p^{2}X^{3}λ_{p}^{G}}{Q_{p,G}^{2}(X_{1})(1 + Y)}) .

It is easy to see, that the expression in the big brackets in (77), is equal to

{((α + α^{3}Np^{2}X^{4})(1 + Y) − α^{2}p^{2}X^{3}λ_{p}^{G}) Q_{p,G}^{2}(X′_{1})
− ((α′ + α′^{3}Np^{2}X^{4})(1 + Y) − α′^{2}p^{2}X^{3}λ_{p}^{G}) Q_{p,G}^{2}(X_{1})}
Q_{p,G}^{2}(X_{1})^{−1}Q_{p,G}^{2}(X′_{1})^{−1}(1 + Y)^{−1}.

Inserting Y = pNX^{2}, N = p^{k−1} and X_{∗} = α_{∗}pN^{−1}X gives us

{(α + αpNX^{2} − α^{2}p^{2}X^{3}λ_{p}^{G} + α^{3}Np^{2}X^{4} + α^{3}N^{2}p^{3}X^{6}) Q_{p,G}^{2}(α′pN^{−1}X)

− (α′ + α′pNX^{2} − α′^{2}p^{2}X^{3}λ_{p}^{G} + α′^{3}Np^{2}X^{4} + α′^{3}N^{2}p^{3}X^{6}) Q_{p,G}^{2}(αpN^{−1}X)}

{Q_{p,G}^{2}(X_{1})Q_{p,G}^{2}(X′_{1})(1 + Y)}^{−1} .

If we multiply out the numerator and rearrange the terms according to the powers of y, then we get

D_{h,G;F}^{(p)} (s) = \frac{⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩_{A}S_{F}^{(2)}(Y)^{prim}}{D_{p,G}^{2,∗}(Y)} \{Q_{p,G}^{2}(X_{1})Q_{p,G}^{2}(X′_{1})(1 + Y)\}^{−1}
×(1 − y)(1 + (2pN − p^{2}N^{−1}A_{2}^{G}) Y + (2 − 2pN^{−2}A_{2}^{G} + pN^{−2}(λ_{P}^{G})^{2}) Y^{2} (78)
+ (2pN − p^{2}N^{−1}A_{2}^{G}) Y^{3} + Y^{4}).

Here A_{2}^{G} = p (λ_{T_{1}(p^{2})}^{G} + (p^{2} + 1)p^{2k−6}), with G|_{k}T_{1}^{(2)}(p^{2}) = λ_{T_{1}(p^{2})}^{G}G. The expression (78) in the last big bracket turns out to be just D_{p,G}^{2}(pNX^{2}).

Lemma 4.8 Let F ∈ S_{2}^{k,M} be a Hecke eigenform with corresponding f ∈ S_{1}^{2k−2}. Then

S_{F}^{(2)}(p^{−2s−k+2})^{factor} = L_{p,f}(p^{−(2s+2k−3)}).

5 THE L-FUNCTION Z_{G⊗H}(S) 40

Proof: We have f|_{k}T (p) = λ_{f}(p) f ⟺ Φ̃_{1}^{F} |_{k}D(T̃_{p}^{EZ}) = λ_{f}(p)Φ̃_{1}^{F} (see for example [K-S89]§2) and T_{p}^{J,r} = p^{3−k}T̃_{p}^{EZ}, which gives the desired result.

Thus we have

Theorem 4.9 Let F, G ∈ S_{2}^{k}, h ∈ S_{1}^{k} be Hecke eigenforms with totally real Fourier coefficients, h normalized and k even. Let F be in the Maass space with corresponding f ∈ S_{1}^{2k−2}. Then we have for s ∈ ℂ with k + 2Re(s) > 6 and X = p^{−(s+k−1)}:

D_{h,G;F} (s) = ⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J} × ∏_{p} \frac{S_{F}^{(2)}(pNX^{2})^{factor}(1 − pNX^{2})}{Q_{p,G}^{2}(αpN^{−1}X)Q_{p,G}^{2}(α′pN^{−1}X)}  (79)

with S_{F}^{(2)}(pNX^{2})^{factor} = L_{p,f}(pX^{2}).

5 The L-function Z_{G⊗h}(s)

In this last section we present our final result, a new integral representation of Z_{G⊗h}(s) for G ∈ S_{2}^{k} and h ∈ S_{1}^{k}. Assuming that the first Fourier-Jacobi coefficient of G is not identically zero enables us to read off properties of the L-function. For example, we show that there is a close connection between the triple product L-function and Z_{G⊗h}(s).

Theorem 5.1 Let F, G ∈ S_{2}^{k} and h ∈ S_{1}^{k} be Hecke eigenforms with totally real algebraic Fourier coefficients. Let k be even and h normalized. Further we assume that F is in the Maass Spezialschar with corresponding f ∈S_{1}^{2k−2}. Then for s ∈ ℂ with 2 Re(s) + k > 6:

⟨⟨⟨E_{5}^{k} ([z_{1}, z_{2}, z_{3}]; s) ; h(z_{1})G(z_{2})F (z_{3})⟩⟩⟩
= a(2, s, k)⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J}
×ζ(2s + k)^{−1}ζ(4s + 2k − 2)^{−1}ζ(4s + 2k − 4)^{−1}
×L_{f}(2s + 2k − 4) Z_{G⊗h}(s + 2k − 3).

Proof: By Theorem 2.7, we know that Φ(h, G, F; s) = a(2, s, k)L_{h,G,F}^{2} (2k + s − 3). Here L_{h,G,F}^{2} (2k + s − 3) is equal to

{ζ(2s + k) ∏_{j=1}^{2} ζ(4s + 2k − 2j)}^{−1} D_{F}^{2} (2s + k − 2) D_{h,G;F} (s).

Since F is a Hecke eigenform in the Maass Spezialschar, we have

D_{F}^{2} (2s + k − 2) = ζ(2s + k − 2) L_{f}(2s + k̃ − 1) L_{f}(2s + k̃ − 2),

with k̃ = 2k − 2 and f ∈ Sk̃ ; 1. Finally, with the explicit form of D_{h,G;F} (s) obtained in the last section the theorem becomes obvious.

Supplementarily we want to note that a(2, s, k) is equal to

2 · 2^{−6s−6k+12}π^{−s−2k+6} \frac{Γ_{2}(k + s − \frac{3}{2})Γ(s + 2k − 3)Γ(s + k − 2)Γ(s + k − 1)}{Γ_{2}(k + s)Γ(2k − 3 + 2s)}. (80)

From now on, we assume that Φ̃_{1}^{G} is not identically zero. It is obvious that one can choose a Hecke eigenform F ∈ S_{2}^{k,M} in such a way (use Definition 3.14 and Theorem 3.15), that

⟨Φ̃_{1}^{F}; Φ̃_{1}^{G}⟩ ≠ 0 and F ∈ S_{2}^{k,M}(ℚ(F)).

Corollary 5.2 The L-function Z_{G⊗h}(s) has an analytic continuation to a meromorphic function on ℂ. If we put

Z_{G⊗h}^{∗}(s) := (2π)^{−4s}Γ(s − k + 1)^{2}Γ(s − k + 2)Γ(s)Z_{G⊗h}(s),  (81)

then we have the functional equation

Z_{G⊗h}^{∗}(s) = Z_{G⊗h}^{∗}(3k − 3 − s).  (82)

This follows from elementary calculations of the already proven general functional equation stated in (31) and the well known functional equation of L_{f}(s). Remark 5.3 It is pleasant that in our formula the help function F has no longer any influence on the functional equation of Z_{G⊗h}(s). Further, in contrast to the situation in the case of the triple product L-function, the value of Z_{G⊗h}(s) at the center of symmetry is not forced to be zero (the minus sign of the functional equation of L_{f}(s) plays a decisive role). Corollary 5.4 Let k > 6 be even, G ∈ S_{2}^{k}(ℚ(G)) and h ∈ S_{1}^{k} be Hecke eigenforms with h normalized. If the first Fourier-Jacobi coefficient of G is not zero, then

Z′_{G⊗h}(2k − 3) := π^{8−5k} \frac{Z_{G⊗h}(2k − 3)}{⟨G; G⟩⟨h; h⟩} ∈ \overline{ℚ}.  (83)

Corollary 5.5 Let h and G be as in Corollary 5.4. Then

(Z′_{G⊗h}(2k − 3))^{σ} = Z′_{G^{σ}⊗h^{σ}}(2k − 3);  ∀σ ∈ Gal(\overline{ℚ}/ℚ).  (84)

Proof: By Theorem 1.9 and Theorem 5.1, it is enough to show that

(\frac{a(2, 0, k)⟨ϕ_{1}^{F}; ϕ_{1}^{G}⟩_{J}L_{f}(2k − 4)}{ζ(2s + k)ζ(4s + 2k − 2)ζ(4s + 2k − 4)⟨F; F ⟩})^{σ}
= \frac{a(2, 0, k)⟨Φ_{1}^{F^{σ}}; Φ_{1}^{G^{σ}}⟩_{J}L_{f^{σ}}(2k − 4)}{ζ(2s + k)ζ(4s + 2k − 2)ζ(4s + 2k − 4)⟨F^{σ}; F^{σ}⟩} ∈ \overline{ℚ}.

We put a ∼ b for a, b ∈ ℂ, b ≠ 0, if ab^{−1} ∈ \overline{ℚ}. Then a(2, 0, k) ∼ π^{6−2k} and (ζ(2s + k)ζ(4s + 2k − 2)ζ(4s + 2k − 4))^{−1} ∼ π^{−5k+6}, thus π^{5k−8}a(2, 0, k) (ζ(2s + k)ζ(4s + 2k − 2)ζ(4s + 2k − 4))^{−1} ∼ π^{4−2k}. By Proposition 3.18 and L_{f}(2k − 3) L_{f}(2k − 4) ≠ 0, it is enough to prove:

(\frac{π^{−3+2k}⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J}}{L_{f}(2k − 3)} )^{σ} = \frac{π^{−3+2k}⟨Φ_{1}^{F^{σ}}; Φ_{1}^{G^{σ}}⟩_{J}}{L_{f^{σ}}(2k − 3)} ∈ \overline{ℚ}.

But this is an easy consequence of the properties 1.12a and 1.12b and Theorem 1 stated in [Sh81]. Of course, we have used the well known translation of Jacobi forms of index one into modular forms of half-index weight, where the corresponding scalar products are equal up to a non-zero rational number [E-Z85], [Za80]).

The L-function Z_{G⊗h}(s) has already been investigated by M. Furusawa [Fu93]. He constructed an integral representation by integrating a Hermitian Klingen-Eisenstein series for h against a Siegel cusp form G. Under the assumption that certain non-vanishing properties of the Fourier coefficients of G hold, he was able to deduce the meromorphic continuation, the functional equation and the algebraicity of Z′_{G⊗h}(2k−3). Because of the not sufficiently known Galois behaviour of the Fourier coefficients of a Hermitian Klingen-Eisenstein series [Ha84], he was not able to study the Galois action at the special value. We want to conclude our work with the following observation. For simplification let F ∈ S_{n}^{k}(ℚ(F)), G ∈ S_{n}^{k}(ℚ(G)) and h ∈ S_{1}^{k}(ℚ(h)) be Hecke eigenforms (and every elliptic cusp form is assumed to be normalized). Let σ ∈ Gal(\overline{ℚ}/ℚ), then

L_{h,G,F}^{n,∗} (s) = L_{h,G,F}^{n,∗} (3k − n − 1 − s)
(\frac{π^{\frac{n^{2}+3n−4k+2}{2}} L_{h,G,F}^{n} (2k − n − 1)}{⟨h; h⟩⟨G; G⟩⟨F; F ⟩})^{σ} = \frac{π^{\frac{n^{2}+3n−4k+2}{2}} L_{h^{σ},G^{σ},F^{σ}}^{n}(2k − n − 1)}{⟨h^{σ}; h^{σ}⟩⟨G^{σ}; G^{σ}⟩⟨F^{σ}; F^{σ}⟩} ∈ ℚ̄.

This enables us to give an unique approach to and description of the triple product Lfunction L_{h,G,F} (s) for h, G, F ∈ S_{1}^{k} (P. Garrett [Ga87]) and the L-function Z_{G⊗h}(s). Let us assume that n is equal to 1 or 2 and that the first Fourier-Jacobi coefficient of G is not identically zero. Then we have

L_{h,G,F}^{n} (s+2k−n−1) = \{ \begin{matrix}ζ(2s + k)^{−1}ζ(4s + 2k − 2)^{−1}L_{h,G,F} (s + 2k − 2) & for n = 1 \\ ζ(2s + k)^{−1}ζ(4s + 2k − 2)^{−1}ζ(4s + 2k − 4)^{−1} \\ ×⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J}L_{f}(2s + 2k − 4)Z_{G⊗h}(s + 2k − 3) & for n = 2, \end{matrix}

and

\begin{aligned}L_{h,G,F}^{∗} (s) & = & (−1)L_{h,G,F}^{∗} (3k − 2 − s) & for n = 1, \\ Z_{G⊗h}^{∗}(s) & = & Z_{G⊗h}^{∗}(3k − 3 − s) & for n = 2 \end{aligned}

with L_{h,G,F}^{∗} (s) = (2π)^{−4s}Γ(s − k + 1)^{3}Γ(s)L_{h,G,F} (s) and Z_{G⊗h}^{∗}(s) = (2π)^{−4s}Γ(s − k + 1)^{2} Γ(s−k+2)Γ(s)Z_{G⊗h}^{∗}(s). The central value of L_{h,G,F}^{n} (s) is equal to \frac{3k−n−1}{2}, which coincides for k = n + 1 with our described special value s = 2k − n − 1. The Galois behaviour shows up as

(\frac{π^{−5k+5}L_{h,G,F} (2k − 2)}{⟨h; h⟩⟨G; G⟩⟨F; F ⟩})^{σ} = \frac{π^{−5k+5}L_{h^{σ},G^{σ},F^{σ}}(2k − 2)}{⟨h^{σ}; h^{σ}⟩⟨G^{σ}; G^{σ}⟩⟨F^{σ}; F^{σ}⟩} ∈ ℚ̄ and

(\frac{π^{−5k+8}Z_{G⊗h}(2k − 3)}{⟨h; h⟩⟨G; G⟩})^{σ} = \frac{π^{−5k+8}Z_{G^{σ}⊗h^{σ}}(2k − 3)}{⟨h^{σ}; h^{σ}⟩⟨G^{σ}; G^{σ}⟩} ∈ ℚ̄.

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————– Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, 53225 Bonn, Germany

E-mail address: heim@mpim-bonn.mpg.de, heim@euklid.math.uni-mannheim.de