L-functions for Jacobi forms and the basis problem

By Bernhard E. Heim

Abstract

This paper deals with Jacobi forms Φ on ℍ×ℂ. The Rankin-Selberg doubling method is employed to study properties of the standard L-function of Hecke-Jacobi eigenforms. It is shown that every analytic Klingen-Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke-Jacobi cusp eigenforms of weight k > 4 and k ≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved.^{1}

Introduction

Jacobi forms are a mixture of modular forms and elliptic functions, examples are Jacobi theta functions and Fourier coefficients of Siegel modular forms.

In the early eighties Eichler and Zagier, stimulated by the proof of the Saito-Kurokawa conjecture, developed a systematic theory of Jacobi forms on ℍ×ℂ [E-Z85]. One knows that Jacobi forms play a role in several contexts:

• the theory of modular forms of half-integral weight

• the theory of Maass lift

• the theory of elliptic cusp forms

• the theory of Heegner points

• the theory of Borcherds products.

Recently Jacobi forms turned out to be very useful in studying properties of Siegel modular forms. Kohnen and Böcherer [B-K93], (see also [Br98]) showed that one can obtain a very good estimate for the Fourier coefficients of a Siegel modular form by analyzing the Fourier coefficients of Jacobi-Poincaré series. In [He96], we established a functional equation of the twisted spinor L-function, i.e. the L-function attached to GSp_{2} × Gl_{2} and proved special value results, by using the Saito-Kurokawa conjecture and the Maass lift.

This paper deals with another aspect and application of the theory of Jacobi forms. We study the standard L-function of a Hecke-Jacobi cusp eigenform of even weight k > 4 on ℍ × ℂ by the Rankin-Selberg doubling method. We prove that every such form can be written as a linear combination of theta series. This generalizes works of Murase [Mu89], [Mu91], Arakawa [Ar94], Sugano [Su95] and Dulinski [Du97] on this problem.

1Mathematics Subject Classification: 11F27, 11F55, 11F67

Parts of this work is based on ideas of Paul Garrett [Ga84], [Ga89] [Ga92] and Siegfried Böcherer [Boe83], [Boe85]. They calculated the Rankin-Selberg convolution of a diagonal-restricted Siegel Eisenstein series and a Siegel cusp form. They obtained an integral expression of the standard L-function and the Klingen Eisenstein series associated to a Siegel cusp eigenform. Then Böcherer used this formula to solve the basis problem for Siegel modular forms of weight k and degree n, where k ≡ 0 mod 4 and k > 2n. He proved that every modular form with this data can be written as a linear combination of theta series (Siegel’s class-invariants).

Murase [Mu89],[Mu91] showed that the ideas of Böcherer and Garrett can also be used to study standard L-functions attached to Jacobi forms Φ on ℍ_{n} × ℂ^{r,n}, where ℍ_{n} is Siegel’s upper half-space of degree n. More precisely, he proved under certain assumptions on the index of the Jacobi form that the standard L-function has a functional equation (for example in the case n = r = 1, only the index 1 is allowed). Then Arakawa [Ar94] used a variation of the formula of Murase to solve the basis problem for Jacobi forms on ℍ_{n} × ℂ^{1,n} of index 1. Supplementary to this, Dulinski was able to generalize the approach of Murase in the case of Jacobi forms on ℍ × ℂ to square free index.

In this paper we prove the following. Let Φ be a Jacobi cusp form on ℍ × ℂ, which is an eigenform with respect to the (full) Hecke-Jacobi algebra. First of all we show that every analytic Klingen Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. This generalizes our previous work [He97]. We prove that Φ can be explicitly written as a linear combination of theta series. Moreover we study standard L-functions L(Φ, s) attached to Φ, when Φ is a newform (in the sense of [S-Z88]) or the index of Φ is square free. One ingredient of our proof is a certain rationality theorem in the Hecke-Jacobi algebra due to Gritsenko [Gr95]. In contrast to the previous works on the standard L-function of a Jacobi forms, our proof does not depend on the nature of the index, because we allow ourselves to work in a Hecke-Jacobi algebra, which is non commutative and has zero divisors. We believe that our method can be used to study standard L-functions of Jacobi forms on ℍ_{n} × ℂ^{1,n}. But much work remains to be done to get a better understanding of the underlying Hecke-Jacobi algebra.

Acknowledgment The author would like to thank the Max-Planck Institut für Mathematik in Bonn for hospitality and the DFG for financial support.

1 On a Dirichlet series

This section deals with the properties of a certain kind of Dirichlet series, attached to the Jacobi symmetric-square operator and a Hecke-Jacobi eigenform Φ. We start with a short description of the underlying Hecke algebra.

1.1 Hecke-Jacobi theory

In this section we present a selfcontained introduction of a rationality theorem in the Hecke-Jacobi algebra attached to the Jacobi group. The theorem was used to study Kohnen and Skoruppa’s [K-S89] approach towards the spinor zeta function of a Siegel modular form of degree 2. We show that such a theorem can also be used to study standard L-functions of Jacobi forms. One of the ingredients is the doubling method in the Rankin-Selberg method [Ar94]. This has the advantage that we do not have to care about the index of the underlying Jacobi forms, because we can mainly work in the Hecke-Jacobi algebra itself. This leads to an uniform approach to standard L-functions of Jacobi forms.

Let G^{+}Sp_{n} denote the group of symplectic similitudes of degree n with positive similitude factor. Let R = ℤ, ℤ_{[ \frac{1}{p}]} or ℚ. We define G_{R}^{J} = \{({β \atop 0_{3}} ∗) ∈ G^{+}Sp_{2}(R) | β > 0} and put Γ^{J} = G_{ℤ}^{J} ∩ Sp_{2}(ℤ). Then we associate to the Hecke pairs (Γ^{J} , G_{ℚ}^{J}) and (Γ^{J} , G_{ℤ_{[ \frac{1}{p}]}}^{J}) the Hecke algebras H^{J} and H_{p}^{J}, respectively. Next we distinguish several

elements in H^{J}. Let [a_{1}, a_{2}, a_{3}, a_{4}] denote a diagonal matrix in ℂ^{4,4}. We define T^{J,r}(n) = Γ^{J}[1, n^{−1}, 1, n]Γ^{J}, Δ_{n} = Γ^{J}[n, n, n, n]Γ^{J}, Λ_{−}^{r}(p) = [p^{−1}, 1, p, 1] and Λ_{+}^{r}(p) = Γ^{J}[p, 1, p^{−1}, 1]Γ^{J}. Further let

∇_{n}^{r} = ∑_{b mod n} Γ^{J} (\begin{matrix}1 0 0 1 & \begin{matrix}\frac{b}{n} & 0 \\ 0 & 0 \end{matrix} \\ 0 & 1 0 0 1 \end{matrix}) Γ^{J} and Ξ_{n}^{r} = ∑_{λ,μ,κ mod n} Γ^{J} (\begin{matrix}\begin{matrix}1 & \frac{λ}{n} \\ 0 & 1 \end{matrix} & \frac{μ}{n} 0^{\frac{κ}{n} \frac{μ}{n}} \\ 0 & \begin{matrix}1 & −\frac{λ}{n} \\ 0 & 1 \end{matrix} \end{matrix}) .

Let H̃_{0}^{J} be a subalgebra of H^{J} generated by T^{J,r}, Δ, ∇^{r}, Ξ^{r} and H̃_{0,p}^{J} = H̃_{0}^{J} ∩ H_{p}^{J}. Then H̃_{0}^{J} = ⊗_{p}H̃_{0,p}^{J}. Now everything is prepared to state the rationality theorem. It seems to have first appeared in a paper of Gritsenko ([Gr95] section 5, Theorem 5.8).

Theorem 1.1 (Gritsenko) The formal power series ∑_{n=o}^{∞} T^{J,r}(p^{n})(\frac{X}{p})^{n} is a rational function with left denominator

Q_{p}^{J}(X) = 1 − (p^{−1}T^{J,r}(p) − (p − 1)p^{−1}∇_{p}^{r})X + p(p^{−2}Ξ_{p}^{r} − p^{−1}∇_{p}^{r} + 1)X^{2}.

More precisely we have

Q_{p}^{J}(X) ∑_{n=0}^{∞} T^{J,r}(p^{n}) (\frac{X}{p} )^{n} = P_{p}^{J}(X), (1)

where P_{p}^{J}(X) = 1 + (p − 1)p^{−1}∇_{p}^{r}X − ((p − 1)p^{−1}∇_{p}^{r} + 1)X^{2}.

In this paper we would like to apply this result to study Arakawa’s [Ar94] approach to the standard L-function of a Jacobi form in more detail. As already practiced in [He96], we simulate the behavior of certain Hecke-Jacobi operators in the underlying local Hecke-Jacobi algebra H̃_{0,p}^{J} to show the various specializations, depending on the index t of the underlying representation space of Jacobi forms. We have to distinguish between three cases: p ∤ t, p|t with p^{2} ∤ t and p^{2}|t. For further simplification we put n ∥ m if n | m and (n, \frac{m}{n}) = 1.

CASE I Let ∇_{p}^{r} = Ξ_{p}^{r} = 0 and T^{J,r}(p) = λ. Then

Q_{p}^{J}(X) = 1 − p^{−1}λX + pX^{2}
P_{p}^{J}(X) = 1 − X^{2}.

CASE II Let ∇_{p}^{r} = p, p^{−2}Ξ_{p}^{r} = ε, where ε = ±1 and T^{J,r}(p) = λ.

Q_{p}^{J}(X) = 1 − (p^{−1}λ − p + 1)X + p ε X^{2}
P_{p}^{J}(X) = (1 + pX) (1 − X).
(a) Let λ = −p(ε + 1).
Q_{p}^{J}(X) = (1 + pX) (1 + ε X)
(b) Let ε = 1 and λ = λ′ − p + p^{2}.
Q_{p}^{J}(X) = 1 − p^{−1}λ′X + pX^{2}

CASE III Let ∇_{p}^{r} = p, p^{−2}Ξ_{p}^{r} = 0 and T^{J,r}(p) = −p.

Q_{p}^{J}(X) = (1 + pX)
P_{p}^{J}(X) = (1 + pX) (1 − X).

In section 3 we will give an explanation for the made assumptions and its connection to L-functions of Jacobi forms.

1.2 Zeta functions of Hecke-Jacobi eigenforms

We denote by J_{k,t}^{cusp} the space of Jacobi cusp forms on D = ℍ×ℂ of weight k and index t with respect to the Jacobi group Γ^{J}. Moreover let ⟨ , ⟩_{J} be the Petersson scalar product. By Ĵ_{k,t}^{cusp} we denote the space of completed cusp forms on ℍ_{2}. Here ℍ_{n} denotes Siegel’s upper half space of degree n. We frequently switch between the notation Φ and Φ̂ for the same Jacobi form. For more details the reader is referred to the standard reference [E-Z85] and [A-H98], section 1.

In [Ar94], Theorem 2.9, Arakawa proved the following relation between analytic Siegel Jacobi Eisenstein series and analytic Klingen-Jacobi Eisenstein series. Roughly speaking, he transfered a well-known theorem of Garrett and Böcherer [He96], Theorem 1.11 into the setting of Jacobi forms, which was a non trivial task. Let k, t, n ∈ ℕ and k even. We denote by E_{n}^{k,t}((τ, z); s) the analytic Siegel-Jacobi Eisenstein series on D_{n,1} = ℍ_{n} ×ℂ^{1,n} of weight k and index t. Let further Φ ∈ J_{k,t}^{cusp}. Then we denote by E_{n,1}^{k,t}((τ, z), Φ, s) the analytic Klingen-Jacobi Eisenstein series attached to Φ on D_{n,1} of weight k and index t. Here s ∈ ℂ with s ≫ 0 (see [Ar94], section 2.4 for a explicit definition and more details).

Let F : ℍ_{n} ⟶ ℂ, k ∈ ℤ and M = \binom{a b}{c d} ∈ G^{+}Sp_{n}(ℝ). Then

(F |_{k}M)(z) := μ(M)^{nk− \frac{n(n+1)}{2}} j(M, z)^{−k}F (M(z)),

where j(M, z) = det(c z + d) and μ(M) is the factor of similitudes of M. Let X ∈ H^{J}. Then we define the action of X = ∑ a_{j} (Γ^{J} g_{j}) on the space of Jacobi forms by

Φ̂|_{k}D(X) = ∑_{j} a_{j} Φ̂|_{k} g_{j}. (2)

Definition 1.2 Let Φ ∈ J_{k,t}^{cusp}. Then Φ is denoted by Hecke-Jacobi eigenform if

Φ̂|_{k}D(T^{J,r}(n)) = λ(Φ, n) Φ̂,

i.e. Φ is an eigenform for all operators T^{J,r}(n) with eigenvalues λ(Φ, n).

Remark 1.3 Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Then Φ is an eigenform with respect to all elements of H̃_{0}^{J}. This follows from the formula

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

Let s ∈ ℂ. The Jacobi symmetric-square operator is formally defined by

(4)  S^{J}(s) = ∑_{n=1}^{∞} T^{J,r}(n) n^{−s}
= ∏_{p} S_{p}^{J}(s),

where S_{p}^{J}(s) = ∑_{j=0}^{∞} T^{J,r}(p^{j}) p^{−js}.

Lemma 1.4 Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Let s ∈ ℂ with Re(s) > 4. Then the following identity is well-defined

Φ̂|_{k}D(S^{J}(s)) = Z^{J}(s, Φ) Φ̂, (5)

where Z^{J}(s, Φ) = ∑_{n=1}^{∞} λ(Φ, n) n^{−s}. Moreover we have the Euler product

Z^{J}(s, Φ) = ∏ Z_{p}^{J}(s, Φ) with Z_{p}^{J}(s, Φ) = ∑_{j=0}^{∞} λ(p^{j}, Φ) p^{−js}.

Here the series Z^{J}(s, Φ) is absolutely convergent.

Proof: This has been proven in [Ar94], section 2.

Theorem 1.5 (Arakawa) Let m ≥ 1 and k an even natural number. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform and k + 2Re(s) > m + 3. Let (τ, z) ∈ D_{m,1}. Then we have

⟨Φ(τ′, z′) ; E_{m+1}^{k,t}( (\begin{matrix}−\overline{τ} & 0 \\ 0 & τ′ \end{matrix}) , (\overline{z}, z′), s)⟩_{J} (6)

(7)  = μ(k, t, 1, s) Z^{J}(2s + k, Φ) E_{m,1}^{k,t}(Φ, (τ, z), s).

In particular if m = 1, then the right hand side of this equation is equal to

μ(k, t, 1, s)Z^{J}(2s + k, Φ) Φ^{♯}(τ, z).

Here μ( ) is a certain constant depending only on k, t and s ([Ar94] (2.4)).

We have proven in [He97] that analytic Siegel-Jacobi Eisenstein series have a meromorphic continuation on the whole complex plane. Thus we have

Proposition 1.6 Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Then Z^{J}(s, Φ) has a meromorphic continuation to the whole complex plane.

At this point we do not know if it can happen that Z^{J}(s, Φ) is identically zero.

2 Non-vanishing of the local L-factors

Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Let s ≫ 0. We are interested in the question if the local L-factor Z_{p}^{J}(s, Φ) can vanish. We have

Φ̂|_{k}D(S_{p}^{J}(2s + k)) = Z_{p}^{J}(2s + k)Φ̂. (8)

On the other hand, we can also calculate the left side of this equation by performing Theorem 1.1. This will show that Z_{p}^{J}(s) is a rational function in p^{−s}, if we assume that Φ is a Hecke-Jacobi eigenform. More precisely: Let X = p^{−(2s+k)}. Then

Φ̂|_{k}D(Q_{p}^{J}(pX)S_{p}^{J}(X)) = Φ̂|_{k}D(P_{p}^{J}(pX)), (9)

by abuse of notation. Here we have interpreted f(X) ∈ H_{0,p}^{J}[X] as an operator in the obvious way.

Let us denote the eigenvalues of Q_{p}^{J}(pX) and P_{p}^{J}(pX)) by Q_{p}^{J,Φ}(pX) and P_{p}^{J,Φ}(pX). Since these eigenvalues can be viewed as non-trivial polynomials with respect to X, we can see that the quotient of these eigenvalues is holomorphic and has no zeros for Re(s) ≫ 0.

Equipped with the absolute convergence of Z^{J}(s, Φ) for Re(s) > 4 we only have to analyze the numerator on Z^{J}(s, Φ) in more detail. We distinguish between two cases: p ∤ t and p | t. This gives either p^{−1}∇_{p}^{r} = 0 or 1. The value of P_{p}^{J,Φ}(pX)) is therefore (1 − p^{2}X^{2}) or (1 − pX)(1 − X). This shows that this is holomorphic in s and P_{p}^{J,Φ}(p^{−2s−k+1}) ≠ 0 for k + 2Re(s) > 4. We have

Φ̂|_{k}D(S_{p}^{J}(X)) = \frac{P_{p}^{J,Φ}(pX))}{Q_{p}^{J,Φ}(pX)} Φ̂. (10)

Now we are in a good situation. We have proven the following global non-vanishing result.

Theorem 2.1 Let k be an even natural number larger than 4. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Let s ∈ ℂ with Re(s) > 4. Then Z_{p}(s, Φ) never vanishes and

Z^{J}(s, Φ) = ∏_{p} Z_{p}^{J}(s, Φ) ≠ 0. (11)

Corollary 2.2 Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform and m ∈ ℕ. Then the analytic Klingen-Jacobi Eisenstein series

E_{m,1}^{k,t}(Φ, (τ, z), s)

associated to Φ on D_{n,1} of weight k and index t has a meromorphic continuation on the whole complex plane.

Corollary 2.3 Let t be square-free and Φ ∈ J_{k,t}^{cusp}. Then E_{m,1}^{k,t}((τ, z), Φ, s) has a meromorphic continuation on the whole complex plane.

Proof: It is well-known that for t square-free the space of Jacobi cusp forms has always a basis of Hecke-Jacobi eigenforms ([Gr95], section 5 & 6).

Corollary 2.4 Let k be an even natural number larger than 4 and Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Then

Z^{J}(k, Φ) ≠ 0.

3 Local computations

Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. We would like to calculate Z_{p}^{J}(2s + k, Φ) explicitly for every prime number p. It is convenient to distinguish between the three cases: p ∤ t, p ∥ t, and p^{2}|t (see section 1.1).

In the last section we have proven for the local factors of the Dirichlet series Z^{J}(s, Φ):

Z_{p}^{J}(2s + k, Φ) = \frac{P_{p}^{J,Φ}(pX)}{Q_{p}^{J,Φ}(pX)}, (12)

where X = p^{−(2s+k)}.

Already in the beginning of this paper, we have simulated the different cases in the Hecke-Jacobi algebra H̃_{0,p}^{J}. Before we refine these formulas, we give some convenient definitions. Let ζ_{p}(s) = 1 − p^{−s}.

3.1 Newforms

We start by introducing some virtual definitions for local L-factors associated to a Hecke-Jacobi form at the unramified primes.

Definition 3.1 Let p ∤ t and let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Let Φ̂|_{k}T^{J,r}(p) = λΦ̂ and λ^{EZ} := p^{k−3}λ. We define

L̃_{p}^{J}(s, Φ) = 1 − λ p^{−s} + p^{3} p^{−2s}
L̃_{p}^{EZ}(s, Φ) = 1 − λ^{EZ} p^{−s} + p^{2k−3} p^{−2s}.

The definition of L̃_{p}^{J}(s, Φ) seems to be natural in the context of L-functions for Jacobi forms, but L̃_{p}^{EZ}(s, Φ) = L̃_{p}^{J}(s − k + 3, Φ) stresses a hidden correspondance between elliptic cusp forms φ of weight 2k − 2 and level t and Jacobi cusp forms Φ of weight k and index t. For example, let t = 1 and let L_{p}(s, φ) be the Hecke polynomial attached to the eigenform φ. Then L̃_{p}^{EZ}(s, Φ) = L_{p}(s, φ).

Next we would like to recall the concept of newforms in the context of Jacobi forms introduced by Skoruppa and Zagier [S-Z88]. The space of Jacobi newforms is defined by

(13)  Ĵ_{k,t}^{cusp,new} = ( ∑_{\begin{matrix}l,d>0, ld^{2}>1 \\ ld^{2}|t \end{matrix}} Ĵ_{k,t/(ld^{2})}^{cusp}|_{k}D(Λ_{−}^{r}(d) T_{−}^{r}(l)))^{⊥} ,

here ⊥ denotes the orthogonal complement with respect to the Petersson scalar product of Jacobi forms. It remains to define the operators T_{−}^{r}(p) and T_{+}^{r}(n). We have T_{−}^{r}(p) = Δ_{p}^{−1}Γ^{J}[1, 1, p, p]Γ^{J} and T_{+}^{r}(p) = Δ_{p}^{−1}Γ^{J}[p, 1, 1, p]Γ^{J}. In [Gr95], page 80 Gritsenko has mentioned that this can be simplified to

(14) Ĵ_{k,t}^{cusp,new} = ⋂_{p|t} Ker ( J_{k,t}^{cusp}|_{k}D(T_{+}^{r}(p))) ∩ Ker ( J_{k,t}^{cusp}|_{k}D(Λ_{+}^{r}(p))) .

\underline{Facts:}

• The space J_{k,t}^{cusp,new} has a basis of Hecke Jacobi eigenforms. •

(15) J_{k,t}^{cusp} = J_{k,t}^{cusp,new} ⊕ ⨁_{\begin{matrix}l,d>0 \\ ld^{2}|t \\ ld^{2}>1 \end{matrix}} J_{k,t/(ld^{2})}^{cusp, new}|_{k}D(Λ_{−}^{r}(d)T_{−}^{r}(l)).

Gritsenko has introduced another interesting subspace of J_{k,t}^{cusp}, the space

(16) J_{k,t}^{cusp, G} := ⋂_{p|t} Ker ( J_{k,t}^{cusp}|_{k}D((T_{+}^{r}(p))^{2})) ∩ Ker ( J_{k,t}^{cusp}|_{k}D(Λ_{+}^{r}(p))) .

This space has some remarkable properties: It has a basis of Hecke-Jacobi eigenforms and in the case of t square-free, it exhausts the whole space,

(17) J_{k,t}^{cusp, G} = J_{k,t}^{cusp}.

For arbitrary t we have

(18) J_{k,t}^{cusp,new} ⊂ J_{k,t}^{cusp, G} ⊂ J_{k,t}^{cusp}.

3.2 Explicit form of Z_{p}^{J}(s, Φ)

Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi newform. We will determine in this section the explicit nature of the p^{th} - local factor of Z^{J}(s, Φ). In addition we will also show in this paper, that if the index t is square free, the assumption on Φ to be a Hecke eigenform is sufficient, to compute Z_{p}^{J}(s, Φ). This improves the results of Murase [Mu89], Arakawa [Ar94] and Dulinski [Du97] for Jacobi forms on ℍ × ℂ. We would like to mention, that our approach and proof is new, even if we restrict ourself to the known cases of the papers stated above. We believe that this method should also work for Jacobi forms on ℍ_{n} × ℂ^{1,n}. But this would involve much more complicated calculations and much work on the underlying Hecke-Jacobi theory. We will come back to this task in another paper.

As usual, in this section λ and ε will be the eigenvalues of Φ ∈ J_{k,t}^{cusp,new} with respect to T^{J,r}(p) and p^{−2}Ξ_{p}^{r}. Let X = p^{−(2s+k)}. In the following we distinguish between three cases:

CASE I: p ∤ t, CASE II: p ∥ t and CASE III: p^{2} | t.

The arguments below will justify the assumptions and computations already done in the Hecke-Jacobi algebra (see section 1.1). The starting point of everything which follows is the formula given in (12). Let us start with the generic

CASE I It is obvious that if p ∤ t, then Φ̂|_{k}D(∇_{p}^{r}) = Φ̂|_{k}D(Ξ_{p}^{r}) = 0. This shows that the assumptions in section 1.1 in CASE I are satisfied. Hence we have

Z_{p}^{J}(2s + k, Φ) = \frac{ζ_{p}(4s + 2k − 2)}{L̃_{p}^{J}(2s + k, Φ)}
= \frac{ζ_{p}(4s + 2k − 2)}{L̃_{p}^{EZ}(2s + 2k − 3, Φ)}.

Next we devote our interest to the situation p | t. Since Φ̂|_{k}D(p^{−1}∇_{p}^{r}) = Φ̂, we have

(19)  P_{p}^{J}(X) = (1 + pX)(1 − X),
(20)  Q_{p}^{J}(X) = 1 − (p^{−1}λ − p + 1)X + p ε X^{2}.

CASE II Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi newform and p ∥ t. The fact that p ∥ t links our operator Ξ_{p}^{r} with the Atkin-Lehner operator W_{p} introduced in [E-Z85], section 5.

Remark 3.2 ([Gr95], Theorem 5.10) Let t = m p and (m, p) = 1 and Φ ∈ J_{k,t}^{cusp}, then

Φ̂|_{k}D(p^{−2}Ξ_{p}^{r}) = Φ̂|_{k}W_{p}, (21)

where W_{p} denotes the involution given in [E-Z85], page 60.

Newforms have the property, that they lie in the kernel of T_{+}^{r}(p). The list given in [He96], (section 3.3, number 31) gives the identity

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

Putting these two informations together shows that pT^{J,r}(p) + p^{2} + Ξ_{p}^{r} is trivial on the space of newforms. Thus we have (by abuse of notation)

T^{J,r}(p) = −p(1 + ε) and (22)

Z_{p}^{J}(2s + k, Φ) = \frac{ζ_{p}(4s + 2k − 2)}{1 + ε p^{−(2s+k−1)})(1 + p^{−(2s+k−1)})},

where ε = ±1.

CASE III Let Φ ∈ J_{k,t}^{cusp,new} be a Hecke-Jacobi newform and p^{2} ∤ t. Then formula (42) given in the list in [He96], section 3.3 for Λ_{+}^{r}(p)Λ_{−}^{r}(p) leads finally to

Φ̂|_{k}D(Ξ_{p}^{r}) = 0.

The formulae given in (20) and (22) lead us to T^{J,r}(p) = −p. Hence we have Q_{p}^{J}(X) = (1 + pX) and

Z_{p}^{J}(2s + k, Φ) = \frac{ζ_{p}(4s + 2k − 2)}{1 + p^{−(2s+k−1)})}.

Finally we obtain the desired theorem on the local nature of Z^{J}(s, Φ).

Theorem 3.3 Let k be an even natural number. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi newform and k + 2Re(s) > 4. Then we have

\frac{Z_{p}^{J}(2s + k, Φ)}{ζ(4s + 2k − 2)} = \{ \begin{matrix}L̃_{p}^{EZ}(2s + 2k − 3, Φ)^{−1} & if p ∤ t \\ (1 + ε p^{−(2s+k−1)})^{−1}(1 + p^{−(2s+k−1)})^{−1} & if p ∥ t \\ (1 + p^{−(2s+k−1)})^{−1} & if p^{2} | t \end{matrix} .

Let S^{k}(Γ_{0}(N)) denote the space of cusp forms of weight k with respect to Γ_{0}(N) = \{\binom{a b}{c d} ∈ SL_{2}(ℤ) | c ≡ 0 mod N\}. The results of Skoruppa and Zagier [S-Z88] on a Hecke-equivariant map between J_{k,t}^{cusp} and a certain subspace M_{2k−2}^{cusp,−}(N) of S^{2k−2}(Γ_{0}(N)) suggests the following definition of a L-function attached to a Hecke-Jacobi eigenform in J_{k,t}^{cusp,new}.

Definition 3.4 Let k be an even natural number and Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi newform. Let Re(s) ≫ 0 and ε_{p}^{Φ} be the eigenvalue with respect to p^{−2}Ξ_{p}^{r}, then the (standard) L-function of Φ is defined by

L(s, Φ) = ∏_{p∥t}(1 + ε_{p}^{Φ} p^{k−2}p^{−s})^{−1} ∏_{p∤t} L̃_{p}^{EZ}(s, Φ)^{−1}. (23)

Remark 3.5 Let φ ∈ S^{2k−2}(Γ_{0}(1)) correspond to the Hecke-Jacobi eigenform Φ ∈ J_{k,1}^{cusp} by the Saito-Kurokawa lift and let L(s, φ) be the Hecke L-function of φ. Then

L(s, Φ) = L(s, φ). (24)

Summarizing everything gives the following main result of this section.

Theorem 3.6 Let k be an even natural number and Re(s) > 4. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi newform. Then we have

Z^{J}(s, Φ) = ζ(2s − 2)^{−1}( ∏_{p|t}(1 + p^{−(s−1)}))^{−1} L(s + k − 3, Φ). (25)

Moreover let k > 4. Then L(2k − 3, Φ) ≠ 0.

Let us specialize the formula given in Theorem 3.6 to t = 1. If we combine this with the identity (24) given in the remark we get

Z^{J}(s, Φ) = \frac{L(s + k − 3, φ)}{ζ(2s − 2)}. (26)

This result was also obtained by Arakawa [Ar94] (section 6.1) with a totally different proof.

3.3 Beyond newforms

As already mentioned, there exists a subspace J_{k,t}^{cusp,G} of J_{k,t}^{cusp} which contains the space of newforms and possesses a basis of Hecke-Jacobi eigenforms. We would like to study the Dirichlet series Z^{J}(s, Φ) attached to Φ ∈ J_{k,t}^{cusp}, where t is square free in more detail. We remind the reader, that in this case J_{k,t}^{cusp} = J_{k,t}^{cusp,G} and

(27) J_{k,t}^{cusp} = J_{k,t}^{cusp,new} ⊕ ⨁_{l|t, l>1} J_{k,t/l}^{cusp, new}|_{k}D(T_{−}^{r}(l)).

The decomposition suggest how one can construct a good basis of the space of Jacobi cusp forms.

Lemma 3.7 Let t be a square free natural number and Ψ ∈ J_{k,t/l}^{cusp}, where l | t, be a Hecke-Jacobi eigenform. Then

Ψ̂|_{k}D(T_{−}^{r}(l))

is also a Hecke-Jacobi eigenform.

Proof: Let p|l. Then is suffices to show that Ψ̂|_{k}D(T_{−}^{r}(p)) is an eigenform with respect to T^{J,r}(p) and p^{−2}Ξ_{p}^{r}. First of all we would like to remark that since t is square free, (t/l, p) = 1 implies Ψ̂|_{k}D(T_{+}^{r}(p)) = Ψ̂|_{k}D(∇_{p}^{r}) = 0.

We have (again we use results of the list given in [He96], section 3.3):

Ψ̂|_{k}D(T_{−}^{r}(p)) |_{k} D(T^{J,r}(p)) = p(p − 1)Ψ̂|_{k}D(T_{−}^{r}(p)) + Ψ̂|_{k}D(T^{J,r}(p)T_{−}^{r}(p),
Ψ̂|_{k}D(T_{−}^{r}(p)) |_{k} D(p^{−2}Ξ_{p}^{r}) = Ψ̂|_{k}D(T_{−}^{r}(p)),

which proves the lemma.

It is obvious, that the arguments employed in the above proof of the lemma, do not work for arbitrary index. With the results of the lemma in mind we can construct a certain distinguished basis of J_{k,t}^{cusp} for t square free. Let us start with the decomposition in (27). For every l | t, we fix a normalized basis (Φ_{j}^{l})_{j=1}^{d_{l}} of Hecke-Jacobi eigenforms of the underlying space of newforms J_{k,t/l}^{cusp,new} (of course this will be only relevant if J_{k,t/l}^{cusp,new} is non-trivial). Then

B = ⋃_{l|t} (Φ̂_{j}^{l} |_{k} D(T_{−}^{r}(l))_{j=1}^{d_{l}} (28)

is a basis of Hecke-Jacobi eigenforms of J_{k,t}^{cusp} for t square free. Let Ψ ∈ B, i.e there exists a l ∈ ℕ and 1 ≤ j ≤ d_{l} such that Ψ̂ = Φ̂_{j}^{l} |_{k} D(T_{−}^{r}(l)). Let l > 1 and p_{1} ·. . .·p_{l_{r}} = l be the prime number decomposition of l. Let λ and λ′ be the eigenvalues of Ψ and Φ_{j}^{l} with respect to T^{J,r}(p_{i}). Then

λ = p_{i}(p_{i} − 1) + λ′,

and Ψ̂ |_{k} D(p_{i}^{−2}Ξ_{p_{i}}^{r}) = Ψ̂.

Definition 3.8 Let k, t be natural numbers. Let k be even and t square free. We call the basis B of J_{k,t}^{cusp} defined in (28), the generalized newform basis of J_{k,t}^{cusp}. Elements are called generalized newforms.

Proposition 3.9 Let t be square free and k even. Let Φ ∈ J_{k,t}^{cusp} be a generalized newform, i.e

Ψ̂ = Φ̂_{j}^{l} |_{k} D(T_{−}^{r}(l))

for suitable Hecke-Jacobi eigenforms Φ_{j}^{l} ∈ J_{k,t/l}^{cusp,new}. Then

\frac{Z_{p}^{J}(2s + k, Φ)}{ζ(4s + 2k − 2)} = \{ \begin{matrix}L̃_{p}^{EZ}(2s + 2k − 3, Φ_{j}^{l})^{−1} & if p ∤ t \\ (1 + ε p^{−(2s+k−1)})^{−1}(1 + p^{−(2s+k−1)})^{−1} & if p | \frac{t}{l} \\ \frac{(1+p^{−(2s+k−2)}}{(1+p^{−(2s+k−1)})} L̃_{p}^{EZ}(2s + 2k − 3, Φ_{j}^{l})^{−1} & if p | l \end{matrix} .

Let p ∤ t. Then L̃_{p}^{EZ}(s, Φ) = L̃_{p}^{EZ}(s, Φ_{j}^{l}). Let p | \frac{t}{l}, then the eigenvalue of Φ and Φ_{j}^{l} with respect to p^{−2}Ξ_{p}^{r} is ε.

Let Ψ̂ = Φ̂_{j}^{l} |_{k} D(T_{−}^{r}(l)) be a generalized newform. Then we associate to Φ the (standard) L-function

L(s, Φ) = ∏_{p|l}(1 + p^{k−1}p^{−s}) ∏_{p|(t/l)}(1 + ε_{p}^{Φ} p^{k−2}p^{−s})^{−1}

∏_{p|l} L̃_{p}^{EZ}(s, Φ_{j}^{l})^{−1}. ∏_{p∤t} L̃_{p}^{EZ}(s, Φ)^{−1}.

This is compatible with the Definition 3.4 for the L-function attached to Φ ∈ J_{k,t}^{cusp,new}.

4 Global results and the basis problem

Let us first summarize what we have proven in the last sections.

Theorem 4.1 Let k be an even natural number and Re(s) > 4. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Then the Dirichlet series Z^{J}(s, Φ), defined in Lemma 1.4, has a meromorphic continuation on the whole complex space and Z^{J}(k, Φ) ≠ 0 for k > 4.

If we assume Φ to be a (generalized) newform, generalized if t is square free. Then we have

Z^{J}(s, Φ) = ζ(2s − 2)^{−1}( ∏_{p|t}(1 + p^{−(s−1)}))^{−1} L(s + k − 3, Φ). (29)

Let k > 4 then

L(2k − 3, Φ) ≠ 0.

4.1 Siegel’s formula for Jacobi forms

Let m, n, t ∈ ℕ. By A_{n}^{+} we denote the set of semi-integral positive definite matrices of size n and A_{n}^{+}(t) = \{\binom{∗ ∗}{∗ t} ∈ A_{n}^{+}\}. Two elements T, T′ ∈ A_{m+1}^{+}(t) are said to be in the same t−class, if there exists γ = \{\binom{u 0}{y 1} with u ∈ SL_{m}(ℤ), y ∈ ℤ^{1,m} such that T′ = T [γ]. Moreover T, T′ are in the same t−genus, if for each prime integer p, γ = \binom{u 0}{y 1} can be choosen in such a way that u ∈ SL_{m}(ℤ_{p}) and y ∈ ℤ_{p}^{1,m}. Each t−genus consists of finitely many t−classes (cf. [Ar93], Proposition 2.1). We denote by H(t) the number of t−classes of the t−genus for given t. This number is called the t−class number of t.

Definition 4.2 Let Q = (\begin{matrix}M & (q/2)^{t} \\ q/2 & t \end{matrix}) ∈ A_{m+1}^{+}(t). We define the theta series

Θ_{Q}^{(n)}(τ, z), where (τ, z) ∈ D_{n,1} by

Θ_{Q}^{(n)}(τ, z) = ∑_{G∈ℤ^{m+1,n}} e{Q[G] τ + z^{t}(q, 2t)G}. (30)

It is known that Θ_{Q}^{(n)}(τ, z) is a Jacobi form on ℍ_{n} × ℂ^{1,n} of index t and weight \frac{m+1}{2}, if 2 Q is unimodular. Let E(Q) be the order of the finite group

\{P = \binom{u 0}{y 1} ∈ ℤ^{m+1,m+1} | u ∈ SL_{m}(ℤ) and Q[P] = Q\} .

Arakawa has proven [Ar93] Theorem 4.3 (Analytic Siegel’s formula ) Assume that m > 2n + 3. Let Q ∈ A_{m+1}^{+}(t) with 2Q unimodular. Let Q_{1}, Q_{2}, …, Q_{H} be a complete set of representatives of the t−classes in the given t−genus of Q. Then

(∑_{j=1}^{H} \frac{Θ_{Q_{j}}^{(n)}(τ, z)}{E(Q_{j})}) / (∑_{j=1}^{H} \frac{1}{E(Q_{j})}) = E_{n}^{\frac{m+1}{2} ,t}(τ, z). (31)

It is useful to introduce the following map. Let

Λ : \{ \begin{matrix}J_{k,t}^{cusp} & ⟶ & J_{k,t}^{cusp} \\ Φ & ↦ & ⟨Φ , E_{2}^{k,t} (\begin{matrix}\overline{τ} & 0 \\ 0 & ∗ \end{matrix}) , (\overline{z}, ∗))⟩_{J}. \end{matrix}

Then Λ is a linear map. Moreover, if Φ is a Hecke-Jacobi eigenform, then it is also an eigenfunction of Λ. Now, let the eigenvalue of Φ with respect to Λ be non zero, then Siegel’s theorem can be used to give an explicit describtion of Φ as a linear combination of theta series [Boe83].

Theorem 4.4 Let k, t be natural numbers and let k ≡ 0 mod 4 and k > 4. Let Φ ∈ J_{k,t}^{cusp} be a Hecke-Jacobi eigenform. Then Φ can be explicitly written as a linear combination of theta series Θ_{Q_{j}}^{(1)}(τ, z), where Q_{1}, Q_{2}, …, Q_{H} is a complete set of representatives of the t−classes in the given t−genus of suitable Q ∈ A_{2k}^{+}(t) with 2Q unimodular.

Φ(τ, z) = (μ(k, t, 1, 0) Z^{J}(k, Φ))^{−1} ∑_{j=1}^{H} \frac{⟨Φ, Θ_{Q_{j}}^{(1)}⟩_{J}}{m_{Q}E(Q_{j})} Θ_{Q_{j}}^{(1)}(τ, z),

where m_{Q} = ∑_{j} 1/E(Q_{j}).

Proof: First of all we have to show hat the assumptions of Siegel’s Theorem are satisfied. Let 2Q_{8} denote an even unimodular positive definite symmetric matrix of size 8 associated with the E_{8} lattice. We may choose Q_{8} such that Q_{8} ∈ A_{8}^{+}(t). Because the classical theta series on ℍ attached to E_{8} is proportional to the elliptic Eisenstein series of weight 4, and hence has positive Fourier coefficients. We have already proven in this paper that Z^{J}(k, Φ) ≠ 0 for Hecke-Jacobi eigenforms Φ. Finally Theorem 1.5 and Siegel’s Theorem complete the proof of this theorem.

Corollary 4.5 Let k, t be natural numbers. Let k > 4, k ≡ 0 mod 4 and t be square free. Then the space J_{k,t} is ℂ− linearly spanned by the theta series Θ_{Q_{j}}^{(1)}(τ, z), (1 ≤ j ≤ H). Here Q_{1}, Q_{2}, …, Q_{H} are given as in the Theorem 4.4.

Proof: This follows from the results of Theorem 4.4 and a structure theorem [Ar94], section 4.2 on the space of Jacobi forms J_{k,t}. Here a certain maximality condition on the index t is essential.

References

[An87] A. N. Andrianov, Quadratic Forms and Hecke Operators, Grundlehren der math. Wissenschaften 286. Berlin, Heidelberg, New York: Springer 1987

[Ar90] T. Arakawa, Real Analytic Eisenstein Series for the Jacobi Group, Abh. Math. Sem.Univ. Hamburg 60 (1990), 131-148

[Ar93] T. Arakawa, Siegel’s formula for Jacobi forms, Int. J. in Math. 4 (1993), 689-719 [Ar94] T. Arakawa, Jacobi Eisenstein series and a basis problem for Jacobi forms, Comm.

Math. Universitatis Sancti Pauli Vol. 43 No.2 (1994), 181-216 [A-H98] T. Arakawa and B. Heim, Real analytic Jacobi Eisenstein series and Dirichlet

series attached to three Jacobi forms, MPI preprint, 1998 (66)

[Boe83] S. Böcherer, Über die Fourier-Jacobi-Entwicklung der Siegelschen Eisensteinreihen , Math. Z.183 (1983), 21-46

[Boe85] S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. reine angew. Math. 362 (1985), 146-168

[B-K93] S. Böcherer and W. Kohnen, Estimates for Fourier coefficients of Siegel cusp

forms, Math. Ann. 297 (1993), 499-517

[Br98] S. Breulmann, Estimates for Fourier coefficients of Siegel cusp forms of genus 3, Math. Ann. 310, no 1 (1998), 129-160

[Du97] J. Dulinski, L-Functions for Jacobi Forms on IH × ℂ, Result. Math. 31 (1997), 75-94

[E-Z85] M. Eichler und D. Zagier, The theory of Jacobi forms, Progress in Mathematics

Vol. 55, Birkhäuser Verlag (1985)

[Ga84] P. Garrett, Pullbacks of Eisenstein series; Applications. In: Automorphic forms of several variables. Taniguchi symposium 1983, Birkhäuser (1984)

[Ga89] P. Garrett, Integral representations of Eisenstein series and L-functions, Number theory, trace formulas and discrete groups, Symposium in honor of Atle Selberg, Academic press (1989)

[Ga92] P. Garrett, On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients, Inv. Math. 107 (1992), 453-481

[Gr95] V. A. Gritsenko, Modulformen zur Paramodulgruppe und Modulräume der Abelschen Varietäten, Mathematica Gottingensis, Schriftreihe des Sonderforschungsbereichs Geometrie und Analysis Heft 12 (1995)

[He96] B. Heim, Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic Lfunctions, to appear in the Proceedings of Symposia of Pure Mathematics, in honor of G. Shimura 1996

[He97] B. Heim, Analytic Jacobi Eisenstein series and the Shimura method, MPI preprint 1998 (1)

[Kl90] H. Klingen, Introductory lectures on Siegel modular forms, Cambridge: Cambridge University Press 1990

[K-S89] W. Kohnen and N.-P. Skoruppa, A certain Dirichlet series attached to Siegel

modular forms of degree two, Invent. Math. 95 (1989), 449-476

[Mu89] A. Murase, L-functions attached to Jacobi forms of degree n, Part I. The basic identity, J. reine Math. 401 (1989), 122-156

[Mu91] A. Murase, L-functions attached to Jacobi forms of degree n, Part II. Functional equation, Math. Ann 290 (1991), 247-276

[M-S91] A. Murase and T. Sugano, Whittaker-Shintani functions on the symplectic group

of Fourier-Jacobi type, Comp. Mathematica 79 (1991), 321-349 [S-Z88] N.-P Skoruppa and D. Zagier, Jacobi forms and a certain space of modular

forms, Inv. Math. 94, No 1. (1988), 113-146

[Su95] T. Sugano, Jacobi forms and the Theta lifting, Comment. Math. Univ. St. Pauli, Vol 44, No1 (1995), 1-58

————– 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