Lifting from two Elliptic Modular Forms

to Siegel Modular Forms of Half-Integral Weight

of even Degree

Shuichi Hayashida

Received: February 24, 2015 Revised: December 12, 2015

Communicated by Don Blasius

Abstract. The aim of this paper is to construct lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree under the assumption that the constructed Siegel modular form is not identically zero. The key ingredient of the proof is a new Maass relation for Siegel modular forms of half-integral weight and any degree.

2010 Mathematics Subject Classification: 11F46 (primary), 11F37, 11F50 (secondary) Keywords and Phrases: Siegel modular forms, Jacobi forms, Maass relation

1 Introduction

1.1

Lifts from two elliptic modular forms to Siegel modular form of halfintegral weight of degree two have been conjectured by Ibukiyama and the author[H-I 05]. In the present article we will give a partial answer for the conjecture in [H-I 05] and shall generalize these lifts as lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of any even degree (Theorem 8.3). The construction of the lift can be regarded as a half-integral weight version of the Miyawaki-Ikeda lift. The Miyawaki-Ikeda lift has been shown by Ikeda [Ik 06]. In the present article we will give a proof to the fact that constructed Siegel modular forms of half-integral weight are eigenforms, if it does not identically vanish. Moreover, we will compute the L-function of the constructed Siegel modular forms of half-integral weight. The key ingredient of the proof of the lift in the present article is to introduce a generalized Maass relation for Siegel modular forms of half-integral weight (Theorem 7.6, 8.2). Generalized Maass relations are relations among Fourier-Jacobi coefficients of Siegel modular forms and are regarded as relations among Fourier coefficients. Theorem 7.6 is a generalization of the Maass relation for generalized Cohen-Eisenstein series, which is a Siegel modular form of half-integral weight of general degree. And Theorem 8.2 is a generalization of the Maass relation for Siegel cusp forms of half-integral weight of odd degree.

1.2

We explain our results more precisely. We denote by M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) the generalized plus-space of weight k − \frac{1}{2} of degree n, which is a subspace of Siegel modular forms of half-integral weight and is a generalization of the Kohnen plus-space (see [Ib 92] or §4.3 for the definition of generalized plus-space). Let F ∈ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) be an eigenform for any Hecke operators. We put

Q_{F,p}(z) = ∏_{i=0}^{n}(1 − μ_{i,p}z)(1 − μ_{i,p}^{−1}z),

where complex numbers {μ_{i,p}^{±}} are p-parameters of F introduced in [Zh 84] if p is an odd prime. If p = 2, then we define {μ_{i,2}^{±}} by using the isomorphism between generalized plus-space and the space of Jacobi forms of index 1. We denote the modified Zhuravlev L-function by

L(s, F) := ∏_{p} Q_{F,p}(p^{−s+k− \frac{3}{2}} ).

The Zhuravlev L-function is originally introduced in [Zh 84] without the Euler 2-factor, which is a generalization of the L-function of elliptic modular forms of half-integral weight introduced in [Sh 73]. We denote by S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) the space of Siegel cusp forms in M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)). The following theorem is the main result of this article.

Theorem 8.3. Let k be an even integer and n be an integer greater than 1. Let h ∈ S_{k−n+ \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)) and g ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)) be eigenforms for all Hecke

operators. Then there exists a F_{h,g} ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−2)}). Under the assumption that F_{h,g} is not identically zero, then F_{h,g} is an eigenform with the L-function which satisfies

L(s, F_{h,g}) = L(s, g) ∏_{i=1}^{2n−3} L(s − i, h).

By numerical computations of Fourier coefficients of F_{h,g} we checked that F_{h,g} does not identically vanish for some (n, k). (See §9 for the detail). Remark that the above theorem was first conjectured by Ibukiyama and the author [H-I 05] in the case of n = 2 not only for even integer k, but also for odd integer k. The construction of F_{h,g} was suggested by T. Ikeda to the author, which is given by a composition of three maps and an inner product. These three maps are a Ikeda lift (Duke-Imamoglu-Ibukiyama-Ikeda lift), a map of the Fourier-Jacobi expansion and an isomorphism between Jacobi forms of index 1 and Siegel modular forms of half-integral weight. In §8 we will explain the detail of the construction of F_{h,g}. To prove Theorem 8.3 we use a generalized Maass relation for generalized Cohen-Eisenstein series (Theorem 7.6). Once we obtain Theorem 7.6, it is not so hard to show Theorem 8.3. The most part of this article is devoted to show Theorem 7.6. We now explain the generalized Maass relation for generalized Cohen-Eisenstein series (Theorem 7.6). Let k be an even integer and H_{k− \frac{1}{2}}^{(n+1)} be the generalized Cohen-Eisenstein series

of degree n+1 of weight k − \frac{1}{2} (see §4.4 for the definition of generalized Cohen-Eisenstein series). The form H_{k− \frac{1}{2}}^{(n+1)} is a Siegel modular form of weight k − \frac{1}{2} of degree n + 1. For integer m, we denote by e_{k− \frac{1}{2} ,m}^{(n)} the m-th Fourier-Jacobi coefficient of

H_{k− \frac{1}{2}}^{(n+1)}:

H_{k− \frac{1}{2}}^{(n+1)}((\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{\begin{matrix}m≥0 \\ m≡0,3 mod 4 \end{matrix}} e_{k− \frac{1}{2} ,m}^{(n)}(τ, z) e^{2π\sqrt{−1}mω}, (1.1)

where τ ∈ ℌ_{n} and ω ∈ ℌ_{1}, and where ℌ_{n} denotes the Siegel upper half space of degree n. We denote by J_{k− \frac{1}{2} ,m}^{(n)} the space of Jacobi forms of degree n of

weight k − \frac{1}{2} of index m (cf. §2.6) and denote by J_{k− \frac{1}{2} ,m}^{(n)∗} (cf. §4.4) a subspace

of J_{k− \frac{1}{2} ,m}^{(n)}. Then, the above form e_{k− \frac{1}{2} ,m}^{(n)} belongs to J_{k− \frac{1}{2} ,m}^{(n)∗}. Because H_{k− \frac{1}{2}}^{(n+1)}

belongs to the generalized plus-space M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4)), we can show that the

form e_{k− \frac{1}{2} ,m}^{(n)} is identically zero unless m ≡ 0, 3 mod 4.

We denote by M_{k}(Γ_{n+2}) the space of Siegel modular forms of weight k of degree n + 2 and denote by J_{k,1}^{(n+1)} the space of Jacobi forms of weight k of index 1

of degree n + 1. We denote by E_{k}^{(n)} ∈ M_{k}(Γ_{n}) the Siegel-Eisenstein series of weight k of degree n (cf. (3.2) in §3) and by E_{k,1}^{(n)} ∈ J_{k,1}^{(n)} the Jacobi-Eisenstein

series of weight k of index 1 of degree n (cf. (3.1) in §3). The form H_{k− \frac{1}{2}}^{(n+1)} is

constructed from E_{k,1}^{(n+1)}. The diagram of the above correspondence is

\xymatrix{E_{k}^{(n+2)} ∈ M_{k}(Γ_{n+2}) \ar[d] \\ E_{k,1}^{(n+1)} ∈ J_{k,1}^{(n+1)} \ar[r] & H_{k− \frac{1}{2}}^{(n+1)} ∈ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4)) \ar[d] \\  & \{e_{k− \frac{1}{2} ,m}^{(n)}\}_{m} ∈ ⨂_{m≡0,3 mod 4} J_{k− \frac{1}{2} ,m}^{(n)∗}.}

In §2.7 (for any odd prime p) and in §4.7 (for p = 2) we will introduce index-shift maps Ṽ_{α,n−α}(p^{2}) (α = 0, ..., n), which are linear maps from J_{k− \frac{1}{2} ,m}^{(n)∗} to the space of holomorphic functions on ℌ_{n} × ℂ^{(n,1)}. If p is odd then Ṽ_{α,n−α}(p^{2}) is a linear map from J_{k− \frac{1}{2} ,m}^{(n)∗} to J_{k− \frac{1}{2} ,mp^{2}}^{(n)}. These maps Ṽ_{α,n−α}(p^{2}) are generalizations of the V_{l}-map in [E-Z 85, p.43] for half-integral weight of general degrees. For any ϕ ∈ J_{k− \frac{1}{2} ,m}^{(n)} and for any integer a we define (ϕ|U_{a})(τ, z) := ϕ(τ, az). The following theorem is a generalization of the Maass relation for the generalized Cohen-Eisenstein series, where we use the symbol

\begin{matrix}e_{k− \frac{1}{2} ,m}^{(n)}|(Ṽ_{0,n}(p^{2}), Ṽ_{1,n−1}(p^{2}), ..., Ṽ_{n,0}(p^{2})) \\ := (e_{k− \frac{1}{2} ,m}^{(n)}|Ṽ_{0,n}(p^{2}), e_{k− \frac{1}{2} ,m}^{(n)}|Ṽ_{1,n−1}(p^{2}), ..., e_{k− \frac{1}{2} ,m}^{(n)}|Ṽ_{n,0}(p^{2})). \end{matrix}

Theorem 7.6. Let e_{k− \frac{1}{2} ,m}^{(n)} be the m-th Fourier-Jacobi coefficient of generalized Cohen-Eisenstein series H_{k− \frac{1}{2}}^{(n+1)}. (See (1.1)). Then we obtain

e_{k− \frac{1}{2} ,m}^{(n)}|(Ṽ_{0,n}(p^{2}), Ṽ_{1,n−1}(p^{2}), ..., Ṽ_{n,0}(p^{2}))
= p^{k(n−1)− \frac{1}{2} (n^{2}+5n−5)} (e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(n)} |U_{p^{2}}, e_{k− \frac{1}{2} ,m}^{(n)}|U_{p}, e_{k− \frac{1}{2} ,mp^{2}}^{(n)})

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ) diag(1, p^{1/2}, ⋯, p^{n/2}).

Here A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ) is a 2 × (n + 1) matrix which is introduced in the beginning of §7 and the both sides of the above identity are vectors of forms. For any prime p we regard e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(n)} as zero, if \frac{m}{p^{2}} is not an integer or \frac{m}{p^{2}} ≢ 0,

3 mod 4. The symbol (\frac{∗}{p}) denotes the Legendre symbol for odd prime p, and (\frac{a}{2}) := 0, 1, −1 accordingly as a is even, a ≡ ±1 mod 8 or a ≡ ±3 mod 8.

Theorem 7.6 gives also a relation among Fourier coefficients of Siegel-Eisenstein series of integral weight. The Fourier coefficients of Ikeda lifts satisfy similar relations to the ones of the Fourier coefficients of Siegel-Eisenstein series (see Theorem 8.2 for the detail). We call these relations of Fourier coefficients of Ikeda lifts also the generalized Maass relations. The generalized Maass relation among Fourier coefficients of the Ikeda lift I_{2n}(h) of h gives a fact that F_{h,g} in Theorem 8.3 is an eigenform for all Hecke operators, since the form F_{h,g} is constructed from I_{2n}(h) (and g). Moreover, the eigenvalues of F_{h,g} are calculated from the generalized Maass relations of Fourier coefficients of I_{2n}(h). This is the reason why we need Theorem 7.6 to show Theorem 8.3. For the detail of the proof of Theorem 8.3 see §8.

1.3 About generalized Cohen-Eisenstein series

We remark that the generalized Cohen-Eisenstein series has been introduced by Arakawa [Ar 98]. These series are Siegel modular forms of half-integral weight. The Cohen-Eisenstein series were originally introduced by Cohen [Co 75] as one variable functions. In the case of degree one, it is known that the Cohen-Eisenstein series correspond to the Eisenstein series with respect to SL(2, ℤ) by the Shimura correspondence. The generalized Cohen-Eisenstein series is defined from the Jacobi-Eisenstein series of index 1 through the isomorphism between Jacobi forms of index 1 and Siegel modular forms of half-integral weight.

1.4 About generalized Maass relations

As for generalizations of the Maass relation, Yamazaki [Yk 86, Yk 89] obtained some relations among Fourier-Jacobi coefficients of Siegel-Eisenstein series of arbitrary degree of integral weight of integer indices. For our purpose we generalize some results in [Yk 86, Yk 89] on Fourier-Jacobi coefficients of Siegel-Eisenstein series of integer indices to indices of half-integral symmetric matrix of size 2. Here the right-lower part or the left-upper part of these matrices of the index is 1. We need to introduce index-shift maps on Jacobi forms of indices of such matrix (cf. §2.7). To calculate the action of index-shift maps on Fourier-Jacobi coefficients of Siegel-Eisenstein series, we use a relation between Fourier-Jacobi coefficients of Siegel-Eisenstein series and Jacobi-Eisenstein series (cf. Proposition 3.3). This relation has been shown by Boecherer [Bo 83, Satz7]. We also need to show a identity relation between Jacobi forms of integral weight of 2 × 2 matrix index and Jacobi forms of half-integral weight of integer index (Lemma 4.2). Moreover, we need to show a compatibility between this identity relation and index-shift maps (cf. Proposition 4.3, 4.4). Through these relations we can show that the generalized Maass relation of generalized Cohen-Eisenstein series (Theorem 7.6) are equivalent to relations among Jacobi-Eisenstein series of integral weight of indices of matrix of size 2 (Proposition 7.4). Finally, to obtain the generalized Maass relation in Theorem 7.6, we need to calculate the action of index-shift maps on Jacobi-Eisenstein series of integral weight of indices of matrix of size 2 (cf. §5).

Remark 1.1 In his paper [Ko 02] Kohnen gives a generalization of the Maass relation for Siegel modular forms of even degree 2n. His result is different from our generalization, since his result is concerned with the Fourier-Jacobi coefficients with (2n − 1) × (2n − 1) matrix index. We remark that some characterizations of the Ikeda lift by using generalized Maass relation in [Ko 02] are obtained by Kohnen-Kojima [KK 05] and by Yamana [Yn 10]. The characterization of the Ikeda lift by using the generalized Maass relation in Theorem 8.2 is open problem.

Remark 1.2 In his paper [Ta 86, §5] Tanigawa has obtained the same identity in Theorem 7.6 for Siegel-Eisenstein series of half-integral weight of degree two with arbitrary level N which satisfies 4|N. He showed the identity by using the formula of local densities under the assumption p |̸N. In our case we treat the generalized Cohen-Eisenstein series of arbitrary degree, which has essentially level 1. Hence our result contains the relation also for p = 2. Moreover, our result is valid for any general degree.

Remark 1.3 To show the generalized Maass relations in Theorem 7.6, 8.2, we treat the following four things:

1. Fourier-Jacobi expansion of Jacobi forms (cf. §4.1),

2. Fourier-Jacobi expansion of Siegel modular forms of half-integral weight (cf. §4.2),

3. An isomorphism between Jacobi forms of matrix index of integral weight and Jacobi forms of integer index of half-integral weight (cf. §4.5)

4. Exchange relations between the Siegel Φ-operator for Jacobi forms and the index-shift map for Jacobi forms of matrix index or of half-integral weight (cf. §6). This is an analogue of the result shown by Krieg [Kr 86] in the case of Siegel modular forms of integral weight.

1.5

This paper is organized as follows: in Sect. 2, the necessary notation and definitions are reviewed. In Sect. 3, the relation among Fourier-Jacobi coefficients of the Siegel-Eisenstein series and the Jacobi-Eisenstein series is derived, which is a modification of the result given by Boecherer [Bo 83] for special cases. In Sect. 4, a map from a subspace of Jacobi forms of integral weight of matrix index to a subspace of Jacobi forms of half-integral weight of integer index is defined. Moreover, the compatibility of this map with index-shift maps is studied. In Sect. 5, we calculate the action of index-shift maps on the Jacobi-Eisenstein series. We express these functions as summations of exponential functions with generalized Gauss sums. In Sect. 6, a commutativity between index-shift maps on Jacobi forms and Siegel Φ-operators is derived. In Sect. 7, a generalized Maass relation for generalized Cohen-Eisenstein series (Theorem 7.6) will be proved, while we will give a generalized Maass relation for Siegel cusp forms of half-integral weight and the proof of the main result (Theorem 8.3) in Sect. 8. We shall explain some numerical examples of the non-vanishing of the lift in Sect. 9.

Acknowledgement. The construction of the lift in this article was suggested by Professor Tamotsu Ikeda to the author at the Hakuba Autumn Workshop 2001. The author wishes to express his hearty gratitude to Professor Ikeda for the suggestion. The author also would like to express his sincere gratitude to Professor Tomoyoshi Ibukiyama for continuous encouragement. The author thanks very much to the referee, whose advice was helpful in improving the original version of the manuscript. This work was supported by JSPS KAKENHI Grant Number 23740018 and 80597766.

2 Notation and definitions

ℝ^{+} : the set of all positive real numbers R^{(n,m)} : the set of n × m matrices with entries in a commutative ring R Sym_{n}^{∗} : the set of all half-integral symmetric matrices of size n Sym_{n}^{+} : all positive definite matrices in Sym_{n}^{∗} tB : the transpose of a matrix B A[B] := ^{t}BAB for two matrices A ∈ R^{(n,n)} and B ∈ R^{(n,m)} 1_{n} (resp. 0_{n}) : identity matrix (resp. zero matrix) of size n tr(S) : the trace of a square matrix S e(S) := e^{2π\sqrt{−1} tr(S)} for a square matrix S rank_{p}(x) : the rank of matrix x ∈ ℤ^{(n,m)} over the finite field ℤ/pℤ

diag(a_{1}, ..., a_{n}) : the diagonal matrix ( ..._{a_{n}}^{a_{1}} ) for square matrices a_{1}, ...,

a_{n} (\frac{∗}{p}) : the Legendre symbol for odd prime p

(\frac{∗}{2}) := 0, 1, −1 accordingly as a is even, a ≡ ±1 mod 8 or a ≡ ±3 mod 8 M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)) : the space of Siegel modular forms of weight k − \frac{1}{2} of degree n

M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) : the plus-space of M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)) (cf. [Ib 92]). ℌ_{n} : the Siegel upper half space of degree n δ(S) := 1 or 0 accordingly as the statement S is true or false.

For any function F and operators T_{1}, T_{2}, ... , T_{n} we put

F |(T_{1}, T_{2}, ..., T_{n}) := (F |T_{1}, F |T_{2}, ..., F |T_{n}).

2.1 Jacobi group

For a positive integer n we define the following groups:

GSp_{n}^{+}(ℝ) := \{g ∈ ℝ^{(2n,2n)} | g ( \begin{matrix}0_{n} & −1_{n} \\ 1_{n} & 0_{n} \end{matrix} )^{t}g = n(g) ( \begin{matrix}0_{n} & −1_{n} \\ 1_{n} & 0_{n} \end{matrix} )
for some n(g) ∈ ℝ^{+}} ,
Sp_{n}(ℝ) := {g ∈ GSp_{n}^{+}(ℝ) | n(g) = 1} ,
Γ_{n} := Sp_{n}(ℝ) ∩ ℤ^{(2n,2n)},
Γ_{∞}^{(n)} := \{(\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Γ_{n} | C = 0_{n}\} ,
Γ_{0}^{(n)}(4) := \{(\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Γ_{n} | C ≡ 0 mod 4\} .

For a matrix g ∈ GSp_{n}^{+}(ℝ), the number n(g) in the above definition of GSp_{n}^{+}(ℝ) is called the similitude of the matrix g. For positive integers n and r, we define a subgroup G_{n,r}^{J} ⊂ GSp_{n+r}^{+}(ℝ) by

G_{n,r}^{J} := \{(\begin{matrix}A & & B \\ & U \\ C & & D \\ & & & V \end{matrix} ) (\begin{matrix}1_{n} & & & μ \\ ^{t}λ & 1_{r} & ^{t}μ & ^{t}λμ + κ \\ & & 1_{n} & −λ \\ & & & 1_{r} \end{matrix}) ∈ GSp_{n+r}^{+}(ℝ)\} ,

where the matrices runs over (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ GSp_{n}^{+}(ℝ), (\begin{matrix}U & 0 \\ 0 & V \end{matrix} ) ∈ GSp_{r}^{+}(ℝ),

λ, μ ∈ ℝ^{(n,r)} and κ = ^{t}κ ∈ ℝ^{(r,r)}.

We will abbreviate such an element ( \begin{matrix}A & & B \\ & U \\ C & & D \\ & & & V \end{matrix} ) ( \begin{matrix}1_{n} & & & μ \\ ^{t}λ & 1_{r} & ^{t}μ & ^{t}λμ+κ \\ & & 1_{n} & −λ \\ & & & 1_{r} \end{matrix} ) as

((\begin{matrix}A & B \\ C & D \end{matrix}) × (\begin{matrix}U & 0 \\ 0 & V \end{matrix} ) , [(λ, μ), κ]) .

We remark that two matrices ( \begin{matrix}A & B \\ C & D \end{matrix}) and ( \begin{matrix}U & 0 \\ 0 & V \end{matrix} ) in the above notation have the same similitude. We will often write

((\begin{matrix}A & B \\ C & D \end{matrix}) , [(λ, μ), κ])

instead of writing (( \begin{matrix}A & B \\ C & D \end{matrix}) × 1_{2r}, [(λ, μ), κ]) for simplicity. We remark that the element (( \begin{matrix}A & B \\ C & D \end{matrix}) , [(λ, μ), κ]) belongs to Sp_{n+r}(ℝ). Similarly, an element

( \begin{matrix}1_{n} & & & μ \\ ^{t}λ & 1_{r} & ^{t}μ & ^{t}λμ+κ \\ & & 1_{n} & −λ \\ & & & 1_{r} \end{matrix} ) ( \begin{matrix}A & & B \\ & U \\ C & & D \\ & & & V \end{matrix} )

will be abbreviated as

([(λ, μ), κ], (\begin{matrix}A & B \\ C & D \end{matrix}) × (\begin{matrix}U & 0 \\ 0 & V \end{matrix} )) ,

and we will abbreviate it as ([(λ, μ), κ], ( \begin{matrix}A & B \\ C & D \end{matrix})) for the case U = V = 1_{r} . We set a subgroup Γ_{n,r}^{J} of G_{n,r}^{J} by

Γ_{n,r}^{J} := {(M, [(λ, μ), κ]) ∈ G_{n,r}^{J} | M ∈ Γ_{n}, λ, μ ∈ ℤ^{(n,r)}, κ ∈ ℤ^{(r,r)}} .

2.2 Groups { \atop GSp_{n}^{+}(ℝ)} and G̃_{n,1}^{J}

We denote by { \atop GSp_{n}^{+}(ℝ)} the group which consists of pairs (M, φ(τ)), where M is a matrix M = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ GSp_{n}^{+}(ℝ), and where φ is any holomorphic function on ℌ_{n} such that |φ(τ)|^{2} = det(M)^{− \frac{1}{2}} | det(Cτ + D)|. The group operation on

{ \atop GSp_{n}^{+}(ℝ)} is given by (M, φ(τ))(M′, φ′(τ)) := (MM′, φ(M′τ)φ′(τ)).

We embed Γ_{0}^{(n)}(4) into the group { \atop GSp_{n}^{+}(ℝ)} via M → (M, θ^{(n)}(Mτ) θ^{(n)}(τ)^{−1}), where θ^{(n)}(τ) := ∑_{p∈ℤ^{(n,1)}} e(τ[p]) is the theta constant. We denote by Γ_{0}^{(n)}(4)^{∗}

the image of Γ_{0}^{(n)}(4) in { \atop GSp_{n}^{+}(ℝ)} by this embedding. We define a Heisenberg group

H_{n,1}(ℝ) := {(1_{2n}, [(λ, μ), κ]) ∈ Sp_{n+1}(ℝ) | λ, μ ∈ ℝ^{(n,1)}, κ ∈ ℝ} .

If there is no confusion, we will write [(λ, μ), κ] for the element (1_{2n}, [(λ, μ), κ]) for simplicity. We define a group

G̃_{n,1}^{J} := { \atop GSp_{n}^{+}(ℝ)} ⋉ H_{n,1}(ℝ)
= \{(M̃, [(λ, μ), κ]) | M̃ ∈ { \atop GSp_{n}^{+}(ℝ)}, [(λ, μ), κ] ∈ H_{n,1}(ℝ)\} .

Here the group operation on G̃_{n,1}^{J} is given by

(M̃_{1}, [(λ_{1}, μ_{1}), κ_{1}]) · (M̃_{2}, [(λ_{2}, μ_{2}), κ_{2}]) := (M̃_{1}M̃_{2}, [(λ′, μ′), κ′])

for (M̃_{i}, [(λ_{i}, μ_{i}), κ_{i}]) ∈ G̃_{n,1}^{J} (i = 1, 2), and where [(λ′, μ′), κ′] ∈ H_{n,1}(ℝ) is the matrix determined through the identity

\begin{matrix}(M_{1} × ( \begin{matrix}n(M_{1}) & 0 \\ 0 & 1 \end{matrix} ) , [(λ_{1}, μ_{1}), κ_{1}])(M_{2} × ( \begin{matrix}n(M_{2}) & 0 \\ 0 & 1 \end{matrix} ) , [(λ_{2}, μ_{2}), κ_{2}]) \\ = (M_{1}M_{2} × ( \begin{matrix}n(M_{1})n(M_{2}) & 0 \\ 0 & 1 \end{matrix} ) , [(λ′, μ′), κ′]) \end{matrix}

in G_{n,1}^{J}. Here n(M_{i}) is the similitude of M_{i}.

134 S.Hayashida

2.3 Action of the Jacobi group The group G_{n,r}^{J} acts on ℌ_{n} × ℂ^{(n,r)} by

γ · (τ, z) := ((\begin{matrix}A & B \\ C & D \end{matrix}) · τ , ^{t}(Cτ + D)^{−1}(z + τλ + μ)^{t}U)

for any γ = (( \begin{matrix}A & B \\ C & D \end{matrix}) × ( \begin{matrix}U & 0 \\ 0 & V \end{matrix} ) , [(λ, μ), κ]) ∈ G_{n,r}^{J} and for any (τ, z) ∈ ℌ_{n}×ℂ^{(n,r)}.

Here (\begin{matrix}A & B \\ C & D \end{matrix}) · τ := (Aτ + B)(Cτ + D)^{−1} is the usual transformation. The group G̃_{n,1}^{J} acts on ℌ_{n} × ℂ^{(n,1)} through the projection G̃_{n,1}^{J} → G_{n,1}^{J}. It means G̃_{n,1}^{J} acts on ℌ_{n} × ℂ^{(n,1)} by

γ̃ · (τ, z) := (M × ( \begin{matrix}n(M) & 0 \\ 0 & 1 \end{matrix} ) , [(λ, μ), κ]) · (τ, z)

for γ̃ = ((M, φ), [(λ, μ), κ]) ∈ G̃_{n,1}^{J} and for (τ, z) ∈ ℌ_{n} × ℂ^{(n,1)}. Here n(M) is the similitude of M ∈ GSp_{n}^{+}(ℝ).

2.4 Factors of automorphy Let k be an integer and let M ∈ Sym_{r}^{+}. For γ = (( \begin{matrix}A & B \\ C & D \end{matrix}) × ( \begin{matrix}U & 0 \\ 0 & V \end{matrix} ) , [(λ, μ), κ]) ∈ G_{n,r}^{J} we define a factor of automorphy

J_{k,M} (γ, (τ, z))

:= det(V)^{k} det(Cτ + D)^{k} e(V^{−1}MU(((Cτ + D)^{−1}C)[z + τλ + μ]))
×e(−V^{−1}MU(^{t}λτλ +^{t}zλ +^{t}λz +^{t}μλ +^{t}λμ + κ)).

We define a slash operator |_{k,M} by

(ϕ|_{k,M}γ)(τ, z) := J_{k,M}(γ, (τ, z))^{−1}ϕ(γ · (τ, z))

for any function ϕ on ℌ_{n} × ℂ^{(n,r)} and for any γ ∈ G_{n,r}^{J}. We remark that

J_{k,M}(γ_{1}γ_{2}, (τ, z)) = J_{k,M}(γ_{1}, γ_{2} · (τ, z))J_{k,V_{1}^{−1}MU_{1}}(γ_{2}, (τ, z)),
ϕ|_{k,M}γ_{1}γ_{2} = (ϕ|_{k,M}γ_{1})|_{k,V_{1}^{−1}MU_{1}}γ_{2}.

for any γ_{i} = (M_{i} × ( \begin{matrix}U_{i} & 0 \\ 0 & V_{i} \end{matrix} ) , [(λ_{i}, μ_{i}), κ_{i}]) ∈ G_{n,r}^{J} (i = 1, 2).

Let k and m be integers. We define a slash operator |_{k− \frac{1}{2} ,m} for any function ϕ on ℌ_{n} × ℂ^{(n,1)} by

ϕ|_{k− \frac{1}{2} ,m}γ̃ := J_{k− \frac{1}{2} ,m}(γ̃, (τ, z))^{−1}ϕ(γ̃ · (τ, z))

for any γ̃ = ((M, φ), [(λ, μ), κ]) ∈ G̃_{n,1}^{J}. Here we define a factor of automorphy

J_{k− \frac{1}{2} ,m}(γ̃, (τ, z)) := φ(τ)^{2k−1}e(n(M)m(((Cτ + D)^{−1}C)[z + τλ + μ]))

×e(−n(M)m(^{t}λτλ +^{t}zλ +^{t}λz +^{t}μλ +^{t}λμ + κ)),

Documenta Mathematica 21 (2016) 125–196

where n(M) is the similitude of M. We remark that

J_{k− \frac{1}{2} ,m}(γ̃_{1}γ̃_{2}, (τ, z)) = J_{k− \frac{1}{2} ,m}(γ̃_{1}, γ̃_{2} · (τ, z))J_{k− \frac{1}{2} ,n(M_{1})m}(γ̃_{2}, (τ, z))
ϕ|_{k− \frac{1}{2} ,m}γ̃_{1}γ̃_{2} = (ϕ|_{k− \frac{1}{2} ,m}γ̃_{1})|_{k− \frac{1}{2} ,n(M_{1})m}γ̃_{2}

for any γ̃_{i} = ((M_{i}, φ_{i}), [(λ_{i}, μ_{i}), κ_{i}]) ∈ G̃_{n,1}^{J} (i = 1, 2).

2.5 Jacobi forms of matrix index

We quote the definition of Jacobi form of matrix index from [Zi 89].

Definition 1. For an integer k and for an matrix M ∈ Sym_{r}^{+}, a ℂ-valued holomorphic function ϕ on ℌ_{n} × ℂ^{(n,r)} is called a Jacobi form of weight k of index M of degree n, if ϕ satisfies the following two conditions:

1. the transformation formula ϕ|_{k,M}γ = ϕ for any γ ∈ Γ_{n,r}^{J},

2. ϕ has the Fourier expansion: ϕ(τ, z) = ∑_{\begin{matrix}N∈Sym_{n}^{∗},R∈Z^{(n,r)} \\ 4N−RM^{−1}^{t}R≥0 \end{matrix}}c(N, R)e(Nτ)e(^{t}Rz).

We remark that the second condition follows from the Koecher principle (cf. [Zi 89, Lemma 1.6]) if n > 1. In the condition (2), if ϕ satisfies c(N, R) = 0 unless 4N − RM^{−1}^{t}R > 0, then ϕ is called a Jacobi cusp form. We denote by J_{k,M}^{(n)} the ℂ-vector space of Jacobi forms of weight k of index M of degree n.

2.6 Jacobi forms of half-integral weight

We set a subgroup Γ_{n,1}^{J∗} of G̃_{n,1}^{J} by

Γ_{n,1}^{J∗} := {(M^{∗}, [(λ, μ), κ]) ∈ G̃_{n,1}^{J} | M^{∗} ∈ Γ_{0}^{(n)}(4)^{∗}, λ, μ ∈ ℤ^{(n,1)}, κ ∈ ℤ}
≅ Γ_{0}^{(n)}(4)^{∗} ⋉ H_{n,1}(ℤ),

where we put H_{n,1}(ℤ) := H_{n,1}(ℝ) ∩ ℤ^{(2n+2,2n+2)}. Here the group Γ_{0}^{(n)}(4)^{∗} was defined in §2.2.

Definition 2. For an integer k and for an integer m, a holomorphic function ϕ on ℌ_{n} ×ℂ^{(n,1)} is called a Jacobi form of weight k− \frac{1}{2} of index m, if ϕ satisfies the following two conditions:

1. the transformation formula ϕ|_{k− \frac{1}{2} ,m}γ^{∗} = ϕ for any γ^{∗} ∈ Γ_{n,1}^{J∗},

2. ϕ^{2}|_{2k−1,2m}γ has the Fourier expansion for any γ ∈ Γ_{n,1}^{J}:

(ϕ^{2}|_{2k−1,2m}γ) (τ, z) = ∑_{\begin{matrix}N∈Sym_{n}^{∗},R∈ℤ^{(n,1)} \\ 4Nm−hR^{t}R≥0 \end{matrix}} C(N, R) e(\frac{1}{h}Nτ) e(^{t}Rz) .

with a integer h > 0, and where the slash operator |_{k− \frac{1}{2} ,m} was defined in §2.4.

In the condition (2), for any γ if ϕ satisfies C(N, R) = 0 unless 4Nm−hR^{t}R > 0, then ϕ is called a Jacobi cusp form. We denote by J_{k− \frac{1}{2} ,m}^{(n)} the ℂ-vector space of Jacobi forms of weight k − \frac{1}{2} of index m of degree n.

2.7 Index-shift maps of Jacobi forms

In this subsection we introduce two kinds of maps. The both maps shift the index of Jacobi forms and these are generalizations of the V_{l}-map in the sense of Eichler-Zagier [E-Z 85]. We define two groups GSp_{n}^{+}(ℤ) := GSp_{n}^{+}(ℝ) ∩ ℤ^{(2n,2n)} and

{ \atop GSp_{n}^{+}(ℤ)} := \{(M, φ) ∈ { \atop GSp_{n}^{+}(ℝ)} | M ∈ GSp_{n}^{+}(ℤ)\} .

First we define index-shift maps for Jacobi forms of integral weight of matrix index. Let M = ( \begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix} ) ∈ Sym_{2}^{+}. Let X ∈ GSp_{n}^{+}(ℤ) be a matrix such that the similitude of X is n(X) = p^{2} with a prime p. For any ϕ ∈ J_{k,M}^{(n)} we define the map

ϕ|V (X)

:= ∑_{u,v∈(ℤ/pℤ)^{(n,1)}} ∑_{M∈Γ_{n}\Γ_{n}XΓ_{n}} ϕ|_{k,M} (M × ( \begin{matrix}p^{2} & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & p \end{matrix} ) , [((0, u), (0, v)), 0_{2}]) ,

where (0, u), (0, v) ∈ (ℤ/pℤ)^{(n,2)} and where 0_{2} is the zero matrix of size 2.

See §2.1 for the symbol of the matrix (M × ( \begin{matrix}p^{2} & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & p \end{matrix} ) , [((0, u), (0, v)), 0_{2}]) .

The above summations are finite sums and do not depend on the choice of the representatives u, v and M. A straightforward calculation shows that ϕ|V (X) belongs to J_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}. Namely V (X) is a map:

V (X) : J_{k,M}^{(n)} → J_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}.

For the sake of simplicity we set

V_{α,n−α}(p^{2}) := V (diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α}))

for any prime p and for any α (0 ≤ α ≤ n). Next we shall define index-shift maps for Jacobi forms of half-integral weight of integer index. We assume that p is an odd prime. Let m be a positive integer.

Let Y = (X, φ) ∈ { \atop GSp_{n}^{+}(ℤ)} with n(X) = p^{2t}, where t is a positive integer. For ψ ∈ J_{k− \frac{1}{2} ,m}^{(n)} we define

ψ|Ṽ(Y) := n(X)^{\frac{n(2k−1)}{4} − \frac{n(n+1)}{2}} ∑_{M̃∈Γ_{0}^{(n)}(4)^{∗}\Γ_{0}^{(n)}(4)^{∗}Y Γ_{0}^{(n)}(4)^{∗}} ψ|_{k− \frac{1}{2} ,m}(M̃, [(0, 0), 0]),

where the above summation is a finite sum and does not depend on the choice of the representatives M̃. A direct computation shows that ψ|Ṽ(Y) belongs to J_{k− \frac{1}{2} ,mp^{2t}}^{(n)}. For the sake of simplicity we set

Ṽ_{α,n−α}(p^{2}) := Ṽ ((diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α}), p^{α/2}))

for any odd prime p and for any α (0 ≤ α ≤ n). As for p = 2, we will introduce index-shift maps Ṽ_{α,n−α}(4) in §4.7, which are maps from a subspace J_{k− \frac{1}{2} ,m}^{(n)∗} of

J_{k− \frac{1}{2} ,m}^{(n)} to J_{k− \frac{1}{2} ,4m}^{(n)}.

2.8 Hecke operators for Siegel modular forms of half-integral weight

The Hecke theory for Siegel modular forms was first introduced by Shimura [Sh 73] for degree n = 1 and by Zhuravlev [Zh 83, Zh 84] for degree n > 1. We quote the definition of Hecke operator from [Zh 83, Zh 84]. Let Y = (X, φ) ∈

{ \atop GSp_{n}^{+}(ℤ)}. Let ϕ ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)). We define

ϕ|T̃(Y) := n(X)^{\frac{n(2k−1)}{4} − \frac{n(n+1)}{2}} ∑_{M̃∈Γ_{0}^{(n)}(4)^{∗}\Γ_{0}^{(n)}(4)^{∗}Y Γ_{0}^{(n)}(4)^{∗}} ϕ|_{k− \frac{1}{2}} M̃,

where (ϕ|_{k− \frac{1}{2}} M̃)(τ) := φ(τ)^{−2k+1}ϕ(M · τ) for M̃ = (M, φ) and n(X) is the similitude of X. For the sake of simplicity we set

T̃_{α,n−α}(p^{2}) := T̃((diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α}), p^{α/2}))

for any odd prime p and for any α (0 ≤ α ≤ n).

2.9 L-function of Siegel modular forms of half-integral weight

In this subsection we review the Hecke theory for Siegel modular forms of half-integral weight which has been introduced by Zhuravlev [Zh 83, Zh 84] and quote the definition of L-function of a Siegel modular form of half-integral weight. Let H̃_{p^{2}}^{(m)} be the local Hecke ring generated by double cosets

K_{α}^{(m)} := Γ_{0}^{(m)}(4)^{∗}(diag(1_{α}, p1_{m−α}, p^{2}1_{α}, p1_{m−α}), p^{α/2})Γ_{0}^{(m)}(4)^{∗} (0 ≤ α ≤ m) and K_{0}^{(m)}^{−1} over ℂ. If p is an odd prime, then it is shown in [Zh 83, Zh 84] that the local Hecke ring H̃_{p^{2}}^{(m)} is commutative and there exists the isomorphism map

Ψ_{m} : H̃_{p^{2}}^{(m)} → R_{m},

where the symbol R_{m} denotes R_{m} := ℂ^{W_{2}}[z_{0}^{±}, z_{1}^{±}, ..., z_{m}^{±}], and where the subring ℂ^{W_{2}}[z_{0}^{±}, z_{1}^{±}, ..., z_{m}^{±}] of ℂ [z_{0}^{±}, z_{1}^{±}, ..., z_{m}^{±}] consists of all W_{2}-invariant polynomials. Here W_{2} is the Weyl group of a symplectic group and the action of W_{2} on ℂ[z_{0}^{±}, ..., z_{m}^{±}] is generated by all permutations of {z_{1}, ..., z_{m}} and by the maps

σ_{i} : z_{0} → z_{0}z_{i}, z_{i} → z_{i}^{−1}, z_{j} → z_{j} (j ≠ i)

for i = 1, ..., m. The isomorphism Ψ_{m} is defined as follows: Let

T = ∑_{i} a_{i}Γ_{0}^{(m)}(4)^{∗}(X_{i}, φ_{i})

be a decomposition of T ∈ H̃_{p^{2}}^{(m)}, where a_{i} ∈ ℂ and (X_{i}, φ_{i}) ∈ { \atop GSp_{n}^{+}(ℤ)}. We

can assume that X_{i} is an upper-triangular matrix X_{i} = ( \begin{matrix}p^{δ_{i}}^{t}D_{i}^{−1} & B_{i} \\ 0 & D_{i} \end{matrix} ) with

D_{i} = (\begin{matrix}d_{i1} & ∗ & ∗ \\ 0 & ... & ∗ \\ 0 & 0 & d_{im} \end{matrix})

and φ_{i} is a constant function. It is known that |φ_{i}|^{−1}φ_{i} is a forth root of unity. Then Ψ_{m}(T) is given by

Ψ_{m}(T) := ∑_{i} a_{i} (\frac{φ_{i}}{|φ_{i}|})^{−2k+1} z_{0}^{δ_{i}} ∏_{j=1}^{m}(p^{−j}z_{j})^{d_{ij}}

with a fixed integer k. For the explicit decomposition of generators K_{α}^{(m)} by left Γ_{0}^{(m)}(4)^{∗}-cosets, see [Zh 83, Prop.7.1]. We define γ_{j} ∈ ℂ[z_{1}^{±}, ..., z_{m}^{±}] (j = 0, ..., 2m) through the identity

∑_{j=0}^{2m} γ_{j}X^{j} = ∏_{i=1}^{m} {(1 − z_{i}X) (1 − z_{i}^{−1}X)} .

Here γ_{j} (j = 0, ..., 2m) is a W_{2}-invariant. There exists γ̃_{i,p} ∈ H̃_{p^{2}}^{(m)} (i = 0, ..., 2m) which satisfies Ψ_{m}(γ̃_{i,p}) = γ_{i} ∈ R_{m}. We remark that γ̃_{i,p} = γ̃_{2m−i,p} and γ̃_{0,p} = K_{0}^{(m)}.

For p = 2 we will introduce in §4.3 the Hecke operators T̃_{α,m−α}(4) (α = 0, ..., m) through the isomorphism between Siegel modular forms of half-integral weight and Jacobi forms of index 1 (see (4.2) in §4.3). We remark that the Hecke operators T̃_{α,m−α}(4) (α = 0, ..., m) are defined for the generalized plus-space, which is a subspace of M_{k− \frac{1}{2}} (Γ_{0}^{(m)}(4)). Through the definition of γ̃_{i,p} for odd prime p, we define γ̃_{i,2} in the same formula by using T̃_{α,m−α}(4) (α = 0, ..., m) as in the case of odd primes. by replacing p by 2. Let F ∈ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(m)}(4)) be an eigenform for any Hecke operator T̃_{α,m−α}(p^{2})

(0 ≤ α ≤ m) and for any prime p. Here M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(m)}(4)) denotes the generalized

plus-space which is a subspace of M_{k− \frac{1}{2}} (Γ_{0}^{(m)}(4)) (see [Ib 92] or §4.3 for the

definition of M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(m)}(4))). We define the Euler p-factor of F by

Q_{F,p}(z) : = ∑_{j=0}^{2m} λ_{F} (γ̃_{j,p})z^{j},

where λ_{F} (γ̃_{j,p}) is the eigenvalue of F with respect to γ̃_{j,p}. There exists a set of complex numbers {μ_{0,p}^{2}, μ_{1,p}^{±}, ...μ_{m,p}^{±}} which satisfies

Q_{F,p}(z) = ∏_{i=1}^{m} {(1 − μ_{i,p}z) (1 − μ_{i,p}^{−1}z)}

and

μ_{0,p}^{2}μ_{1,p} ⋯ μ_{m,p} = p^{m(2k−1)/2−m(m+1)/2},

since γ_{2m−j} = γ_{j} (j = 0, ..., m − 1), Q_{F,p}(z^{−1}) = z^{−2m}Q_{F,p}(z) and Q_{F,p}(0) = 1 ≠ 0. Following Zhuravlev [Zh 84] we call the set {μ_{0,p}^{2}, μ_{1,p}^{±}, ..., μ_{m,p}^{±}} the p-parameters of F. The L-function of F is defined by

L(s, F) := ∏_{p} Q_{F,p}(p^{−s+k−3/2})^{−1}.

3 Fourier-Jacobi expansion of Siegel-Eisenstein series with matrix index

In this section we assume that k is an even integer. Let r be a non-negative integer. For M ∈ Sym_{r}^{+} and for an even integer k we define the Jacobi-Eisenstein series of weight k of index M of degree n by

E_{k,M}^{(n)} := ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}} ∑_{λ∈ℤ^{(n,r)}} 1|_{k,M}([(λ, 0), 0_{r}], M).  (3.1)

The above sum converges for k > n + r + 1 (cf. [Zi 89]). The Siegel-Eisenstein series E_{k}^{(n)} of weight k of degree n is defined by

E_{k}^{(n)}(Z) := ∑_{( \begin{matrix}∗ & ∗ \\ C & D \end{matrix})∈Γ_{∞}^{(n)}\Γ_{n}} det(CZ + D)^{−k},  (3.2)

where Z ∈ ℌ_{n}. We denote by e_{k,M}^{(n−r)} the M-th Fourier-Jacobi coefficient of

E_{k}^{(n)}, it means that

E_{k}^{(n)}(( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{M∈Sym_{r}^{∗}} e_{k,M}^{(n−r)}(τ, z) e(Mω)  (3.3)

is a Fourier-Jacobi expansion of the Siegel-Eisenstein series E_{k}^{(n)} of weight k of degree n, where τ ∈ ℌ_{n−r}, ω ∈ ℌ_{r} and z ∈ ℂ^{(n−r,r)}. The explicit formula for the Fourier-Jacobi expansion of Siegel-Eisenstein series is given in [Bo 83, Satz 7] for arbitrary degree. The purpose of this section is to express the Fourier-Jacobi coefficient e_{k,M}^{(n−2)} for M = ( \begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix} ) ∈ Sym_{2}^{+} as a summation of Jacobi-Eisenstein series of matrix index (Proposition 3.3). We first obtain the following lemma.

Lemma 3.1. For any M ∈ Sym_{r}^{+} and for any A ∈ GL_{r}(ℤ) we have

E_{k,M}^{(n)}(τ, z) = E_{k,M[A^{−1}]}^{(n)}(τ, z^{t}A)

and

e_{k,M}^{(n)}(τ, z) = e_{k,M[A^{−1}]}^{(n)}(τ, z^{t}A).

Proof. The first identity follows directly from the definition. The transforma-

tion formula E_{k}^{(n+r)} ((\begin{matrix}1_{n} \\ & A \end{matrix}) (\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}) (\begin{matrix}1_{n} \\ & ^{t}A \end{matrix})) = E_{k}^{(n+r)} ((\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}))

gives the second identity. ⊓ ⊔

Let m be a positive integer. We denote by D_{0} the discriminant of ℚ(\sqrt{−m}), and we put f := \sqrt{\frac{m}{|D_{0}|}}. We note that f is a positive integer if −m ≡ 0, 1

mod 4. We denote by h_{k− \frac{1}{2}} (m) the m-th Fourier coefficient of the Cohen-Eisenstein series of weight k − \frac{1}{2} (cf. Cohen [Co 75]). The following formula is known (cf. [Co 75], [E-Z 85]):

h_{k− \frac{1}{2}} (m)

= \begin{cases}h_{k− \frac{1}{2}} (|D_{0}|) m^{k− \frac{3}{2}} ∑_{d|f} μ(d) ( \frac{D_{0}}{d} ) d^{1−k}σ_{3−2k}( \frac{f}{d} ) & if − m ≡ 0, 1 mod 4, \\ 0 & otherwise, \end{cases}

where we define σ_{a}(b) := ∑_{d|b} d^{a} and μ is the Möbis function.

We assume −m ≡ 0, 1 mod 4. Let D_{0} and f be as above. For the sake of simplicity we define

g_{k}(m) := ∑_{d|f} μ(d) h_{k− \frac{1}{2}} (\frac{m}{d^{2}} ) .

We will use the following lemma for the proof of Proposition 7.5.

Lemma 3.2. Let m be a natural number such that −m ≡ 0, 1 mod 4. Then for any prime p we have

g_{k}(p^{2}m) = (p^{2k−3} − (\frac{−m}{p}) p^{k−2}) g_{k}(m).

Proof. Let D_{0}, f be as above. By using the formula of h_{k− \frac{1}{2}} (m) we obtain

h_{k− \frac{1}{2}} (m) = h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} ∏_{q|f} \{σ_{2k−3}(q^{l_{q}}) − ( \frac{D_{0}}{q} ) q^{k−2}σ_{2k−3}(q^{l_{q}−1})\} ,

where q runs over all primes which divide f, and where we put l_{q} := ord_{q}(f). In particular, the function h_{k− \frac{1}{2}} (m)(h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} )^{−1} is multiplicative as function of f. Hence, for any prime q, we have

\begin{matrix}h_{k− \frac{1}{2}} (|D_{0}|q^{2l_{q}}) − h_{k− \frac{1}{2}} (|D_{0}|q^{2l_{q}−2}) \\ = h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} (q^{(2k−3)l_{q}} − (\frac{D_{0}}{q} ) q^{k−2+(2k−3)(l_{q}−1)}) , \end{matrix}

Thus

g_{k}(m) = h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} ∑_{d|f} μ(d)\frac{h_{k− \frac{1}{2}} (\frac{m}{d^{2}} )}{h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}}}

= h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} ∏_{q|f} \frac{h_{k− \frac{1}{2}} (|D_{0}|q^{2l_{q}}) − h_{k− \frac{1}{2}} (|D_{0}|q^{2l_{q}−2})}{h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}}}

= h_{k− \frac{1}{2}} (|D_{0}|)|D_{0}|^{k− \frac{3}{2}} ∏_{q|f} (q^{(2k−3)l_{q}} − (\frac{D_{0}}{q} ) q^{k−2+(2k−3)(l_{q}−1)}) .

The lemma follows from this identity, since (\frac{−m}{p}) = 0 if p|f; (\frac{−m}{p}) = (\frac{D_{0}}{p} ) if p |̸f. ⊓ ⊔

By using the function g_{k}(m), we obtain the following proposition.

Proposition 3.3. For M = (\begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix}) ∈ Sym_{2}^{+} we put m = det(2M). Let D_{0}, f be as above, which depend on the integer m. If k > n + 1, then

e_{k,M}^{(n−2)}(τ, z) = ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{k,M[W_{d}^{−1}]}^{(n−2)}(τ, z^{t}W_{d}),

where we chose a matrix W_{d} ∈ GL_{2}(ℚ) ∩ ℤ^{(2,2)} for each d which satisfies the conditions det(W_{d}) = d, ^{t}W_{d}^{−1}MW_{d}^{−1} ∈ Sym_{2}^{+} and ^{t}W_{d}^{−1}MW_{d}^{−1} = (\begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix}). Remark that the matrix W_{d} is not uniquely determined, but the above summation does not depend on the choice of W_{d}.

Proof. We use the terminology and the Satz 7 in [Bo 83] for this proof. For M′ ∈ Sym_{n}^{+} we denote by a_{2}^{k}(M′) the M′-th Fourier coefficient of Siegel-Eisenstein series of weight k of degree 2. We put

M_{2}^{n}(ℤ)^{∗} := \{N ∈ ℤ^{(2,2)} | det(N) ≠ 0 and there exists V = ( \begin{matrix}N & ∗ \\ ∗ & ∗ \end{matrix}) ∈ GL_{n}(ℤ)\} .

A matrix N ∈ ℤ^{(n,2)} is called primitive if there exists a matrix V ∈ GL_{n}(ℤ) such that V = (N ∗). From [Bo 83, Satz 7] we have

e_{k,M}^{(n−2)}(τ, z) = ∑_{\begin{matrix}N_{1}∈M_{2}^{n}(ℤ)^{∗}/GL_{2}(ℤ) \\ N_{1}^{−1}M^{t}N_{1}^{−1}∈Sym_{2}^{+} \end{matrix}} a_{2}^{k}(M[^{t}N_{1}^{−1}]) ∑_{\begin{matrix}N_{3}∈ℤ^{(n−2,2)} \\ ( \begin{matrix}N_{1} \\ N_{3} \end{matrix} ):primitive \end{matrix}} f(M, N_{1}, N_{3}; τ, z),

where

f(M, N_{1}, N_{3}; τ, z)
= ∑_{( \begin{matrix}A & B \\ C & D \end{matrix})∈Γ_{∞}^{(n−2)}\Γ_{n−2}} det(Cτ + D)^{−k}
× e(M {−^{t}z(Cτ + D)^{−1}Cz +^{t}z(Cτ + D)^{−1}N_{3}N_{1}^{−1}
+^{t}N_{1}^{−1}^{t}N_{3}^{t}(Cτ + D)^{−1}z
+^{t}N_{1}^{−1}^{t}N_{3}(Aτ + B)(Cτ + D)^{−1}N_{3}N_{1}^{−1}}) .

For any positive integer l such that l^{2}|m, we chose a matrix W_{l} ∈ ℤ^{(2,2)} which satisfies three conditions det(W_{l}) = l, ^{t}W_{l}^{−1}MW_{l}^{−1} ∈ Sym_{2}^{+} and tW_{l}^{−1}MW_{l}^{−1} = (\begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix}). By virtue of these conditions, W_{l} has the form

W_{l} = (\begin{matrix}l & 0 \\ x & 1 \end{matrix}) with some x ∈ ℤ. The set ^{t}W_{l} GL_{2}(ℤ) is uniquely determined for each positive integer l such that l^{2}|m. If N_{1} = ^{t}W_{l} = (\begin{matrix}l & x \\ 0 & 1 \end{matrix}), then

∑_{\begin{matrix}N_{3}∈ℤ^{(n−2,2)} \\ ( \begin{matrix}N_{1} \\ N_{3} \end{matrix} ):primitive \end{matrix}} f(M, N_{1}, N_{3}; τ, z) = ∑_{a|l} μ(a) ∑_{N_{3}∈ℤ^{(n−2,2)}} f(M, N_{1}, N_{3} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) ; τ, z).

Thus

e_{k,M}^{(n−2)}(τ, z)

= ∑_{\begin{matrix}l \\ l^{2}|m \end{matrix}} a_{2}^{k}(M[W_{l}^{−1}]) ∑_{a|l} μ(a) ∑_{N_{3}∈ℤ^{(n−2,2)}} f(M,^{t}W_{l}, N_{3} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) ; τ, z)

= ∑_{\begin{matrix}l \\ l^{2}|m \end{matrix}} a_{2}^{k}(M[W_{l}^{−1}]) ∑_{a|l} μ(a)

× ∑_{N_{3}∈ℤ^{(n−2,2)}} f(M[W_{l}^{−1} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} )], 1_{2}, N_{3}; τ, z^{t}W_{l} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} )^{−1}).

Therefore

e_{k,M}^{(n−2)}(τ, z)

= ∑_{\begin{matrix}l \\ l^{2}|m \end{matrix}} a_{2}^{k}(M[W_{l}^{−1}]) ∑_{a|l} μ(a) E_{k,M[W_{l}^{−1}( \begin{matrix}a \\ & 1 \end{matrix} )]}^{(n−2)}(τ, z^{t}W_{l} ( \begin{matrix}a^{−1} \\ & 1 \end{matrix} ))

= ∑_{\begin{matrix}d \\ d^{2}|m \end{matrix}} E_{k,M[W_{d}^{−1}]}^{(n−2)}(τ, z^{t}W_{d}) ∑_{\begin{matrix}a \\ a^{2}| \frac{m}{d^{2}} \end{matrix}} μ(a) a_{2}^{k}(M[W_{d}^{−1} ( \begin{matrix}a^{−1} \\ & 1 \end{matrix} )]).

Here we have a_{2}^{k}(M′) = h_{k− \frac{1}{2}} (det(2M′)) for any M′ = ( \begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix} ) ∈ Sym_{2}^{+}. Moreover, if m ≢ 0, 3 mod 4, then h_{k− \frac{1}{2}} (m) = 0. Hence

e_{k,M}^{(n−2)}(τ, z) = ∑_{\begin{matrix}d \\ d|f \end{matrix}} E_{k,M[W_{d}^{−1}]}^{(n−2)}(τ, z^{t}W_{d}) ∑_{\begin{matrix}a \\ a| \frac{f}{d} \end{matrix}} μ(a) h_{k− \frac{1}{2}}(\frac{m}{a^{2}d^{2}} ) .

Therefore this proposition follows. ⊓ ⊔

4 Relation between Jacobi forms of half-integral weight of integer index and Jacobi forms of integral weight of matrix index

In this section we fix a positive definite half-integral symmetric matrix M ∈

Sym_{2}^{+}, and we assume that M has the form M = (\begin{matrix}l & \frac{1}{2}r \\ \frac{1}{2}r & 1 \end{matrix}) with integers l

and r.

The purpose of this section is to give a map ι_{M} which is a linear map from a subspace of holomorphic functions on ℌ_{n} ×ℂ^{(n,2)} to a subspace of holomorphic functions on ℌ_{n} × ℂ^{(n,1)}. A restriction of ι_{M} gives a map from a subspace J_{k, M}^{(n)∗} of J_{k, M}^{(n)} to a subspace J_{k− \frac{1}{2} , det(2M)}^{(n)∗} of J_{k− \frac{1}{2} , det(2M)}^{(n)} (cf. Lemma 4.2). Moreover, we shall show a compatibility between the map ι_{M} and index-shift maps (cf. Proposition 4.3 and Proposition 4.4). Furthermore, we define indexshift maps Ṽ_{α,n−α}(p^{2}) for J_{k− \frac{1}{2} , det(2M)}^{(n)∗} at p = 2 through the map ι_{M} (cf. §4.7). By virtue of the map ι_{M} and by the results in this section, we can translate some relations among Jacobi forms of half-integral weight of integer index to relations among Jacobi forms of integral weight of matrix index.

4.1 An expansion of Jacobi forms of integer index

In this subsection we consider an expansion of Jacobi forms of integer index and shall introduce a subspace J_{k,M}^{(n)∗} ⊂ J_{k,M}^{(n)}.

The symbol J_{k,1}^{(n+1)} denotes the space of Jacobi forms of weight k of index 1 of degree n + 1 (cf. §2.5). Let ϕ_{1}(τ, z) ∈ J_{k,1}^{(n+1)} be a Jacobi form. We regard ϕ_{1}(τ, z) e(ω) as a holomorphic function on ℌ_{n+2}, where τ ∈ ℌ_{n+1}, z ∈ ℂ^{(n+1,1)} and ω ∈ ℌ_{1} such that ( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}) ∈ ℌ_{n+2}. We have an expansion

ϕ_{1}(τ, z)e(w) = ∑_{\begin{matrix}S∈Sym_{2}^{+} \\ S=( \begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix} ) \end{matrix}} ϕ_{S}(τ′, z′)e(Sω′),

where τ′ ∈ ℌ_{n}, z′ ∈ ℂ^{(n,2)} and ω′ ∈ ℌ_{2} which satisfy ( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}) = ( \begin{matrix}τ′ & z′ \\ ^{t}z′ & ω′ \end{matrix} ) ∈ ℌ_{n+2}.

Because the group Γ_{n,2}^{J} (cf. § 2.1) is a subgroup of Γ_{n+1,1}^{J}, the form ϕ_{S} belongs to J_{k,S}^{(n)}. We denote this map by FJ_{1,S}, it means that we have a map

FJ_{1,S} : J_{k,1}^{(n+1)} → J_{k,S}^{(n)}.

By an abuse of language, we call the map FJ_{1,S} the Fourier-Jacobi expansion with respect to S. The ℂ-vector subspace J_{k,M}^{(n)∗} of J_{k,M}^{(n)} denotes the image of J_{k,1}^{(n+1)} by FJ_{1,M}, where M is a half-integral symmetric matrix of size 2.

4.2 Fourier-Jacobi expansion of Siegel modular forms of halfintegral weight

The purpose of this subsection is to show the following lemma.

Lemma 4.1. Let F (( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{m∈ℤ} ϕ_{m}(τ, z)e(mω) be a Fourier-Jacobi expansion of F ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(n+1)}(4)), where τ ∈ ℌ_{n}, ω ∈ ℌ_{1} and z ∈ ℂ^{(n,1)}. Then

ϕ_{m} ∈ J_{k− \frac{1}{2} ,m}^{(n)} for any natural number m.

Proof. Due to the definition of J_{k− \frac{1}{2} ,m}^{(n)}, we only need to show the identity

θ^{(n+1)}(γ · ( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) θ^{(n+1)}(( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}))^{−1} = θ^{(n)}(( \begin{matrix}A & B \\ C & D \end{matrix}) · τ) θ^{(n)}(τ)^{−1}

for any γ = (( \begin{matrix}A & B \\ C & D \end{matrix}) , [(λ, μ), κ]) ∈ Γ_{n,1}^{J} and for any ( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}) ∈ ℌ_{n+1} such that τ ∈ ℌ_{n}, ω ∈ ℌ_{1}. Here θ^{(n+1)} and θ^{(n)} are the theta constants (cf. §2.2). For any M = ( \begin{matrix}A′ & B′ \\ C′ & D′ \end{matrix} ) ∈ Γ_{0}^{(n+1)}(4) it is known that

(θ^{(n+1)}(M · Z) θ^{(n+1)}(Z)^{−1})^{2} = det(C′Z + D′) (\frac{−4}{det D′} ) ,

where Z ∈ ℌ_{n+1}. Here (\frac{−4}{det D′} ) is the quadratic symbol and it is known the

identity (\frac{−4}{det D′} ) = (−1)^{\frac{det D′−1}{2}} . Hence, for any γ = (( \begin{matrix}A & B \\ C & D \end{matrix}) , [(λ, μ), κ]) ∈ Γ_{n,1}^{J}, we obtain

(θ^{(n+1)}(γ · Z) θ^{(n+1)}(Z)^{−1})^{2} = det(Cτ + D) (\frac{−4}{det D} ) ,

where Z = ( \begin{matrix}τ & z \\ ^{t}z & ω \end{matrix}) ∈ ℌ_{n+1} with τ ∈ ℌ_{n}. In particular, the holomorphic function \frac{θ^{(n+1)}(γ·Z)}{θ^{(n+1)}(Z)} does not depend on the choice of z ∈ ℂ^{(n,1)} and of ω ∈ ℌ_{1}. We

substitute z = 0 into \frac{θ^{(n+1)}(γ·Z)}{θ^{(n+1)}(Z)} and a straightforward calculation gives

\frac{θ^{(n+1)}(γ · ( \begin{matrix}τ & 0 \\ 0 & ω \end{matrix}))}{θ^{(n+1)}(( \begin{matrix}τ & 0 \\ 0 & ω \end{matrix}))} = \frac{θ^{(n)}(( \begin{matrix}A & B \\ C & D \end{matrix}) · τ)}{θ^{(n)}(τ)}.

Hence we conclude this lemma. ⊓ ⊔

4.3 The map σ and the Hecke operator T̃_{α,n−α}(p^{2})

In this subsection we review the isomorphism between the space of Jacobi forms of index 1 and a subspace of Siegel modular forms of half-integral weight, which has been shown by Eichler-Zagier[E-Z 85] for degree one and by Ibukiyama[Ib 92] for general degree. Let M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) be the generalized plus-space introduced in [Ib 92, page 112], which is a generalization of the Kohnen plus-space for higher degrees:

M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) := \{F ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)) |\begin{matrix}the coefficients A(N) = 0 unless \\ N + (−1)^{k}R^{t}R ∈ 4 Sym_{n}^{∗} \\ for some R ∈ ℤ^{(n,1)} \end{matrix} \} .

146 S.Hayashida

A form F ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)) is called a Siegel cusp form if F^{2} is a Siegel cusp form of weight 2k − 1. We denote by S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) the space of all Siegel cusp forms in M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)). For any even integer k, the isomorphism between J_{k,1}^{(n)} (the space of Jacobi forms of weight k of index 1 of degree n) and M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) is shown in [E-Z 85, Theorem 5.4] for n = 1 and in [Ib 92, Theorem 1] for n > 1. We call this isomorphism the Eichler-Zagier-Ibukiyama correspondence and denote this linear map by σ which is a bijection from J_{k,1}^{(n)} to M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) as modules over the ring of Hecke operators. By the map σ the space S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) is isomorphic to the space of Jacobi cusp forms J_{k,1}^{(n) cusp}. The map

σ : J_{k,1}^{(n)} → M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4))

is given as follows: if

ϕ(τ, z) = ∑_{\begin{matrix}N∈Sym_{n}^{∗}, R∈ℤ^{(n,1)} \\ 4N−R^{t}R≥0 \end{matrix}} C(N, R) e(Nτ + R^{t}z)

is a Jacobi form which belongs to J_{k,1}^{(n)}, then σ(ϕ) ∈ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) is defined by

σ(ϕ)(τ) := ∑_{\begin{matrix}R mod (2ℤ)^{(n,1)} \\ R∈ℤ^{(n,1)} \end{matrix}} ∑_{\begin{matrix}N∈Sym_{n}^{∗} \\ 4N−R^{t}R≥0 \end{matrix}} C(N, R) e((4N − R^{t}R)τ).

For the double coset Γ_{n}diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n} and for ϕ ∈ J_{k,1}^{(n)}, the Hecke operator T_{α,n−α}^{J}(p^{2}) is defined by

ϕ|T_{α,n−α}^{J}(p^{2}) := ∑_{λ,μ∈(ℤ/pℤ)^{n}} ∑_{M} ϕ|_{k,1} (M × ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) , [(λ, μ), 0]) .

Here, in the second summation of the RHS, the matrix M runs over all representatives of Γ_{n}\Γ_{n} diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n}. Let T̃_{α,n−α}(p^{2}) be the Hecke operator introduced in §2.8 for odd prime p, which acts on the space M_{k− \frac{1}{2}} (Γ_{0}^{(n)}(4)). For any odd prime p the identity

σ(ϕ)|T̃_{α,n−α}(p^{2}) = p^{α/2+k(2n+1)−(2n+7)n/2}σ(ϕ|T_{α,n−α}^{J}(p^{2})). (4.1)

has been obtained in [Ib 92]. Through the identity (4.1) the Hecke operator T̃_{α,n−α}(4) for M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) is defined. It means that we define

σ(ϕ)|T̃_{α,n−α}(4) := 2^{α/2+k(2n+1)−(2n+7)n/2}σ(ϕ|T_{α,n−α}^{J}(4)). (4.2)

Lifting to Siegel Modular Forms … 147

4.4 A generalization of Cohen-Eisenstein series and the subspace

J_{k−1/2}^{(n)∗} In this subsection we will introduce a subspace J_{k− \frac{1}{2} ,m}^{(n)∗} ⊂ J_{k− \frac{1}{2} ,m}^{(n)} for any integer n. Moreover, we will introduce a generalized Cohen-Eisenstein series H_{k− \frac{1}{2}}^{(n+1)} and will consider the Fourier-Jacobi expansion of H_{k− \frac{1}{2}}^{(n+1)} for any integer n. Let e_{k,1}^{(n+1)} be the first Fourier-Jacobi coefficient of Siegel-Eisenstein series E_{k}^{(n+2)} (see (3.3) in §3 for the definition of e_{k,1}^{(n+1)}). It is known that e_{k,1}^{(n+1)} coincides with the Jacobi-Eisenstein series E_{k,1}^{(n+1)} of weight k of index 1 of degree n + 1 (cf. [Bo 83, Satz 7]. See (3.1) in §3 for the definition of E_{k,1}^{(n+1)}). We define the generalized Cohen-Eisenstein series H_{k− \frac{1}{2}}^{(n+1)} of weight k − \frac{1}{2} of degree n + 1 by

H_{k− \frac{1}{2}}^{(n+1)} := σ(E_{k,1}^{(n+1)}).

Because E_{k,1}^{(n+1)} ∈ J_{k,1}^{(n+1)}, we have H_{k− \frac{1}{2}}^{(n+1)} ∈ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4)) for any integer n. For any integer m we denote by F̃J_{m} the linear map from M_{k− \frac{1}{2}} (Γ_{0}^{(n+1)}(4)) to J_{k− \frac{1}{2} ,m}^{(n)} obtained by the Fourier-Jacobi expansion with respect to the index m. It means that if G ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(n+1)}(4)), then G has the expansion

G((\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{m∈ℤ} ϕ_{m}(τ, z)e(mω)

and we define F̃J_{m}(G) := ϕ_{m}. We remark ϕ_{m} ∈ J_{k− \frac{1}{2} ,m}^{(n)} due to Lemma 4.1. We denote by J_{k− \frac{1}{2} ,m}^{(n)∗} the image of M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4)) by the map F̃J_{m}. We denote by e_{k− \frac{1}{2} ,m}^{(n)} the m-th Fourier-Jacobi coefficient of the generalized Cohen-Eisenstein series H_{k− \frac{1}{2}}^{(n+1)} (see (1.1) in §1 for the definition of e_{k− \frac{1}{2} ,m}^{(n)}). We remark e_{k− \frac{1}{2} ,m}^{(n)} ∈ J_{k− \frac{1}{2} ,m}^{(n)∗} for any integer n.

4.5 The map ι_{M} We recall M = ( \begin{matrix}l & r/2 \\ r/2 & 1 \end{matrix} ) ∈ Sym_{2}^{+}. In this subsection we shall introduce a map

ι_{M} : H_{M}^{(n)} → Hol(ℌ_{n} × ℂ^{(n,1)} → ℂ),

where H_{M}^{(n)} is a subspace of holomorphic functions on ℌ_{n} × ℂ^{(n,2)}, which will be defined below, and where Hol(ℌ_{n} × ℂ^{(n,1)} → ℂ) denotes the space of all

Documenta Mathematica 21 (2016) 125–196

holomorphic functions on ℌ_{n} × ℂ^{(n,1)}. We will show that the restriction of ι_{M} gives a linear isomorphism between J_{k,M}^{(n)∗} and J_{k− \frac{1}{2} ,m}^{(n)∗} (cf. Lemma 4.2).

Let ψ be a holomorphic function on ℌ_{n} × ℂ^{(n,2)}. We assume that ψ has a Fourier expansion

ψ(τ, z) = ∑_{\begin{matrix}N∈Sym_{n}^{∗},R∈ℤ^{(n,1)} \\ 4N−RM^{−1}^{t}R≥0 \end{matrix}} A(N, R) e(Nτ +^{t} Rz)

for (τ, z) ∈ ℌ_{n} × ℂ^{(n,2)}, and assume that ψ satisfies the following condition on the Fourier coefficients: if

(\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & M \end{matrix}) = (\begin{matrix}N′ & \frac{1}{2}R′ \\ \frac{1}{2}^{t}R′ & M \end{matrix}) [\binom{1_{n}}{^{t}T 1_{2}}]

with some T = (0, λ) ∈ ℤ^{(n,2)} and some λ ∈ ℤ^{(n,1)}, then A(N, R) = A(N′, R′). The symbol H_{M}^{(n)} denotes the ℂ-vector space consists of all holomorphic functions which satisfy the above condition. We remark J_{k,M}^{(n)∗} ⊂ J_{k,M}^{(n)} ⊂ H_{M}^{(n)} for any even integer k.

Now we shall define a map ι_{M}. For ψ(τ′, z′) = ∑ A(N, R)e(Nτ′+R^{t}z′) ∈ H_{M}^{(n)} we define a holomorphic function ι_{M}(ψ) on ℌ_{n} × ℂ^{(n,1)} by

ι_{M}(ψ)(τ, z) := ∑_{\begin{matrix}M∈Sym_{n}^{∗}, S∈ℤ^{(n,1)} \\ 4Mm−S^{t}S≥0 \end{matrix}} C(M, S)e(Mτ + S^{t}z)

for (τ, z) ∈ ℌ_{n} × ℂ^{(n,1)}, where we define C(M, S) := A(N, R) if there exist matrices N ∈ Sym_{2}^{∗} and R = (R_{1}, R_{2}) ∈ ℤ^{(n,2)} (R_{1}, R_{2} ∈ ℤ^{(n,1)}) which satisfy

(\begin{matrix}M & \frac{1}{2}S \\ \frac{1}{2}^{t}S & det(2M) \end{matrix}) = 4 (\begin{matrix}N & \frac{1}{2}R_{1} \\ \frac{1}{2}^{t}R_{1} & l \end{matrix}) − (\begin{matrix}R_{2} \\ r \end{matrix}) (^{t}R_{2}, r) ,

C(M, S) := 0 otherwise. We remark that the identity

4 (\begin{matrix}N & \frac{1}{2}R_{1} \\ \frac{1}{2}^{t}R_{1} & l \end{matrix}) − (\begin{matrix}R_{2} \\ r \end{matrix}) (^{t}R_{2}, r) = 4 (\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & M \end{matrix}) [(\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & 1 \\ −\frac{1}{2}^{t}R_{2} & −\frac{1}{2}r \end{matrix})]

holds and remark that the coefficient C(M, S) does not depend on the choice of the matrices N and R. The proof of these facts are as follows. The first fact of the identity follows from a straightforward calculation. As for the second fact, if

4 (\begin{matrix}N & \frac{1}{2}R_{1} \\ \frac{1}{2}^{t}R_{1} & l \end{matrix}) − (\begin{matrix}R_{2} \\ r \end{matrix}) (^{t}R_{2}, r) = 4 (\begin{matrix}N′ & \frac{1}{2}R′_{1} \\ \frac{1}{2}^{t}R′_{1} & l \end{matrix}) − (\begin{matrix}R′_{2} \\ r \end{matrix}) (^{t}R′_{2}, r) ,

then 4N − R_{2}^{t}R_{2} = 4N′ − R′_{2}^{t}R′_{2}. Hence R_{2}^{t}R_{2} ≡ R′_{2}^{t}R′_{2} mod 4. Thus there exists a matrix λ ∈ ℤ^{(n,1)} such that R′_{2} = R_{2} + 2λ. Therefore, by straightforward calculation we have

(\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & M \end{matrix}) = (\begin{matrix}N′ & \frac{1}{2}R′ \\ \frac{1}{2}^{t}R′ & M \end{matrix}) [(\begin{matrix}1_{n} & 0 \\ ^{t}T & 1_{2} \end{matrix})]

with T = (0, λ), R = (R_{1}, R_{2}) and R′ = (R′_{1}, R′_{2}). Because ψ belongs to H_{M}^{(n)}, we have A(N, R) = A(N′, R′). Hence the above definition of C(M, S) is well-defined.

Lemma 4.2. Let k be an even integer. We put m = det(2M). Then we have the commutative diagram:

J_{k,1}^{(n+1)} ⟶^{σ} M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4))
FJ_{1,M}↓  ↓F̃J_{m}
J_{k,M}^{(n)∗} ⟶^{ι_{M}}  J_{k− \frac{1}{2} ,m}^{(n)∗},

where two maps FJ_{1,M} and F̃J_{m} have been introduced in §4.1 and §4.4. Moreover, the restriction of the linear map ι_{M} on J_{k,M}^{(n)∗} gives the bijection between

J_{k,M}^{(n)∗} and J_{k− \frac{1}{2} ,m}^{(n)∗}.

Proof. Let ψ ∈ J_{k,1}^{(n+1)} be a Jacobi form. Due to the definition of σ (cf. §4.3) and ι_{M}, it is not difficult to check the identity ι_{M}(F J_{1,M}(ψ)) = F̃J_{m}(σ(ψ)). Namely, we have the above commutative diagram. Since the restriction of the map F̃J_{m} on M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n+1)}(4)) is surjective, and

since σ is an isomorphism and since ι_{M}(F J_{1,M}(ψ)) = F̃J_{m}(σ(ψ)), the restricted map ι_{M}|_{J_{k,M}^{(n)∗}} : J_{k,M}^{(n)∗} → J_{k− \frac{1}{2} ,m}^{(n)∗} is surjective. The injectivity of the restricted

map ι_{M}|_{J_{k,M}^{(n)∗}} follows directly from the definition of the map ι_{M}. ⊓ ⊔

4.6 Compatibility between index-shift maps and ι_{M}

In this subsection we shall show a compatibility between the map ι_{M} and some index-shift maps. For function ψ on ℌ_{n} × ℂ^{(n,2)} and for L ∈ ℤ^{(2,2)} we define the function ψ|U_{L} on ℌ_{n} × ℂ^{(n,2)} by

(ψ|U_{L})(τ, z) := ψ(τ, z^{t}L).

It is not difficult to check that if ψ belongs to J_{k,M}^{(n)}, then ψ|U_{L} belongs to

J_{k,M[L]}^{(n)}.

For function ϕ on ℌ_{n} × ℂ^{(n,1)} and for integer a we define the function ϕ|U_{a} on ℌ_{n} × ℂ^{(n,1)} by

(ϕ|U_{a})(τ, z) := ϕ(τ, az).

We have ϕ|U_{a} ∈ J_{k− \frac{1}{2} ,ma^{2}}^{(n)} if ϕ ∈ J_{k− \frac{1}{2} ,m}^{(n)}.

Proposition 4.3. For any ψ ∈ J_{k,M}^{(n)∗} and for any L = \binom{a}{b 1} ∈ ℤ^{(2,2)} we obtain

ι_{M[L]}(ψ|U_{L}) = ι_{M}(ψ)|U_{a}.

In particular, for any prime p we have ι_{M[( \begin{matrix}p \\ & 1 \end{matrix} )]} (ψ |U( \begin{matrix}p \\ & 1 \end{matrix} )) = ι_{M}(ψ)|U_{p}.

Proof. We put m = det(2M). Let ψ(τ, z′) = ∑_{\begin{matrix}N∈Sym_{n}^{∗}, R∈ℤ^{(n,2)} \\ 4N−RM^{−1}^{t}R≥0 \end{matrix}} A(N, R)e(Nτ + R^{t}z′)

be a Fourier expansion of ψ. Let

ι_{M}(ψ)(τ, z) = ∑_{\begin{matrix}M∈Sym_{n}^{∗}, S∈ℤ^{(n,1)} \\ 4Mm−S^{t}S≥0 \end{matrix}} C(M, S) e(Mτ + S^{t}z),

ι_{M[L]}(ψ|U_{L})(τ, z) = ∑_{\begin{matrix}M∈Sym_{n}^{∗}, S∈ℤ^{(n,1)} \\ 4Mma^{2}−S^{t}S≥0 \end{matrix}} C_{1}(M, S) e(Mτ + S^{t}z)

and

(ι_{M}(ψ)|U_{a})(τ, z) = ∑_{\begin{matrix}M∈Sym_{n}^{∗}, S∈ℤ^{(n,1)} \\ 4Mma^{2}−S^{t}S≥0 \end{matrix}} C_{2}(M, S) e(Mτ + S^{t}z)

be Fourier expansions. It is enough to show C_{1}(M, S) = C_{2}(M, S). We have C_{2}(M, S) = C(M, a^{−1}S). Moreover, we obtain C_{1}(M, S) = A(N, RL^{−1}) with N ∈ Sym_{n}^{∗} and R ∈ ℤ^{(n,2)} which satisfy

(\begin{matrix}M & \frac{1}{2}S \\ \frac{1}{2}^{t}S & ma^{2} \end{matrix}) = 4 (\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & M[L] \end{matrix}) [(\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & 1 \\ −\frac{1}{2}^{t}(R ( \begin{matrix}0 \\ 1 \end{matrix} )) & −\frac{1}{2}ra − b \end{matrix})] .

For the above matrices N, R, M and S we have

(\begin{matrix}M & \frac{1}{2}a^{−1}S \\ \frac{1}{2}a^{−1}^{t}S & m \end{matrix})

= 4 (\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & M[L] \end{matrix}) [(\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & 1 \\ −\frac{1}{2}^{t}(R ( \begin{matrix}0 \\ 1 \end{matrix} )) & −\frac{1}{2}ra − b \end{matrix}) (\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & a^{−1} \end{matrix})]

= 4 (\begin{matrix}N & \frac{1}{2}RL^{−1} \\ \frac{1}{2}^{t}(RL^{−1}) & M \end{matrix}) [(\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & 1 \\ −\frac{1}{2}^{t}(R ( \begin{matrix}0 \\ 1 \end{matrix} )) & −\frac{1}{2}r \end{matrix})]

= 4 (\begin{matrix}N & \frac{1}{2}RL^{−1} \\ \frac{1}{2}^{t}(RL^{−1}) & M \end{matrix}) [(\begin{matrix}1_{n} & 0 ⋮ 0 \\ 0 ⋯ 0 & 1 \\ −\frac{1}{2}^{t}(RL^{−1} ( \begin{matrix}0 \\ 1 \end{matrix} )) & −\frac{1}{2}r \end{matrix})] .

Thus C_{2}(M, S) = C(M, a^{−1}S) = A(N, RL^{−1}) = C_{1}(M, S). ⊓ ⊔

Proposition 4.4. For odd prime p and for 0 ≤ α ≤ n, let Ṽ_{α,n−α}(p^{2}) and V_{α,n−α}(p^{2}) be index-shift maps defined in §2.7. Then, for any ψ ∈ J_{k,M}^{(n)∗} we have

ι_{M}(ψ)|Ṽ_{α,n−α}(p^{2}) = p^{k(2n+1)−n(n+ \frac{7}{2} )+ \frac{1}{2} α} ι_{M[( \begin{matrix}p \\ & 1 \end{matrix} )]}(ψ|V_{α,n−α}(p^{2})).(4.3)

Proof. The proof is similar to the case of Jacobi forms of index 1 (cf. [Ib 92, Theorem 2]). However, we remark that the maps Ṽ_{α,n−α}(p^{2}) and V_{α,n−α}(p^{2}) in the present article change the indices of Jacobi forms. To prove this proposition, we compare the Fourier coefficients of the both sides of (4.3). Let

ψ(τ, z′) = ∑_{N,R} A_{1}(N, R)e(Nτ + R^{t}z′),
(ψ|V_{α,n−α}(p^{2}))(τ, z′) = ∑_{N,R} A_{2}(N, R)e(Nτ + R^{t}z′),
(ι_{M}(ψ))(τ, z) = ∑_{M,S} C_{1}(M, S)e(Mτ + S^{t}z)

and

(ι_{M}(ψ)|Ṽ_{α,n−α}(p^{2}))(τ, z) = ∑_{M,S} C_{2}(M, S)e(Mτ + S^{t}z)

152 S.Hayashida be Fourier expansions, where τ ∈ ℌ_{n}, z′ ∈ ℂ^{(n,2)} and z ∈ ℂ^{(n,1)}. For the sake of simplicity we put U = ( \begin{matrix}p^{2} \\ & p \end{matrix} ). Then ψ|V_{α,n−α}(p^{2})

= \begin{aligned}& ∑ \\ ( & \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} & ) \end{aligned} ∑_{λ_{2},μ_{2}∈(ℤ/pℤ)^{(n,1)}}

× ψ|_{k,M} (( \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} ) × ( \begin{matrix}U \\ & p^{2}U^{−1} \end{matrix} ) , [((0, λ_{2}), (0, μ_{2})), 0_{2}])

= \begin{aligned}& ∑ \\ ( & \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} & ) \end{aligned} ∑_{λ_{2},μ_{2}∈(ℤ/pℤ)^{(n,1)}} ∑_{N,R} A(N, R)

× e(Nτ + R^{t}z)|_{k,M} (( \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} ) × ( \begin{matrix}U \\ & p^{2}U^{−1} \end{matrix} ) , [((0, λ_{2}), (0, μ_{2})), 0_{2}]) ,

where ( \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} ) runs over a set of all representatives of

Γ_{n}\Γ_{n}diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n},

and where the slash operator |_{k,M} is defined in §2.4. We put λ = (0, λ_{2}), μ = (0, μ_{2}) ∈ ℤ^{(n,2)}, then we obtain

\begin{matrix}e(Nτ + R^{t}z)|_{k,M} (( \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} ) × ( \begin{matrix}U \\ & p^{2}U^{−1} \end{matrix} ) , [(λ, μ), 0_{2}]) \\ = p^{−k} det(D)^{−k}e(N̂τ + R̂^{t}z + NBD^{−1} + RU^{t}μD^{−1}), \end{matrix}

where

N̂ = p^{2}D^{−1}N^{t}D^{−1} + D^{−1}RU^{t}λ + \frac{1}{p^{2}} λUMU^{t}λ

and

R̂ = D^{−1}RU + \frac{2}{p^{2}} λUMU.

Thus

N = \frac{1}{p^{2}} D ((N̂ − \frac{1}{4}R̂_{2}^{t}R̂_{2}) + \frac{1}{4}(R̂_{2} − 2λ_{2})^{t}(R̂_{2} − 2λ_{2}))^{t}D

and

R = D (R̂ − \frac{2}{p^{2}} λUMU) U^{−1},

Documenta Mathematica 21 (2016) 125–196

where R̂_{2} = R̂ ( \begin{matrix}0 \\ 1 \end{matrix} ). Hence, for any N̂ ∈ Sym_{n}^{∗} and for any R̂ ∈ ℤ^{(n,2)}, we have

A_{2}(N̂, R̂)

= p^{−k} \begin{aligned}& ∑ \\ ( & \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} & ) \end{aligned} det(D)^{−k} ∑_{λ_{2}∈(ℤ/pℤ)^{(n,1)}} ∑_{μ_{2}∈(ℤ/pℤ)^{(n,1)}} A_{1}(N, R)

× e(NBD^{−1} + RU^{t}(0, μ_{2})D^{−1})

= p^{−k+n} \begin{aligned}& ∑ \\ ( & \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} & ) \end{aligned} det(D)^{−k} ∑_{λ_{2}∈(ℤ/pℤ)^{(n,1)}} A_{1}(N, R) e(NBD^{−1}),

where N and R are the same symbols as above, which are determined by N̂, R̂ and λ_{2}, and where ( \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} ) runs over a complete set of representatives of

Γ_{n}\Γ_{n}diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n}.

On the RHS of the above first identity the matrix D^{−1}RU belongs to ℤ^{(n,2)}, since R̂ ∈ ℤ^{(n,2)}. We remark that A_{1}(N, R) = 0 unless N ∈ Sym_{n}^{∗} and R ∈ ℤ^{(n,2)}. Due to the definition of ι_{M}, for N ∈ Sym_{n}^{∗} and R ∈ ℤ^{(n,2)} we have the identity

A_{1}(N, R) = C_{1}(4N − R ( \begin{matrix}0 \\ 1 \end{matrix} )^{t} (R ( \begin{matrix}0 \\ 1 \end{matrix} )), 4R ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rR ( \begin{matrix}0 \\ 1 \end{matrix} )).

Here

4N − R ( \begin{matrix}0 \\ 1 \end{matrix} )^{t}(R ( \begin{matrix}0 \\ 1 \end{matrix} )) = \frac{1}{p^{2}} D (4N̂ − R̂_{2}^{t}R̂_{2})^{t}D

and

4R ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rR ( \begin{matrix}0 \\ 1 \end{matrix} ) = \frac{1}{p^{2}} D(4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2}).

Hence we have

A_{2}(N̂, R̂)  (4.4)
= p^{−k+n}\begin{aligned}& ∑ \\ ( & \begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix} & ) \end{aligned}det(D)^{−k}

×C_{1}(\frac{1}{p^{2}} D (4N̂ − R̂_{2}^{t}R̂_{2})^{t}D, \frac{1}{p^{2}} D(4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2}))

×e(\frac{1}{p^{2}} (N̂ − \frac{1}{4}R̂_{2}^{t}R̂_{2})^{t}DB) ∑_{λ_{2}} e(\frac{1}{4p^{2}} (R̂_{2} − 2λ_{2})^{t}(R̂_{2} − 2λ_{2})^{t}DB) ,

where λ_{2} runs over a complete set of representatives of (ℤ/pℤ)^{(n,1)} such that

D (R̂ − \frac{2}{p^{2}} (0, λ_{2})UMU) U^{−1} ∈ ℤ^{(n,2)}.

Let 𝔖_{α} be a complete set of representative of Γ_{n}\Γ_{n} ( \begin{matrix}1_{α} \\ & p1_{n−α} \\ & & p^{2}1_{α} \\ & & & p1_{n−α} \end{matrix} ) Γ_{n}.

Now we quote a complete set of representatives 𝔖_{α} from [Zh 84]. We put

δ_{i,j} := diag(1_{i}, p1_{j−i}, p^{2}1_{n−j})

for 0 ≤ i ≤ j ≤ n. We set

𝔖_{α} := \{(\begin{matrix}p^{2}δ_{i,j}^{−1} & b_{0} \\ 0_{n} & δ_{i,j} \end{matrix}) (\begin{matrix}^{t}u^{−1} & 0_{n} \\ 0_{n} & u \end{matrix}) | i, j, b_{0}, u\} ,

where i and j run over all non-negative integers such that j − i − n + α ≥ 0, and where u runs over a complete set of representatives of (δ_{i,j}^{−1}GL_{n}(ℤ)δ_{i,j} ∩ GL_{n}(ℤ))\GL_{n}(ℤ), and b_{0} runs over all matrices in the set

𝔗 := \{ (\begin{matrix}0_{i} & 0 & 0 \\ 0 & a_{1} & pb_{1} \\ 0 & ^{t}b_{1} & b_{2} \end{matrix})| \begin{matrix}b_{1} ∈ (ℤ/pℤ)^{(j−i,n−j)}, b_{2} = ^{t}b_{2} ∈ (ℤ/p^{2}ℤ)^{(n−j,n−j)}, \\ a_{1} = ^{t}a_{1} ∈ (ℤ/pℤ)^{(j−i,j−i)}, rank_{p}(a_{1}) = j − i − n + α \end{matrix}\} .

For a matrix g = (\begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix}) = (\begin{matrix}p^{2}δ_{i,j}^{−1} & b_{0} \\ 0_{n} & δ_{i,j} \end{matrix}) (\begin{matrix}^{t}u^{−1} & 0_{n} \\ 0_{n} & u \end{matrix}) ∈ 𝔖_{α} with a

matrix b_{0} = (\begin{matrix}0_{i} & 0 & 0 \\ 0 & a_{1} & pb_{1} \\ 0 & ^{t}b_{1} & b_{2} \end{matrix}) ∈ 𝔗, we define ε(g) := (\frac{−4}{p})^{rank_{p}(a_{1})/2} (\frac{det a′_{1}}{p} ),

where a′_{1} ∈ GL_{j−i−n+α}(ℤ/pℤ) is a matrix such that a_{1} ≡ ( \begin{matrix}a′_{1} & 0 \\ 0 & 0_{n−α} \end{matrix} ) [v] mod p

with some v ∈ GL_{j−i}(ℤ). Under the assumption

\frac{1}{p^{2}} D(4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2}) ∈ ℤ^{(n,1)},

the condition D(R̂ − 2p^{−2}(0, λ_{2})UMU)U^{−1} ∈ ℤ^{(n,2)} is equivalent to the condition

u(R̂_{2} − 2λ_{2}) ∈ ( \begin{matrix}p1_{i} & 0 \\ 0 & 1_{n−i} \end{matrix} ) ℤ^{(n,1)}.

Hence the last summation in (4.4) is

∑_{λ_{2}} e(\frac{1}{4p^{2}} (R̂_{2} − 2λ_{2})^{t}(R̂_{2} − 2λ_{2})^{t}DB)

= p^{n−j} ∑_{λ′∈(ℤ/pℤ)^{(j−i,1)}} e(\frac{1}{p}^{t}λ′a_{1}λ′)

= p^{n−i−rank_{p}(a_{1})} ((\frac{−4}{p}) p)^{rank_{p}(a_{1})/2} (\frac{det a′_{1}}{p} )
= p^{n−i− \frac{rankp(a_{1})}{2}} ε(g)
= p^{n+(n−i−j−α)/2}ε(g).

Thus (4.4) is

A_{2}(N̂, R̂)

= p^{−k+2n} ∑_{g} p^{−k(2n−i−j)+(n−i−j−α)/2}ε(g) e(p^{−2} (4N̂ − R̂_{2}^{t}R̂_{2})^{t}DB)

× C_{1}(p^{−2}D(4N̂ − R̂_{2}^{t}R̂_{2})^{t}D, p^{−2}D(4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2})) ,

where g = (\begin{matrix}p^{2}^{t}D^{−1} & B \\ 0_{n} & D \end{matrix}) = (\begin{matrix}p^{2}δ_{i,j}^{−1} & b_{0} \\ 0_{n} & δ_{i,j} \end{matrix}) (\begin{matrix}^{t}u^{−1} & 0_{n} \\ 0_{n} & u \end{matrix}) runs over all ele-

ments in the set 𝔖_{α}. Now we shall express C_{2}(M, S) as a linear combination of Fourier coefficients

C_{1}(M, S) of ι_{M}(ψ). For Y = (diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α}), p^{α/2}) ∈ { \atop GSp_{n}^{+}(ℤ)}

a complete set of representatives of Γ_{0}^{(n)}(4)^{∗}\Γ_{0}^{(n)}(4)^{∗}Y Γ_{0}^{(n)}(4)^{∗} is given by elements

g̃ = (g, ε(g)p^{(n−i−j)/2}) ∈ { \atop GSp_{n}^{+}(ℤ)},

where g runs over all elements in the set 𝔖_{α}, and ε(g) is defined as above (cf. [Zh 84, Lemma 3.2]). Hence

(ι_{M}(ψ)|Ṽ_{α,n−α}(p^{2}))(τ, z)

= p^{n(2k−1)/2−n(n+1)} ∑_{M,S} ∑ ; g̃ p^{(−k+1/2)(n−i−j)}ε(g) C_{1}(M, S)

× e(M(p^{2}^{t}D^{−1}τ + B)D^{−1} + p^{2}S^{t}zD^{−1})

= p^{n(2k−1)/2−n(n+1)} ∑_{M̂,Ŝ} ∑_{g∈𝔖_{α}} p^{(−k+1/2)(n−i−j)}ε(g) C_{1}(p^{−2}DM̂^{t}D, p^{−2}DŜ)

× e(M̂τ + Ŝ^{t}z + p^{−2}M̂^{t}DB).

Thus

\begin{matrix}C_{2}(M̂, Ŝ) \\ = ∑_{g} p^{−n(n+1)+(k−1/2)(i+j)}ε(g) C_{1}(p^{−2}DM̂^{t}D, p^{−2}DŜ) e(p^{−2}M̂^{t}DB). \end{matrix}

Now we put M̂ = 4N̂ − R̂_{2}^{t}R̂_{2} and Ŝ = 4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2}, then

C_{2}(4N̂ − R̂_{2}^{t}R̂_{2}, 4R̂ ( \begin{matrix}1 \\ 0 \end{matrix} ) − 2rpR̂_{2}) = p^{2nk+k−n^{2}− \frac{7}{2} n+ \frac{1}{2} α}A_{2}(N̂, R̂).

The proposition follows from this identity. ⊓ ⊔

4.7 Index-shift maps at p = 2

For p = 2 we define the map

Ṽ_{α,n−α}(4) : J_{k− \frac{1}{2} ,m}^{(n)∗} → Hol(ℌ_{n} × ℂ^{(n,1)} → ℂ)

through an analogue of the identity (4.3), it means that we define

ϕ|Ṽ_{α,n−α}(4) := 2^{k(2n+1)−n(n+ \frac{7}{2} )+ \frac{1}{2} α} ι_{M[( \begin{matrix}2 \\ & 1 \end{matrix} )]}(ψ|V_{α,n−α}(4))

for any ϕ ∈ J_{k− \frac{1}{2} ,m}^{(n)∗}, and where ψ ∈ J_{k,M}^{(n)∗} is the Jacobi form which satisfies ι_{M}(ψ) = ϕ. Here the map V_{α,n−α}(4) is defined in §2.7 and the map ι_{M} is defined in §4.5.

5 Action of index-shift maps on Jacobi-Eisenstein series

In this section we fix a positive definite half-integral symmetric matrix M ∈ Sym_{2}^{+} and we assume that the right-lower part of M is 1, it means M =

(\begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix}).

The purpose of this section is to show that the form E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) is writ-

ten as a linear combination of three forms E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}, E_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} and

E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, where E_{k,M}^{(n)} is the Jacobi-Eisenstein series

of index M (cf. §3), and where V_{α,n−α}(p^{2}) and U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} are index-shift maps

(cf. §2.7 and §4.6). Here X = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) is a matrix. First we will calculate functions K_{i,j}^{β} (cf. Lemma 5.2) which appear in an expression of E_{k,M}^{(n)}|V_{α,n−α}(p^{2}), and after that, we will express E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) as a summation of functions K̃_{i,j}^{β} (cf. Proposition 5.3). The calculation in this section is an analogue to the one given in [Yk 89] for the case of index M = 1. However, we need to modify his calculation for Jacobi-Eisenstein series E_{k,1}^{(n)} of index 1 to our case for E_{k,M}^{(n)} with M = ( \begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix} ) ∈ Sym_{2}^{+}. This calculation is not obvious, since we need to calculate the action of the matrices of type [((0, u_{2}), (0, v_{2})), 0_{2}].

5.1 The function K_{i,j}^{β}

The purpose of this subsection is to introduce a function K_{i,j}^{β} and to express E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) as a summation over K_{i,j}^{β}. Moreover, we shall calculate K_{i,j}^{β} explicitly (cf. Lemma 5.2). We put δ_{i,j} := diag(1_{i}, p1_{j−i}, p^{2}1_{n−j}). For x = diag(0_{i}, x′, 0_{n−i−j}) with

x′ = ^{t}x′ ∈ ℤ^{(j−i,j−i)}, we set δ_{i,j}(x) := (\begin{matrix}p^{2}δ_{i,j}^{−1} & x \\ 0 & δ_{i,j} \end{matrix}) and Γ(δ_{i,j}(x)) :=

Γ_{n} ∩ δ_{i,j}(x)^{−1}Γ_{∞}^{(n)}δ_{i,j}(x). For x = diag(0_{i}, x′, 0_{n−i−j}) and for y = diag(0_{i}, y′, 0_{n−i−j}) with x′ = ^{t}x′, y′ = ty′ ∈ ℤ^{(j−i,j−i)}, following [Yk 89] we say that x and y are equivalent, if there exists a matrix u ∈ GL_{n}(ℤ) ∩ δ_{i,j}GL_{n}(ℤ)δ_{i,j}^{−1} which has a form u = ( \begin{matrix}u_{1} & ∗ & ∗ \\ ∗ & u_{2} & ∗ \\ ∗ & ∗ & u_{3} \end{matrix} ) satisfying x′ ≡ u_{2} y′ ^{t}u_{2} mod p, where u_{2} ∈ ℤ^{(j−i,j−i)}, u_{1} ∈ ℤ^{(i,i)} and u_{3} ∈ ℤ^{(n−j,n−j)}. We denote by [x] the equivalence class of x. We quote the following lemma from [Yk 89].

Lemma 5.1. The double coset Γ_{n}diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n} is written as a disjoint union

Γ_{n} ( \begin{matrix}1_{α} \\ & p1_{n−α} \\ & & p^{2}1_{α} \\ & & & p1_{n−α} \end{matrix} ) Γ_{n} = ⋃_{\begin{matrix}i,j \\ 0≤i≤j≤n \end{matrix}} ⋃_{[x]} Γ_{∞}^{(n)}δ_{i,j}(x)Γ_{n},

where [x] runs over all equivalence classes which satisfy rank_{p}(x) = j − i − n + α ≥ 0.

Proof. The reader is referred to [Yk 89, Corollary 2.2]. ⊓ ⊔

We put U := (\begin{matrix}p^{2} & 0 \\ 0 & p \end{matrix}). By the definition of index-shift map V_{α,n−α}(p^{2}) and of

the Jacobi-Eisenstein series E_{k,M}^{(n)}, we have

E_{k,M}^{(n)}|V_{α,n−α}(p^{2})

= ∑_{u,v∈ℤ^{(n,1)}} ∑_{M′∈Γ_{n}\Γ_{n}diag(1_{α},p1_{n−α},p^{2}1_{α},p1_{n−α})Γ_{n}} ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}} ∑_{λ∈ℤ^{(n,2)}}

× 1|_{k,M}([(λ, 0), 0_{2}], MM′ × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)), 0_{2}]

= ∑_{u,v∈ℤ^{(n,1)}} ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}diag(1_{α},p1_{n−α},p^{2}1_{α},p1_{n−α})Γ_{n}} ∑_{λ∈ℤ^{(n,2)}}

× 1|_{k,M}([(λ, 0), 0_{2}], M × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)), 0_{2}].

Hence, due to Lemma 5.1, we have

E_{k,M}^{(n)}|V_{α,n−α}(p^{2})

= ∑_{u,v∈ℤ^{(n,1)}} ∑_{\begin{matrix}i,j \\ 0≤i≤j≤n \end{matrix}} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=j−i−n+α \end{matrix}} ∑_{M∈Γ_{∞}^{(n)}\δ_{i,j}(x)Γ_{n}} ∑_{λ∈ℤ^{(n,2)}}

× 1|_{k,M}([(λ, 0), 0_{2}], M × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)), 0_{2}]

= ∑_{u,v∈ℤ^{(n,1)}} ∑_{\begin{matrix}i,j \\ 0≤i≤j≤n \end{matrix}} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=j−i−n+α \end{matrix}} ∑_{M∈Γ(δ_{i,j}(x))\Γ_{n}} ∑_{λ∈ℤ^{(n,2)}}

× 1|_{k,M}([(λ, 0), 0_{2}], δ_{i,j}(x)M × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)), 0_{2}].

158 S.Hayashida For β ≤ j − i we define a function

K_{i,j}^{β}(τ, z)

:= K_{i,j,M,p}^{β}(τ, z)

= ∑_{\begin{matrix}[x] \\ rank_{p}(x)=β \end{matrix}} ∑_{M∈Γ(δ_{i,j}(x))\Γ_{n}}

× ∑_{λ∈ℤ^{(n,2)}} \{1|_{k,M}([(λ, 0), 0_{2}], δ_{i,j}(x)M × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))\} (τ, z).

Then we obtain

E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) = ∑_{\begin{matrix}i,j \\ 0≤i≤j≤n \end{matrix}} ∑_{u,v∈ℤ^{(n,1)}} K_{i,j}^{α−i−n+j}|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)), 0_{2}].

We define

L_{i,j} := L_{i,j,M,p} =\{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) | \begin{matrix}λ_{1} ∈ (pℤ)^{(i,2)} , λ_{2} ∈ ℤ^{(j−i,2)} , λ_{3} ∈ (p^{−1}ℤ)^{(n−j,2)} \\ 2λ_{2}M^{t}λ_{3} ∈ ℤ^{(j−i,n−j)} , λ_{3}M^{t}λ_{3} ∈ ℤ^{(n−j,n−j)} \end{matrix}\} .

Moreover, we define a subgroup Γ(δ_{i,j}) of Γ_{∞}^{(n)} by

Γ(δ_{i,j}) := \{(\begin{matrix}A & B \\ 0_{n} & ^{t}A^{−1} \end{matrix}) ∈ Γ_{∞}^{(n)} | A ∈ δ_{i,j}GL_{n}(ℤ)δ_{i,j}^{−1}\} .

Lemma 5.2. Let K_{i,j}^{β} be as above. We obtain K_{i,j}^{β}(τ, z)

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)} ∑_{M∈Γ(δ_{i,j})\Γ_{n}}

× ∑_{λ∈L_{i,j}} 1|_{k,M}([(λ, 0), 0_{2}], M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) ∑_{\begin{matrix}x=^{t}x∈(ℤ/pℤ)^{(n,n)} \\ x=diag(0_{i},x′,0_{n−j}) \\ rank_{p}(x′)=β \end{matrix}}e (\frac{1}{p}M^{t}λxλ) ,

where x runs over a complete set of representatives of (ℤ/pℤ)^{(n,n)} such that x = ^{t}x, rank_{p}(x) = β and x = diag(0_{i}, x′, 0_{n−j}) with some x′ ∈ (ℤ/pℤ)^{(j−i,j−i)}. Proof. We proceed as in [Yk 89, Proposition 3.2]. The inside of the last sum-

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mation of the definition of K_{i,j}^{β}(τ, z) is

(1|_{k,M}([(λ, 0), 0_{2}], δ_{i,j}(x)M × ( \begin{matrix}U & 0 \\ 0 & p^{2}U^{−1} \end{matrix} ))) (τ, z)

= det(p^{2}U^{−1})^{−k} det(δ_{i,j})^{−k}

× (e(M(^{t}λ(p^{2}δ_{ij}^{−1}τ + x)δ_{ij}^{−1}λ + 2^{t}λδ_{ij}^{−1}z (\begin{matrix}p^{2} \\ & p \end{matrix})))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}M) (τ, z)

= p^{−k(2n−i−j+1)}

× (((1|_{k,M}([(pδ_{i,j}^{−1}λ, 0), 0_{2}], (\begin{matrix}1 & p^{−1}x \\ 0 & 1 \end{matrix})))(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))) |_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}M) (τ, z)

= p^{−k(2n−i−j+1)} (1|_{k,M}([(pδ_{i,j}^{−1}λ, 0), 0_{2}], (\begin{matrix}1 & p^{−1}x \\ 0 & 1 \end{matrix}) M)) (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )).

Here we used the identity δ_{i,j}x = δ_{i,j}diag(0_{i}, x′, 0_{n−j}) = px. Thus

K_{i,j}^{β}(τ, z) = p^{−k(2n−i−j+1)} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=β \end{matrix}} ∑_{M∈Γ(δ_{i,j}(x))\Γ_{n}}

× ∑_{λ∈ℤ^{n}} 1|_{k,M} ([(pδ_{i,j}^{−1}λ, 0), 0_{2}], (\begin{matrix}1 & p^{−1}x \\ 0 & 1 \end{matrix}) M) (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )).

We put

U := \{(\begin{matrix}1_{n} & s \\ 0_{n} & 1_{n} \end{matrix}) | s = ^{t}s ∈ ℤ^{(n,n)}\} .

Then the set

V := \{(\begin{matrix}1_{n} & s \\ 0_{n} & 1_{n} \end{matrix}) | ^{s=( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & s_{2} \\ 0 & ^{t}s_{2} & s_{3} \end{matrix} ), s_{2}∈(ℤ/pℤ)^{(j−i,n−j)}, s_{3}=^{t}s_{3}∈(ℤ/pℤ)^{(n−j,n−j)}} \}

is a complete set of representatives of Γ(δ_{i,j}(x))\Γ(δ_{i,j}(x))U. Therefore

K_{i,j}^{β}(τ, z)

= p^{−k(2n−i−j+1)} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=β \end{matrix}} ∑_{M∈(Γ(δ_{i,j}(x))U)\Γ_{n}} ∑_{λ∈ℤ^{(n,2)}} ∑_{( \begin{matrix}1_{n} & s \\ 0 & 1_{n} \end{matrix} )∈V}

× 1|_{k,M}([(pδ_{i,j}^{−1}λ, 0), 0_{2}], (\begin{matrix}1_{n} & p^{−1}x \\ 0 & 1_{n} \end{matrix}) (\begin{matrix}1_{n} & s \\ 0 & 1_{n} \end{matrix}) M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

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160 S.Hayashida Hence

K_{i,j}^{β}(τ, z) = p^{−k(2n−i−j+1)} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=β \end{matrix}} ∑_{M∈(Γ(δ_{i,j}(x))U)\Γ_{n}} ∑_{λ∈ℤ^{(n,2)}}

× 1|_{k,M}([(pδ_{i,j}^{−1}λ, 0), 0_{2}], (\begin{matrix}1_{n} & p^{−1}x \\ 0 & 1_{n} \end{matrix}) M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

× ∑_{( \begin{matrix}1_{n} & s \\ 0 & 1_{n} \end{matrix} )∈V} e (p^{2}M^{t}λδ_{i,j}^{−1}sδ_{i,j}^{−1}λ) .

The last summation of the RHS of the above identity is

∑_{( \begin{matrix}1_{n} & s \\ 0 & 1_{n} \end{matrix} )∈V} e (p^{2}M^{t}λδ_{i,j}^{−1}sδ_{i,j}^{−1}λ)

= \begin{cases}p^{(n−j)(n−i+1)} & if λ_{3}M^{t}λ_{3} ≡ 0 mod p^{2} and 2λ_{3}M^{t}λ_{2} ≡ 0 mod p, \\ 0 & otherwise, \end{cases}

where λ = ( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ ℤ^{(n,2)} with λ_{1} ∈ ℤ^{(i,2)}, λ_{2} ∈ ℤ^{(j−i,2)} and λ_{3} ∈ ℤ^{(n−j,2)}.

Thus

K_{i,j}^{β}(τ, z) = p^{−k(2n−i−j+1)+(n−j)(n−i+1)} ∑_{\begin{matrix}[x] \\ rank_{p}(x)=β \end{matrix}} ∑_{M∈(Γ(δ_{i,j}(x))U)\Γ_{n}}

× ∑_{λ∈L_{i,j}} 1|_{k,M}([(λ, 0), 0_{2}], ( \begin{matrix}1_{n} & p^{−1}x \\ 0 & 1_{n} \end{matrix} ) M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )).

Now Γ(δ_{i,j}(x))U is a subgroup of Γ(δ_{i,j}). For any ( \begin{matrix}A & B \\ 0_{n} & ^{t}A^{−1} \end{matrix} ) ∈ Γ(δ_{i,j}) we have

1|_{k,M}([(λ, 0), 0_{2}], ( \begin{matrix}1_{n} & p^{−1}x \\ 0_{n} & 1_{n} \end{matrix} ) ( \begin{matrix}A & B \\ 0_{n} & ^{t}A^{−1} \end{matrix} ) M)

= 1|_{k,M}([(λ, 0), 0_{2}], ( \begin{matrix}A & B \\ 0_{n} & ^{t}A^{−1} \end{matrix} ) ( \begin{matrix}1_{n} & p^{−1}A^{−1}x^{t}A^{−1} \\ 0_{n} & 1_{n} \end{matrix} ) M)

= 1|_{k,M}([(^{t}Aλ,^{t}Bλ), 0_{2}], ( \begin{matrix}1_{n} & p^{−1}A^{−1}x^{t}A^{−1} \\ 0_{n} & 1_{n} \end{matrix} ) M)

= 1|_{k,M}([(^{t}Aλ, 0), 0_{2}], ( \begin{matrix}1_{n} & p^{−1}A^{−1}x^{t}A^{−1} \\ 0_{n} & 1_{n} \end{matrix} ) M),

and ^{t}AL_{i,j} = L_{i,j}. Moreover, when ( \begin{matrix}A & B \\ 0_{n} & ^{t}A^{−1} \end{matrix} ) runs over all elements in a complete set of representatives of Γ(δ_{i,j}(x))U\Γ(δ_{i,j}), then A^{−1}x^{t}A^{−1} runs over all elements in the equivalence class [x] (cf. [Yk 89, proof of Proposition 3.2]).

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Therefore we have

K_{i,j}^{β}(τ, z)

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)} ∑_{\begin{matrix}x=^{t}x∈(ℤ/pℤ)^{(n,n)} \\ x=diag(0_{i},x′,0_{n−j}) \\ rank_{p}(x′)=β \end{matrix}} ∑_{M∈Γ(δ_{i,j})\Γ_{n}}

× ∑_{λ∈L_{i,j}} 1|_{k,M}([(λ, 0), 0_{2}], ( \begin{matrix}1_{n} & p^{−1}x \\ 0 & 1_{n} \end{matrix} ) M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)} ∑_{M∈Γ(δ_{i,j})\Γ_{n}}

× ∑_{λ∈L_{i,j}} 1|_{k,M}([(λ, 0), 0_{2}], M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) ∑_{\begin{matrix}x=^{t}x∈(ℤ/pℤ)^{(n,n)} \\ x=diag(0_{i},x′,0_{n−j}) \\ rank_{p}(x′)=β \end{matrix}} e (\frac{1}{p}M^{t}λxλ) .

⊓ ⊔

5.2 The function K̃_{i,j}^{β}

The purpose of this subsection is to introduce a function K̃_{i,j}^{β} and to express E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) as a summation of K̃_{i,j}^{β}. Moreover, we shall show that K̃_{i,j}^{β} is a summation of exponential functions with generalized Gauss sums (cf. Proposition 5.3). We define

L_{i,j}^{∗} := L_{i,j,M,p}^{∗}
= \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ L_{i,j} | 2λ_{3}M ( \begin{matrix}0 \\ 1 \end{matrix} ) ∈ ℤ^{(n−j,1)}\}

= \{(\begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix}) ∈ (p^{−1}ℤ)^{(n,2)} | \begin{matrix}λ_{1} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} ∈ ℤ^{(i,2)}, λ_{2} ∈ ℤ^{(j−i,2)} \\ λ_{3} ∈ (p^{−1}ℤ)^{(n−j,2)}, 2λ_{2}M^{t}λ_{3} ∈ ℤ^{(j−i,n−j)} \\ λ_{3}M^{t}λ_{3} ∈ ℤ^{(n−j,n−j)}, 2λ_{3}M ( \begin{matrix}0 \\ 1 \end{matrix} ) ∈ ℤ^{(n−j,1)} \end{matrix}\}

and define a generalized Gauss sum

G_{M}^{j−i,l}(λ_{2}) := ∑_{\begin{matrix}x′=^{t}x′∈(ℤ/pℤ)^{(j−i,j−i)} \\ rank_{p}(x′)=j−i−l \end{matrix}} e (\frac{1}{p}M^{t}λ_{2}x′λ_{2})

for λ_{2} ∈ ℤ^{(j−i,2)}. We define

K̃_{i,j}^{β}(τ, z) := K̃_{i,j,M,p}^{β}(τ, z)

= ∑_{u,v∈(ℤ/pℤ)^{(n,1)}} (K_{i,j}^{β}|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)) , 0_{2}]) (τ, z).

162 S.Hayashida Proposition 5.3. Let the notation be as above. Then we obtain

(E_{k,M}^{(n)}|V_{α,n−α}(p^{2})) (τ, z) = ∑_{\begin{matrix}i,j \\ 0≤i≤j≤n \\ j−i≥n−α \end{matrix}} K̃_{i,j}^{α−i−n+j}(τ, z),

where

K̃_{i,j}^{α−i−n+j}(τ, z) = p^{−k(2n−i−j+1)+(n−j)(n−i+1)+2n−j}
× ∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ=( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} )∈L_{i,j}^{∗}} \{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

× ∑_{u_{2}∈(ℤ/pℤ)^{(j−i,1)}} G_{M}^{j−i,n−α}(λ_{2} + (0, u_{2})).

Proof. From the definition of K̃_{i,j}^{β} and Lemma 5.2 we obtain

(5.1)

K̃_{i,j}^{α−i−n+j}(τ, z)

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)} ∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ=( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} )∈L_{i,j}} G_{M}^{j−i,n−α}(λ_{2})

× ∑_{u,v∈(ℤ/pℤ)^{(n,1)}} (1|_{k,M}([(λ, 0), 0_{2}], M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))) |_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)) , 0_{2}],

where λ_{1} ∈ ℤ^{(i,2)}, λ_{2} ∈ ℤ^{(j−i,2)} and λ_{3} ∈ ℤ^{(n−j,2)} satisfy ( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ L_{i,j}, and

where the n × 2 matrix λ = ( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) runs over the set L_{i,j}. By a straightforward calculation we have

\begin{matrix}(1|_{k,M}([(λ, 0), 0_{2}], M))(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) \\ = (1|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}([(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} , 0), 0_{2}], M))(τ, z). \end{matrix}

Thus the last summation of (5.1) is

∑_{u,v∈(ℤ/pℤ)^{(n,1)}}\{1|_{k,M}([(λ, 0), 0_{2}], M)(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)) , 0_{2}]\}(τ, z)

= ∑_{u,v∈(ℤ/pℤ)^{(n,1)}}

× \{1|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}([(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} , 0), 0_{2}], M)|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}[((0, u), (0, v)) , 0_{2}]\}(τ, z)

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= ∑_{u′,v′∈(ℤ/pℤ)^{(n,1)}} \{1|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}([(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} + (0, u′), (0, v′)), 0_{2}], M)\} (τ, z)
= ∑_{u′,v′∈(ℤ/pℤ)^{(n,1)}} \{1|_{k,M}([(λ + (0, u′), (0, v′)), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= ∑_{u′∈(ℤ/pℤ)^{(n,1)}}\{1|_{k,M}([(λ + (0, u′), 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) ∑_{v′∈(ℤ/pℤ)^{(n,1)}} e(2M^{t}λ(0, v′)),

where, in the second identity, we used

(M, [((0, u), (0, v)), 0_{2}]) = ([((0, u′), (0, v′)), 0_{2}], M)

with ( \begin{matrix}u′ \\ v′ \end{matrix} ) = ( \begin{matrix}D & −C \\ −B & A \end{matrix} ) ( \begin{matrix}u \\ v \end{matrix}) for M = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ Γ_{n}. For λ = ( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ L_{i,j} we now

have

∑_{v′∈(ℤ/pℤ)^{(n,1)}} e(2M^{t}λ(0, v′)) = \begin{cases}p^{n} & if 2λ_{3}M ( \begin{matrix}0 \\ 1 \end{matrix} ) ∈ ℤ^{(n−j,1)}, \\ 0 & otherwise. \end{cases}

Therefore K̃_{i,j}^{α−i−n+j}(τ, z)

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)+n} ∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{\begin{matrix}λ=( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} )∈L_{i,j} \\ 2λ_{3}M( \begin{matrix}0 \\ 1 \end{matrix} )∈ℤ^{(n−j,1)} \end{matrix}} G_{M}^{j−i,n−α}(λ_{2})

× ∑_{u∈(ℤ/pℤ)^{(n,1)}} \{1|_{k,M}([(λ + (0, u), 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )).

Thus

K̃_{i,j}^{α−i−n+j}(τ, z)
= p^{−k(2n−i−j+1)+(n−j)(n−i+1)+n} ∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ=( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} )∈L_{i,j}^{∗}}

× \{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

× p^{n−j} ∑_{u_{2}∈(ℤ/pℤ)^{(j−i,1)}} G_{M}^{j−i,n−α}(λ_{2} + (0, u_{2})),

where L_{i,j}^{∗} is defined as before. ⊓ ⊔ We put

g_{p}(n, α) := ∏_{j=1}^{α} {(p^{n−j+1} − 1)(p^{j} − 1)^{−1}}

It is not difficult to see g_{p}(n, n − α) = g_{p}(n, α).

Documenta Mathematica 21 (2016) 125–196

164 S.Hayashida Lemma 5.4. For any λ = (λ_{1}, λ_{2}) ∈ ℤ^{(n,2)} and for any prime p, we have

∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}(λ + (0, u_{2}))

= \begin{cases}\begin{matrix}p^{\frac{1}{4} (n−α−1)^{2}+ \frac{1}{2} (n−α−1)+α+n} (\frac{−m}{p}) \\ ×g_{p}(n − 1, α) ∏_{\begin{matrix}j=1 \\ j : odd \end{matrix}}^{n−α−2}(p^{j} − 1) \end{matrix} & if n − α ≡ 1 mod 2 and λ_{1} ≢ 0 mod p, \\ 0 & if n − α ≡ 1 mod 2 and λ_{1} ≡ 0 mod p, \\ p^{\frac{1}{4} (n−α)^{2}+ \frac{1}{2} (n−α)+α}g_{p}(n, α) ∏_{\begin{matrix}j=1 \\ j : odd \end{matrix}}^{n−α−1}(p^{j} − 1) & if n − α ≡ 0 mod 2. \end{cases}

Here m = det(2M) and we regard the product ∏_{\begin{matrix}j=1 \\ j : odd \end{matrix}}^{c} (p^{j} − 1) as 1, if c is less than 1. Proof. This calculation is similar to the calculation of

∑_{\begin{matrix}x=^{t}x∈(ℤ/pℤ)^{n} \\ rank_{p}x=n−α \end{matrix}} e (\frac{1}{p}m^{t}λ_{1}xλ_{1})

for λ_{1} ∈ ℤ^{(n,1)} and for m ∈ ℤ which is in [Yk 89, Lemma 3.1]. If p is an odd prime and if λ_{1} ≢ 0 mod p, then

∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}(λ + (0, u_{2}))

= ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} ∑_{\begin{matrix}x′=^{t}x′∈(ℤ/pℤ)^{(n,n)} \\ rank_{p}(x′)=n−α \end{matrix}} e (\frac{1}{p}M^{t}(λ_{1}, u_{2})x′(λ_{1}, u_{2}))

By diagonalizing the matrices x′ we have

∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}(λ + (0, u_{2}))

= ∑_{i=0,1} p^{n−1} |GL_{n−1}(ℤ/pℤ)| |O(x_{i})|^{−1}
× ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} ∑_{\begin{matrix}η∈(ℤ/pℤ)^{(n,1)} \\ η≢0 mod p \end{matrix}} e (\frac{1}{p}M^{t}(η, u_{2})x_{i}(η, u_{2})) ,

where x_{i} = ( \begin{matrix}y_{i} & 0 \\ 0 & 0 \end{matrix} ) ∈ ℤ^{(n,n)}, y_{0} = 1_{n−α}, y_{1} = ( \begin{matrix}1_{n−α−1} & 0 \\ 0 & γ \end{matrix} ) ∈ ℤ^{(n−α,n−α)} and γ

is an integer such that (\frac{γ}{p} ) = −1. Here O(x_{i}) is the orthogonal group of x_{i}:

O(x_{i}) := {g ∈ GL_{n}(ℤ/pℤ) | gx_{i}^{t}g = x_{i}} .

If we diagonalize the matrix M as M ≡ ^{t}X ( \begin{matrix}m & 0 \\ 0 & 1 \end{matrix} ) X mod p with X = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ), then

∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}(λ + (0, u_{2}))

= ∑_{i=0,1} p^{n−1} |GL_{n−1}(ℤ/pℤ)| |O(x_{i})|^{−1}

× ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} ∑_{\begin{matrix}η∈(ℤ/pℤ)^{(n,1)} \\ η≢0 mod p \end{matrix}} e (\frac{1}{p} (mη^{t}η + u_{2}^{t}u_{2}) x_{i}) .

The rest of the calculation is an analogue to [Yk 89, Lemma 3.1]. For the case of p = 2 or λ_{1} ≡ 0 mod p, the calculation is similar. If p = 2, we need to calculate the case that M = ^{t}X ( \begin{matrix}m′ & \frac{1}{2} \\ \frac{1}{2} & 1 \end{matrix} ) X, but it is not difficult. We leave the detail to the reader. ⊓ ⊔

We set

S_{M}^{n,α}(0) := ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}((0, u_{2}))

and

S_{M}^{n,α}(1) := ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α} ((( 1 0 ⋮ 0 ) , u_{2})) .

Due to Lemma 5.4, we have that ∑_{u_{2}∈(ℤ/pℤ)^{(n,1)}} G_{M}^{n,α}(λ + (0, u_{2})) equals S_{M}^{n,α}(0)

or S_{M}^{n,α}(1), according as λ ∈ ℤ^{(n,2)} (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) or λ ∉ ℤ^{(n,2)} (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}).

Proposition 5.5. The form E_{k,M}^{(n)}|V_{α,n−α}(p^{2}) is a linear combination of three

forms E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}, E_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} and E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}. Here

the index-shift map U_{L} is defined in §4.6, and X = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) is a matrix in ℤ^{(2,2)} such that M = ^{t}X ( \begin{matrix}m+1 & 1 \\ 1 & 1 \end{matrix} ) X if p = 2 and \frac{det(2M)}{4} ≡ 3 mod 4, or M ≡ tX ( \begin{matrix}m & 0 \\ 0 & 1 \end{matrix} ) X mod p otherwise, and where m = det(2M).

Proof. By virtue of Proposition 5.3 we only need to show that the form K̃_{i,j}^{α−i−n+j}(τ, z) is a linear combination of the above three forms.

Because of the conditions λ_{3}M^{t}λ_{3} ∈ ℤ^{(n−j,n−j)} and 2λ_{3}M ( \begin{matrix}0 \\ 1 \end{matrix} ) ∈ ℤ^{(n−j,1)} in

the definition of L_{i,j}^{∗}, we obtain

L_{i,j}^{∗} = \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ (\frac{1}{p}ℤ)^{(n,2)} | \begin{matrix}λ_{1} ∈ ℤ^{(i,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , λ_{2} ∈ ℤ^{(j−i,2)}, \\ λ_{3}^{t}X ∈ ℤ^{(n−j,2)} ( \begin{matrix}p^{−1} & 0 \\ 0 & 1 \end{matrix} ) \end{matrix}\} (5.2)

for the case p|f, and

L_{i,j}^{∗} = \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ ℤ^{(n,2)} | λ_{1} ∈ ℤ^{(i,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , λ_{2} ∈ ℤ^{(j−i,2)}, λ_{3} ∈ ℤ^{(n−j,2)}\} (5.3)

for the case p |̸f. Here f is a natural number such that D_{0}f^{2} = − det(2M) and D_{0} is a fundamental discriminant, and where the matrix X is stated in this proposition. We now assume p|f. If p is an odd prime, then the matrix X = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) ∈ ℤ^{(2,2)} satisfies M ≡ ^{t}X ( \begin{matrix}m & 0 \\ 0 & 1 \end{matrix} ) X mod p and p^{2}|m. If p = 2, then the matrix X =

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) ∈ ℤ^{(2,2)} satisfies M = ^{t}X ( \begin{matrix}m & 0 \\ 0 & 1 \end{matrix} ) X with 4|m, or M = ^{t}X ( \begin{matrix}m′ & 1 \\ 1 & 1 \end{matrix} ) X with

4|m′. We remark that M [X^{−1} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}] is a half-integral symmetric matrix.

We put

L_{0} := \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ L_{i,j}^{∗} | λ_{2} ∈ ℤ^{(j−i,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )\} ,
L_{1} := \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) ∈ L_{i,j}^{∗} | λ_{2} ∉ ℤ^{(j−i,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )\} .

and set

L′_{i,j} := \{( \begin{matrix}λ_{1} \\ λ_{2} \\ λ_{3} \end{matrix} ) | λ_{1} ∈ ℤ^{(i,2)} ( \begin{matrix}p^{2} & 0 \\ 0 & 1 \end{matrix} ) , λ_{2} ∈ ℤ^{(j−i,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , λ_{3} ∈ ℤ^{(n−j,2)}\} .

By using the identity

\{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= \{1 |_{k,M[X^{−1}]}([(λ^{t}X, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X)

= \{1 |_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) ,

Lifting to Siegel Modular Forms … 167

we have

K̃_{i,j}^{α−i−n+j}(τ, z)

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)+2n−j}

× ∑_{M∈Γ(δ_{i,j})\Γ_{n}} \{S_{M}^{j−i,n−α}(0) ∑_{λ∈L_{0}} \{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+ S_{M}^{j−i,n−α}(1) ∑_{λ∈L_{1}} \{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))\}

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)+2n−j} ∑_{M∈Γ(δ_{i,j})\Γ_{n}}{ (S_{M}^{j−i,n−α}(0) − S_{M}^{j−i,n−α}(1))

× ∑_{λ∈L_{0}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+S_{M}^{j−i,n−α}(1)

× ∑_{λ∈L_{i,j}^{∗}}\{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))\}

= p^{−k(2n−i−j+1)+(n−j)(n−i+1)+2n−j} ∑_{M∈Γ(δ_{i,j})\Γ_{n}}{ (S_{M}^{j−i,n−α}(0) − S_{M}^{j−i,n−α}(1))

× ∑_{λ∈L′_{j,j}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+S_{M}^{j−i,n−α}(1)

× ∑_{λ∈L′_{i,j}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))\}.

We now calculate the sum

∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ∈L′_{j,j}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )).

We set

H_{i,j} := δ_{i,j} GL_{n}(ℤ) δ_{i,j}^{−1} ∩ GL_{n}(ℤ).

If {A_{l}}_{l} is a complete set of representatives of H_{i,j}\GL_{n}(ℤ), then one can say that the set \{(\begin{matrix}A_{l} & 0 \\ 0 & ^{t}A_{l}^{−1} \end{matrix})\}_{l} is a complete set of representatives of

Documenta Mathematica 21 (2016) 125–196

Γ(δ_{i,j})\Γ_{∞}^{(n)}. Thus

∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ∈L′_{j,j}} {1|_{k,M}([(λ, 0), 0_{2}], M)} (τ, z)

= ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}} ∑_{A∈H_{i,j}\GL_{n}(ℤ)} ∑_{λ∈L′_{j,j}} \{1|_{k,M}([(λ, 0), 0_{2}], ( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) M)\} (τ, z)

= ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}} ∑_{A∈H_{i,j}\GL_{n}(ℤ)} ∑_{λ∈L′_{j,j}} {1|_{k,M}([(^{t}Aλ, 0), 0_{2}], M)} (τ, z).

If B(λ) is a function on λ ∈ ℤ^{(n,2)}. Then

∑_{A∈H_{i,j}\GL_{n}(ℤ)} ∑_{λ∈L′_{j,j}} B(^{t}Aλ)

= [H_{j,j} : H_{i,j}] ∑_{A∈H_{j,j}\GL_{n}(ℤ)} ∑_{λ∈L′_{j,j}} B(^{t}Aλ)

= [H_{j,j} : H_{i,j}] (a_{0} ∑_{λ∈ℤ^{(n,2)}}B(λ) + a_{1} ∑_{λ∈ℤ^{(n,2)}}B(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) + a_{2} ∑_{λ∈ℤ^{(n,2)}}B(λ ( \begin{matrix}p^{2} & 0 \\ 0 & 1 \end{matrix} )))

with numbers a_{0}, a_{1} and a_{2} under the assumption that the summations converges absolutely. The values a_{0}, a_{1} and a_{2} are independent of the choice of the function B. For the exact values of a_{0}, of a_{1} and of a_{2} the reader is referred to [H 13, Lemma 3.7]. Hence we have

∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ∈L′_{j,j}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= [H_{j,j} : H_{i,j}] ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}}

×(a_{0} ∑_{λ∈ℤ^{(n,2)}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+a_{1} ∑_{λ∈ℤ^{(n,2)}}\{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+a_{2} ∑_{λ∈ℤ^{(n,2)}}

× \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ ( \begin{matrix}p^{2} & 0 \\ 0 & 1 \end{matrix} ) , 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )))

= [H_{j,j} : H_{i,j}] ∑_{M∈Γ_{∞}^{(n)}\Γ_{n}}

×(a_{0} ∑_{λ∈ℤ^{(n,2)}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+a_{1} ∑_{λ∈ℤ^{(n,2)}} \{1|_{k,M}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+a_{2} ∑_{λ∈ℤ^{(n,2)}} \{1|_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}([(λ, 0), 0_{2}], M)\} (τ, z))

= [H_{j,j} : H_{i,j}] (a_{0}E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+a_{1}E_{k,M}^{(n)}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) + a_{2}E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}(τ, z)).

Similarly, the summation

∑_{M∈Γ(δ_{i,j})\Γ_{n}} ∑_{λ∈L′_{i,j}} \{1|_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}([(λ, 0), 0_{2}], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

is a linear combination of E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{t}X ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )),

E_{k,M}^{(n)}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) and E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}(τ, z).

Therefore, if p|f, then the form K̃_{i,j}^{α−i−n+j}(τ, z) is a linear combination of the above three forms. The proof for the case p |̸f is similar to the case p|f. If p |̸f, then K̃_{i,j}^{α−i−n+j}(τ, z) is a linear combination of two forms E_{k,M}^{(n)}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) and

E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}(τ, z). We omit the detail of the calculation here. ⊓ ⊔

6 Commutativity with the Siegel operators

In [Kr 86] an explicit commutative relation between the generators of Hecke operators for Siegel modular forms and Siegel Φ-operator has been given. In this section we shall give a similar relation in the frameworks of Jacobi forms of matrix index and of Jacobi forms of half-integral weight.

Let M = (\begin{matrix}l & \frac{r}{2} \\ \frac{r}{2} & 1 \end{matrix}) be a 2 × 2 matrix and put m = det(2M) as before.

For any Jacobi form ϕ ∈ J_{k,M}^{(n)}, or ϕ ∈ J_{k− \frac{1}{2} ,m}^{(n)} we define the Siegel Φ-operator

Φ(ϕ)(τ′, z′) := lim_{t→+∞} ϕ ((\begin{matrix}τ′ & 0 \\ 0 & \sqrt{−1}t \end{matrix}) , (\begin{matrix}z′ \\ 0 \end{matrix}))

for (τ′, z′) ∈ ℌ_{n−1} × ℂ^{(n−1,2)}, or for (τ′, z′) ∈ ℌ_{n−1} × ℂ^{(n−1,1)}. This Siegel Φoperator is a map from J_{k,M}^{(n)} to J_{k,M}^{(n−1)}, or from J_{k− \frac{1}{2} ,m}^{(n)} to J_{k− \frac{1}{2} ,m}^{(n−1)}, respectively.

Proposition 6.1. For any Jacobi form ϕ ∈ J_{k,M}^{(n)} and for any prime p, we have

Φ(ϕ|V_{α,n−α}(p^{2})) = Φ(ϕ)|V_{α,n−α}(p^{2})^{∗},

where V_{α,n−α}(p^{2})^{∗} is a map V_{α,n−α}(p^{2})^{∗} : J_{k,M}^{(n−1)} → J_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n−1)} given by

V_{α,n−α}(p^{2})^{∗} = p^{α+2−k}V_{α,n−α−1}(p^{2})
+p(1 + p^{2n+1−2k})V_{α−1,n−α}(p^{2})
+(p^{2n−2α+2} − 1)p^{α−k}V_{α−2,n−α+1}(p^{2}).

Proof. We shall first show that there exists a linear combination of index-shift map V_{α,n−α}(p^{2})^{∗} which satisfies Φ(ϕ|V_{α,n−α}(p^{2})) = Φ(ϕ)|V_{α,n−α}(p^{2})^{∗}.

We set U = (\begin{matrix}p^{2} & 0 \\ 0 & p \end{matrix}). Let

ϕ(τ, z) = ∑_{N,R} A_{1}(N, R)e(Nτ + R^{t}z),
(ϕ|V_{α,n−α}(p^{2}))(τ, z) = ∑_{N̂,R̂} A_{2}(N̂, R̂)e(N̂τ + R̂^{t}z)

be the Fourier expansions. Let \{(\begin{matrix}p^{2}^{t}D_{j}^{−1} & B_{(j,l)} \\ 0_{n} & D_{j} \end{matrix})\}_{(j,l)} be a complete set

of representatives of Γ_{n}\Γ_{n}diag(1_{α}, p1_{n−α}, p^{2}1_{α}, p1_{n−α})Γ_{n}. Then the Fourier coefficients A_{2}(N̂, R̂) have been calculated in the proof of Proposition 4.4:

A_{2}(N̂, R̂) = p^{−k+n} ∑_{j} det(D_{j})^{−k} ∑_{λ_{2}∈(ℤ/pℤ)^{(n,1)}} A_{1}(N, R) ∑_{l} e(NB_{(j,l)}D_{j}^{−1}). (6.1)

Here N and R are determined by

N = \frac{1}{p^{2}} D_{j} ((N̂ − \frac{1}{4}R̂_{2}^{t}R̂_{2}) + \frac{1}{4}(R̂_{2} − 2λ_{2})^{t}(R̂_{2} − 2λ_{2}))^{t}D_{j}, (6.2)
R = D_{j} (R̂ − \frac{2}{p^{2}} λUMU) U^{−1},

where we put R̂_{2} = R̂ ( \begin{matrix}0 \\ 1 \end{matrix} ) and λ = (0 λ_{2}) ∈ ℤ^{(n,2)}. By the definition of V_{α,n−α}(p^{2}) there exists {γ_{i}}_{i} such that ϕ|V_{α,n−α}(p^{2}) = ∑_{i} ϕ|_{k,M}γ_{i}. We can take γ_{i} as a form

γ_{i} = γ_{(j,l,λ_{2},μ_{2})} = ((\begin{matrix}p^{2}^{t}D_{j}^{−1} & B_{(j,l)} \\ 0_{n} & D_{j} \end{matrix}) × (\begin{matrix}U & 0_{2} \\ 0_{2} & p^{2}U^{−1} \end{matrix}) , [((0 λ_{2}), (0 μ_{2})), 0_{2}]) ,

where B_{(j,l)} = ( \begin{matrix}B_{(j,l)}^{∗} & b_{1} \\ ^{t}b_{3} & b_{2} \end{matrix} ), D_{j} = ( \begin{matrix}D_{j}^{∗} & 𝔡 \\ 0 & d_{j} \end{matrix} ), λ_{2} = ( \begin{matrix}λ^{∗} \\ λ_{3} \end{matrix} ), μ_{2} = ( \begin{matrix}μ^{∗} \\ μ_{3} \end{matrix} ) with

( \begin{matrix}p^{2}^{t}D_{j}^{∗}^{−1} & B_{(j,l)}^{∗} \\ 0_{n−1} & D_{j}^{∗} \end{matrix} ) ∈ GSp_{n−1}^{+}(ℤ), λ^{∗}, μ^{∗} ∈ ℤ^{(n−1,1)}, and d_{j}, λ_{3}, μ_{3} ∈ ℤ. We

set

γ_{i}^{∗} := γ_{(j,l,λ^{∗},μ^{∗})}^{∗} = ((\begin{matrix}p^{2}^{t}D_{j}^{∗}^{−1} & B_{(j,l)}^{∗} \\ 0_{n−1} & D_{j}^{∗} \end{matrix}) × (\begin{matrix}U & 0_{2} \\ 0_{2} & p^{2}U^{−1} \end{matrix}) , [((0 λ^{∗}), (0 μ^{∗})), 0_{2}]) .

By the definition of Siegel Φ-operator we have

Φ(∑_{i} ϕ|_{k,M}γ_{i})(τ^{∗}, z^{∗}) = Φ(ϕ|V_{α,n−α}(p^{2}))(τ^{∗}, z^{∗})
= ∑_{N̂,R̂} A_{2}(N̂, R̂)e(N̂^{∗}τ^{∗} + R̂^{∗}^{t}z^{∗}),

where τ^{∗} ∈ ℌ_{n−1}, z^{∗} ∈ ℂ^{(n−1,2)}, N̂ = ( \begin{matrix}N̂^{∗} & 0 \\ 0 & 0 \end{matrix} ) ∈ Sym_{n}^{∗}, N̂^{∗} ∈ Sym_{n−1}^{∗}, R̂ =

( R̂^{∗} ; 0 ) ∈ ℤ^{(n,2)} and R̂^{∗} ∈ ℤ^{(n−1,2)}.

Hence we need to calculate A_{2}(N̂, R̂) for N̂ = ( \begin{matrix}N̂^{∗} & 0 \\ 0 & 0 \end{matrix} ) and R̂ = ( R̂^{∗} ; 0 ) ∈ ℤ^{(n,2)}. From the identity (6.1) we need to calculate

∑_{j} det(D_{j})^{−k} ∑_{λ_{2}∈(ℤ/pℤ)^{(n,1)}} A_{1}(N, R) ∑_{l} e(NB_{(j,l)}D_{j}^{−1}).  (6.3)

We remark that the value A_{1}(N, R) depends on the choice of N̂, R̂, D_{j} and λ_{3}. Under the conditions N̂ ∈ Sym_{n}^{∗} and R̂ ∈ ℤ^{(n,2)} and by the identity (6.2) we can assume dλ_{3} ∈ pℤ, since A_{1}(N, R) = 0 unless N ∈ Sym_{n}^{∗}. It is known that the value A_{1}(N, R) depends only on 4N − RM^{−1}^{t}R and on R mod 2M. We now have

4N − RM^{−1}^{t}R = \frac{1}{p^{2}} D_{j} (4N̂ − p^{2}R̂U^{−1}M^{−1}U^{−1}^{t}R̂)^{t}D_{j}.

We set

R′ = D_{j} (R̂ − \frac{2}{p}(0 λ_{2})MU) U^{−1} + \frac{2}{p} (\begin{matrix}0 & 0 \\ 0 & d_{j}λ_{3} \end{matrix}) M

and

N′ = \frac{1}{4p^{2}} D_{j} (4N̂ − p^{2}R̂U^{−1}M^{−1}U^{−1}^{t}R̂)^{t}D_{j} + \frac{1}{4}R′M^{−1}^{t}R′.

We remark that the last row of R′ is zero, and the last row and the last column of N′ are also zero. Because 4N − RM^{−1}^{t}R = 4N′ − R′M^{−1}^{t}R′ and because

R−R′ ∈ 2ℤ^{(n−1,2)}M, we have A_{1}(N, R) = A_{1}(N′, R′). We write N′ = ( \begin{matrix}N^{′∗} \\ & 0 \end{matrix} )

with N^{′∗} ∈ Sym_{n−1}^{∗}. We have

R′ = D_{j} (R̂ − \frac{2}{p} (0 λ′_{2}) MU) U^{−1},

where λ′_{2} = ( \begin{matrix}λ^{∗}−D_{j}^{∗−1}𝔡λ_{3} \\ 0 \end{matrix} ) ∈ ℚ^{(n,1)}. Now we will show λ^{∗}−D_{j}^{∗−1}𝔡λ_{3} ∈ ℤ^{(n−1,1)}

if ∑_{l} e(NB_{(j,l)}D_{j}^{−1}) ≠ 0 in the sum (6.3). We remark d_{j} = 1, p or p^{2}. Because p^{2}D_{j}^{−1} ∈ ℤ^{(n,n)} we have p^{2}D_{j}^{∗}^{−1}𝔡d_{j}^{−1} ∈ ℤ^{(n−1,1)}. If d_{j} = 1, then we can take 𝔡 = 0 ∈ ℤ^{(n−1,1)} as a representative. If d_{j} = p^{2}, then D_{j}^{∗}^{−1}𝔡 ∈ ℤ^{(n−1,1)}. We now assume d_{j} = p. Then pD_{j}^{∗}^{−1}𝔡 ∈ ℤ^{(n−1,1)}. By using the identity ^{t}B_{(j,l)}D_{j} = ^{t}D_{j}B_{(j,l)} we have

e(NB_{(j,l)}D_{j}^{−1})

= e(N′B_{(j,l)}D_{j}^{−1} − \frac{d_{j}λ_{3}}{2p} R ( \begin{matrix}0 & ··· & 0 & 0 \\ 0 & ··· & 0 & 1 \end{matrix} ) B_{(j,l)}D_{j}^{−1} − \frac{d_{j}λ_{3}}{2p} ( \begin{matrix}0 & 0 \\ ⋮ & ⋮ \\ 0 & 0 \\ 0 & 1 \end{matrix} )^{t}RB_{(j,l)}D_{j}^{−1}

−\frac{d_{j}^{2}λ_{3}^{2}}{p^{2}} ( \begin{matrix}0_{n−1} \\ & 1 \end{matrix} ) B_{(j,l)}D_{j}^{−1})

= e(N^{′∗}B_{(j,l)}^{∗}D_{j}^{∗}^{−1}) e(−\frac{d_{j}λ_{3}}{p^{2}} (R̂_{2}^{∗} − 2λ^{∗} − D_{j}^{∗}^{−1}𝔡_{𝔧}λ_{3})^{t}b_{3}) e(\frac{d_{j}λ_{3}^{2}}{p^{2}} b_{2}) .

Hence, if d_{j} = p, then ∑_{b_{2} mod p} e(\frac{d_{j}λ_{3}^{2}}{p^{2}} b_{2}) is zero unless λ_{3} ≡ 0 mod p. Thus,

for any d_{j} ∈ {1, p, p^{2}}, we conclude ∑_{l} e(NB_{(j,l)}D_{j}^{−1}) = 0 in the sum (6.3) unless D_{j}^{∗}^{−1}𝔡λ_{3} ∈ ℤ^{(n−1,1)}. Hence λ^{∗} − D_{j}^{∗−1}𝔡λ_{3} ∈ ℤ^{(n−1,1)} and λ′_{2} ∈ ℤ^{(n,1)}, if ∑_{l} e(NB_{(j,l)}D_{j}^{−1}) ≠ 0. Therefore there exists a set of complex numbers {C_{γ_{i}}}_{i} := {C_{γ_{i},k,M}}_{i} which satisfies

Φ (∑_{i} ϕ|_{k,M}γ_{i}) = ∑_{i} C_{γ_{i}^{∗}}Φ(ϕ)|_{k,M}γ_{i}^{∗}.

By a well-known argument we have ∑_{i} C_{γ_{i}^{∗}}γ_{i}^{∗}γ = ∑_{i} C_{γ_{i}^{∗}}γ_{i}^{∗} for any γ ∈ Γ_{n−1,2}^{J}. Hence there exists an index-shift map V_{α,n−α}(p^{2})^{∗} which satisfies the identity Φ(ϕ|V_{α,n−α}(p^{2})) = Φ(ϕ)|V_{α,n−α}(p^{2})^{∗}. For a fixed α (0 ≤ α ≤ n) the index-shift map V_{α,n−α}(p^{2})^{∗} is a linear combination of V_{β,n−1−β}(p^{2}) (β = 0, ..., n − 1). We need to determine these coefficients of the linear combination. This calculation is similar to the case of Siegel modular forms [Kr 86, page 325]. We leave the details to the reader.

⊓ ⊔

Now for integers l (2 ≤ l), β (0 ≤ β ≤ l − 1) and α (0 ≤ α ≤ l), we put

b_{β,α} := b_{β,α,l,p}(X) = \begin{cases}(p^{l+1−α} − p^{−l−1+α})p^{\frac{1}{2}} & if β = α − 2 \\ (X + X^{−1})p & if β = α − 1 \\ p^{−l+α+ \frac{3}{2}} & if β = α \\ 0 & otherwise \end{cases}

and set a matrix

B_{l,l+1}(X) := (b_{β,α})_{\begin{matrix}β=0,...,l−1 \\ α=0,...,l \end{matrix}} = (\begin{matrix}b_{0,0} & ⋯ & b_{0,l} \\ ⋮ & ... & ⋮ \\ b_{l−1,0} & ⋯ & b_{l−1,l} \end{matrix})

with entries in ℂ[X + X^{−1}]. For any ϕ ∈ J_{k,M}^{(l)}, due to Proposition 6.1, we obtain

\begin{matrix}Φ(ϕ)|(V_{0,l}(p^{2})^{∗}, ⋯, V_{l,0}(p^{2})^{∗}) \\ = p^{−k+l+ \frac{1}{2}} (Φ(ϕ)|(V_{0,l−1}(p^{2}), ⋯, V_{l−1,0}(p^{2}))) B_{l,l+1}(p^{k−l− \frac{1}{2}} ). \end{matrix}  (6.4)

Here Φ(ϕ)|(V_{0,l}(p^{2})^{∗}, ⋯, V_{l,0}(p^{2})^{∗}) denotes the row vector

Φ(ϕ)|(V_{0,l}(p^{2})^{∗}, ⋯, V_{l,0}(p^{2})^{∗}) := (Φ(ϕ)|V_{0,l}(p^{2})^{∗}, ..., Φ(ϕ)|V_{l,0}(p^{2})^{∗}) .

Let J_{k− \frac{1}{2} ,m}^{(n)∗} be the subspace of J_{k− \frac{1}{2} ,m}^{(n)} introduced in §4.4.

Corollary 6.2. For any Jacobi form ϕ ∈ J_{k− \frac{1}{2} ,m}^{(n)∗} and for any prime p, we have

Φ(ϕ|Ṽ_{α,n−α}(p^{2})) = Φ(ϕ)|Ṽ_{α,n−α}(p^{2})^{∗},

where Ṽ_{α,n−α}(p^{2})^{∗} is a map Ṽ_{α,n−α}(p^{2})^{∗} : J_{k− \frac{1}{2} ,m}^{(n−1)∗} → J_{k− \frac{1}{2} ,mp^{2}}^{(n−1)} given by

Ṽ_{α,n−α}(p^{2})^{∗} = p^{k−n− \frac{1}{2}} \{p^{−n+α}Ṽ_{α,n−α−1}(p^{2})
+(p^{−k+n+ \frac{1}{2}} + p^{k−n− \frac{1}{2}} )Ṽ_{α−1,n−α}(p^{2})
+(p^{n+1−α} − p^{−n−1+α})Ṽ_{α−2,n−α+1}(p^{2})}.

Proof. By a straightforward calculation we get the fact that ι_{M} and Φ is commutative. The rest of the proof of this corollary follows from Proposition 6.1 and Proposition 4.4. ⊓ ⊔

Let H̃_{p^{2}}^{(m)} be the local Hecke ring and let R_{m} be the subring of a polynomial ring

both defined in §2.9. The isomorphism Ψ_{m} : H̃_{p^{2}}^{(m)} ≅ R_{m} has been obtained in [Zh 83, Zh 84] (see §2.9).

Proposition 6.3. Let p be an odd prime. For any m ≥ 2, the image of generators K_{α}^{(m)} of H̃_{p^{2}}^{(m)} by Ψ_{m} are expressed as a vector

\begin{matrix}(Ψ_{m}(K_{0}^{(m)}), Ψ_{m}(K_{1}^{(m)}), ⋯, Ψ_{m}(K_{m}^{(m)})) \\ = p^{− \frac{3}{2} (m−1)} z_{0}^{2}z_{1} ⋯ z_{m}(p^{−1}, z_{1} + z_{1}^{−1}) (\begin{matrix}1 & 0 \\ 0 & p^{\frac{1}{2}} \end{matrix} )^{−1} \\ × \{∏_{l=2}^{m} B_{l,l+1}(z_{l})\} diag(1, p^{\frac{1}{2}} , ..., p^{\frac{m}{2}} ). \end{matrix}  (6.5)

Here B_{l,l+1}(X) is the l × (l + 1)-matrix introduced in above, and where

∏_{l=2}^{m} B_{l,l+1}(z_{l}) = B_{2,3}(z_{2})B_{3,4}(z_{3}) ⋯ B_{m,m+1}(z_{m})

is a 2 × (m + 1) matrix with entries in ℂ[z_{2}^{±}, ⋯, z_{m}^{±}]. We remark that

Ψ_{m}(K_{0}^{(m)}) = p^{− \frac{m(m+1)}{2}} z_{0}^{2}z_{1} ⋯ z_{m}.

Proof. Let k be an even integer and let F ∈ M_{k− \frac{1}{2}} (Γ_{0}^{(m)}(4)) be a Siegel modular form such that Φ^{S}(F) ≢ 0. Here Φ^{S} denotes the Siegel Φ-operator for Siegel modular forms. Let T ∈ H̃_{p^{2}}^{(m)} and let f_{T} (z_{0}, ..., z_{m}) := Ψ_{m}(T) ∈ R_{m}. Then f_{T} (z_{0}, ...z_{m−1}, p^{k−m− \frac{1}{2}} ) ∈ R_{m−1} and Ψ_{m−1}^{−1}(f_{T} (z_{0}, ..., z_{m−1}, p^{k−m− \frac{1}{2}} )) ∈ H̃_{p^{2}}^{(m−1)}. It is known by Oh-Koo-Kim [OKK 89, Theorem 5.1] that

Φ^{S}(F |T) = Φ^{S}(F)|Ψ_{m−1}^{−1}(f_{T} (z_{0}, ..., z_{m−1}, p^{k−m− \frac{1}{2}} )).  (6.6)

Let ϕ ∈ J_{k− \frac{1}{2} ,a}^{(m)} be a Jacobi form with index a ∈ ℤ such that Φ(ϕ) ≢ 0. Here Φ is the Siegel Φ-operator. If k is large enough, then there exists such ϕ. Due to Corollary 6.2 we have

Φ(ϕ|Ṽ_{α,m−α}(p^{2})) = Φ(ϕ)|Ṽ_{α,m−α}(p^{2})^{∗}.

Let 𝕎 : J_{k− \frac{1}{2} ,a}^{(m)} → M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(m)}(4)) be the Witt operator which is defined by

𝕎(ϕ)(τ) := ϕ(τ, 0)

for any ϕ(τ, z) ∈ J_{k− \frac{1}{2} ,a}^{(m)}. By a straightforward calculation, for any ϕ ∈ J_{k− \frac{1}{2} ,a}^{(m)} we have

𝕎(ϕ|Ṽ_{α,m−α}(p^{2})) = 𝕎(ϕ)|T̃_{α,m−α}(p^{2})

and

𝕎(Φ(ϕ)) = Φ^{S}(𝕎(ϕ)).

Lifting to Siegel Modular Forms … 175

We set

T̃_{α,m−α}(p^{2})^{∗} = p^{k−m− \frac{1}{2}} \{p^{−m+α}T̃_{α,m−α−1}(p^{2})
+(p^{−k+m+ \frac{1}{2}} + p^{k−m− \frac{1}{2}} )T̃_{α−1,m−α}(p^{2})
+(p^{m+1−α} − p^{−m−1+α})T̃_{α−2,m−α+1}(p^{2})}.

If we put F = 𝕎(ϕ), then

Φ^{S}(F |T̃_{α,m−α}(p^{2})) = Φ^{S}(𝕎(ϕ|Ṽ_{α,m−α}(p^{2})))
= 𝕎(Φ(ϕ|Ṽ_{α,m−α}(p^{2})))
= 𝕎(Φ(ϕ)|Ṽ_{α,m−α}(p^{2})^{∗})
= Φ^{S}(F)|T̃_{α,m−α}(p^{2})^{∗}.

Hence if we put T = T̃_{α,m−α}(p^{2}) in (6.6) we have

f_{T} (z_{0}, ..., z_{m−1}, p^{k−m− \frac{1}{2}} ) = p^{k−m− \frac{1}{2}} \{p^{−m+α}Ψ_{m−1}(K_{α}^{(m−1)})

+(p^{−k+m+ \frac{1}{2}} + p^{k−m− \frac{1}{2}} )Ψ_{m−1}(K_{α−1}^{(m−1)}) +(p^{m+1−α} − p^{−m−1+α})Ψ_{m−1}(K_{α−2}^{(m−1)})}.

Since this identity is true for infinitely many k, we have

Ψ_{m}(K_{α}^{(m)}) = f_{T} (z_{0}, ..., z_{m−1}, z_{m})
= z_{m}{p^{−m+α}Ψ_{m−1}(K_{α}^{(m−1)})
+(z_{m} + z_{m}^{−1})Ψ_{m−1}(K_{α−1}^{(m−1)})
+(p^{m+1−α} − p^{−m−1+α})Ψ_{m−1}(K_{α−2}^{(m−1)})}.

Hence (Ψ_{m}(K_{0}^{(m)}), Ψ_{m}(K_{1}^{(m)}), ⋯, Ψ_{m}(K_{m}^{(m)})) = p^{−3/2} z_{m} (Ψ_{m−1}(K_{0}^{(m−1)}), Ψ_{m−1}(K_{1}^{(m−1)}), ⋯, Ψ_{m−1}(K_{m−1}^{(m−1)}))

× diag(1, p^{\frac{1}{2}} , ..., p^{\frac{m−1}{2}} )^{−1}B_{m,m+1}(z_{m}) diag(1, p^{\frac{1}{2}} , ..., p^{\frac{m}{2}} ). Moreover, we have

Ψ_{1}(K_{0}^{(1)}) = p^{−1}z_{0}^{2}z_{1},
Ψ_{1}(K_{1}^{(1)}) = z_{0}^{2}(1 + z_{1}^{2})

Documenta Mathematica 21 (2016) 125–196

by the definition of Ψ_{1}. This proposition follows from these identities and the recursion with respect to m. ⊓ ⊔

7 Maass relation for generalized Cohen-Eisenstein series

We put a 2 × (n + 1)-matrix

A_{2,n+1}^{p}(X) := ∏_{l=2}^{n} B_{l,l+1}(p^{\frac{n+2}{2} −l}X)
= B_{2,3}(p^{\frac{n+2}{2} −2}X)B_{3,4}(p^{\frac{n+2}{2} −3}X) ⋯ B_{n,n+1}(p^{\frac{n+2}{2} −n}X),

where B_{l,l+1}(X) is the l × (l + 1)-matrix introduced in §6.

Lemma 7.1. All components of the matrix A_{2,2n−1}^{p}(X) belong to ℂ[X + X^{−1}].

Proof. We assume p is an odd prime. Let R_{m} be the symbol introduced in §2.9. Because Ψ_{2n−2}(K_{α}^{(2n−2)}) belongs to R_{2n−2} and because of Proposition 6.3 we have relations B_{l,l+1}(z_{l}) = B_{l,l+1}(z_{l}^{−1}) (l = 2, ..., 2n − 2) and

B_{2,3}(z_{2})B_{3,4}(z_{3}) ⋯ B_{2n−2,2n−1}(z_{2n−2})
= B_{2,3}(z_{2n−2})B_{3,4}(z_{2n−3}) ⋯ B_{2n−2,2n−1}(z_{2}).

Hence

A_{2,2n−1}^{p}(X) = B_{2,3}(p^{n−2}X)B_{3,4}(p^{n−3}X) ⋯ B_{2n−2,2n−1}(p^{−n+2}X)
= B_{2,3}(p^{−n+2}X^{−1})B_{3,4}(p^{−n+3}X^{−1}) ⋯ B_{2n−2,2n−1}(p^{n−2}X^{−1})
= B_{2,3}(p^{n−2}X^{−1})B_{3,4}(p^{n−3}X^{−1}) ⋯ B_{2n−2,2n−1}(p^{−n+2}X^{−1})
= A_{2,2n−1}^{p}(X^{−1}).

The relation A_{2,2n−1}^{p}(X) = A_{2,2n−1}^{p}(X^{−1}) holds for infinitely many p. Hence if we regard that the components of the matrix A_{2,2n−1}^{p}(X) are Laurent-polynomials of variables X and p^{1/2}, then we obtain A_{2,2n−1}^{p}(X) = A_{2,2n−1}^{p}(X^{−1}). Hence we have also A_{2,2n−1}^{p}(X) = A_{2,2n−1}^{p}(X^{−1}) for p = 2. ⊓ ⊔

Let M, m, D_{0} and f be the symbols used in the previous sections, it means

that M = (\begin{matrix}∗ & ∗ \\ ∗ & 1 \end{matrix}) is a 2×2 half-integral symmetric-matrix, m = det(2M), D_{0}

is the discriminant of ℚ(\sqrt{−m}) and f is a non-negative integer which satisfies m = D_{0}f^{2}.

For any prime p we set

(\begin{matrix}a_{0,m,p,k} \\ a_{1,m,p,k} \\ a_{2,m,p,k} \end{matrix}) := \begin{cases}(\begin{matrix}p^{−3k+4} \\ 0 \\ p^{−k+1} \end{matrix}) & if p|f, \\ (\begin{matrix}0 \\ p^{−3k+4} + p^{−2k+2} (\frac{−m}{p}) \\ p^{−k+1} − p^{−2k+2} (\frac{−m}{p}) \end{matrix}) & if p |̸f. \end{cases}

Lemma 7.2. For the Jacobi-Eisenstein series E_{k,M}^{(1)} of weight k of index M of degree 1, we have the identity

E_{k,M}^{(1)}|(V_{0,1}(p^{2}), V_{1,0}(p^{2}))

= (E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)})

× (\begin{matrix}0 & a_{0,m,p,k} \\ p^{−2k+2} & a_{1,m,p,k} \\ 0 & a_{2,m,p,k} \end{matrix}) ,

where X = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) ∈ ℤ^{(2,2)} is a matrix such that M[X^{−1} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}] ∈ Sym_{2}^{+}. Here, if p |̸f, there does not exist such matrix X and we regard

E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(1)} as zero.

Proof. From Proposition 5.3 and due to (5.2), (5.3) in the proof of Proposition 5.5, we have

E_{k,M}^{(1)}|V_{0,1}(p^{2}) = K̃_{0,1}^{0}
= p^{−2k+1} ∑_{M∈Γ(δ_{0,1})\Γ_{1}} ∑_{λ_{2}∈L_{0,1}^{∗}} \{1|_{k,M}([(λ_{2}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

× ∑_{u_{2}∈ℤ/pℤ} G_{M}^{1,1}(λ_{2} + (0, u_{2}))

= p^{−2k+2} ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ_{2}∈ℤ^{(1,2)}} \{1|_{k,M}([(λ_{2}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= p^{−2k+2}E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}.

From Proposition 5.3 we also have

E_{k,M}^{(1)}|V_{1,0}(p^{2}) = K̃_{1,1}^{0} + K̃_{0,1}^{1} + K̃_{0,0}^{0}.

178 S.Hayashida Here

K̃_{1,1}^{0} = p^{−k+1} ∑_{M∈Γ(δ_{1,1})\Γ_{1}} ∑_{λ_{1}∈L_{1,1}^{∗}} \{1|_{k,M}([(λ_{1}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= p^{−k+1} ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ_{1}∈pℤ×ℤ} \{1|_{k,M}([(λ_{1}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= p^{−k+1} ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ∈ℤ^{(1,2)}} \{1|_{k,M}([(λ ( \begin{matrix}p \\ & 1 \end{matrix} ) , 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= p^{−k+1} ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ∈ℤ^{(1,2)}} \{1|_{k,M[( \begin{matrix}p \\ & 1 \end{matrix} )]}([(λ, 0), 0], M)\} (τ, z)
= p^{−k+1}E_{k,M[( \begin{matrix}p \\ & 1 \end{matrix} )]}^{(1)}(τ, z).

Now we shall calculate K̃_{0,1}^{1}. First, due to Lemma 5.4 we have

∑_{u_{2}∈ℤ/pℤ} G_{M}^{1,0}(λ + (0, u_{2})) = \begin{cases}0 & if λ ∈ ℤ^{(1,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , \\ (\frac{−m}{p}) p & if λ ∉ ℤ^{(1,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) \end{cases}

for any λ ∈ ℤ^{(1,2)}. Thus

K̃_{0,1}^{1}

= p^{−2k+2} ∑_{M∈Γ(δ_{0,1})\Γ_{1}} ∑_{λ_{2}∈L_{0,1}^{∗}} \{1|_{k,M}([(λ_{2}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

× ∑_{u_{2}∈ℤ/pℤ} G_{M}^{1,0}(λ_{2} + (0, u_{2}))

= −p^{−2k+2} ( \frac{−m}{p} ) ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ∈ℤ^{(1,2)}} \{1|_{k,M}([(λ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) , 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

+p^{−2k+2} ( \frac{−m}{p} ) ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ∈ℤ^{(1,2)}} \{1|_{k,M}([(λ, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= −p^{−2k+2} ( \frac{−m}{p} ) E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)}(τ, z) + p^{−2k+2} ( \frac{−m}{p} ) E_{k,M}(τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= −p^{−2k+2} ( \frac{−m}{p} ) E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)} + p^{−2k+2} ( \frac{−m}{p} ) E_{k,M}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}.

We shall calculate K̃_{0,0}^{0}. Due to (5.2) and due to (5.3) we have

L_{0,0}^{∗} = \begin{cases}ℤ^{(1,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} ^{t}X^{−1} & if p|f, \\ ℤ^{(1,2)} & if p |̸f. \end{cases}

Documenta Mathematica 21 (2016) 125–196

Thus if p|f, then

K̃_{0,0}^{0} = p^{−3k+4} ∑_{M∈Γ(δ_{0,0})\Γ_{1}} ∑_{λ_{3}∈ℤ^{(1,2)}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}^{t}X^{−1}} \{1|_{k,M}([(λ_{3}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))

= p^{−3k+4} ∑_{M∈Γ_{∞}^{(1)}\Γ_{1}} ∑_{λ_{3}∈ℤ^{(1,2)}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}^{t}X^{−1}}

× \{1|_{k,M[X^{−1}( \begin{matrix}p \\ & 1 \end{matrix} )^{−1}]}([(λ_{3}^{t}X ( \begin{matrix}p \\ & 1 \end{matrix} ) , 0), 0], M)\} (τ, z ( \begin{matrix}p \\ & 1 \end{matrix} )^{t}X ( \begin{matrix}p \\ & 1 \end{matrix} ))

= p^{−3k+4}E_{k,M[X^{−1}( \begin{matrix}p \\ & 1 \end{matrix} )^{−1}]}^{(1)}(τ, z ( \begin{matrix}p \\ & 1 \end{matrix} )^{t}X ( \begin{matrix}p \\ & 1 \end{matrix} )),

and if p |̸f, then

K̃_{0,0}^{0} = p^{−3k+4} ∑_{M∈Γ(δ_{0,0})\Γ_{1}} ∑_{λ_{3}∈ℤ^{(1,2)}} \{1|_{k,M}([(λ_{3}, 0), 0], M)\} (τ, z ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ))
= p^{−3k+4}E_{k,M}^{(1)}(τ, z ( \begin{matrix}p \\ & 1 \end{matrix} )).

Hence we obtain the formula for K̃_{0,0}^{0}. Because E_{k,M}^{(1)}|V_{1,0}(p^{2}) = K̃_{1,1}^{0} + K̃_{0,1}^{1} + K̃_{0,0}^{0}, we conclude the lemma. ⊓ ⊔

Lemma 7.3. The three forms E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}

and E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)} are linearly independent.

Proof. We first assume that M ∈ Sym_{g}^{+} is a positive-definite half-integral symmetric matrix of size g. Let

E_{k,M}^{(1)}(τ, z) = ∑_{\begin{matrix}n∈ℤ,R∈ℤ^{(1,g)} \\ 4n−RM^{−1}^{t}R≥0 \end{matrix}} C_{k,M}(n, R) e(nτ + R^{t}z)

be the Fourier expansion of the Jacobi-Eisenstein series E_{k,M}^{(1)}. For any pair n ∈ ℤ and R ∈ ℤ^{(1,g)} which satisfy 4n − RM^{−1}^{t}R > 0, we now show that C_{k,M}(n, R) ≠ 0. The Fourier coefficients of Jacobi-Eisenstein series of degree 1 of integer index have been calculated in [E-Z 85, pp.17–22]. If 4n−RM^{−1}^{t}R > 0, by an argument similar to [E-Z 85] we have

C_{k,M}(n, R) = \frac{(−1)^{\frac{k}{2}} π^{k− \frac{g}{2}}}{2^{k−2}Γ(k − \frac{g}{2})} \frac{(4n − RM^{−1}^{t}R)^{k− \frac{g}{2} −1}}{det(M)^{\frac{1}{2}}} ζ(k − g)^{−1} ∑_{a=1}^{∞} \frac{N_{a}(Q)}{a^{k−1}},

where N_{a}(Q) := |{λ ∈ (ℤ/aℤ)^{(1,g)} | λM^{t}λ + R^{t}λ + n ≡ 0 mod a}|. Hence we conclude C_{k,M}(n, R) ≠ 0.

Documenta Mathematica 21 (2016) 125–196

We now assume M = ( \begin{matrix}l & \frac{r}{2} \\ \frac{r}{2} & 1 \end{matrix} ) ∈ Sym_{2}^{+}. The (n, R)-th Fourier coefficient of

two Jacobi forms E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} and E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)} are C_{k,M}(n, R ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}) and

C_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}(n, R), respectively. If R∉ℤ^{(1,2)} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ), then C_{k,M}(n, R ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}) =

0 and C_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}(n, R) ≠ 0. Hence E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} and E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)} are lin-

early independent. The proof for the linear independence of the three forms of the lemma is similar. We omitted the detail here. ⊓ ⊔

Proposition 7.4. We obtain the identity

E_{k,M}^{(n)}| (V_{0,n}(p^{2}), ⋯, V_{n,0}(p^{2}))
= (E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)})

× (\begin{matrix}0 & a_{0,m,p,k} \\ p^{−2k+2} & a_{1,m,p,k} \\ 0 & a_{2,m,p,k} \end{matrix}) p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)}A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ),

where the 2 × (n + 1)-matrix A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ) is introduced in the beginning of this section.

Proof. Let Φ be the Siegel Φ-operator introduced in §6. From the definition of Jacobi-Eisenstein series, we have Φ(E_{k,M}^{(l)}) = E_{k,M}^{(l−1)}. From the identity (6.4) in §6 and from Lemma 7.2, we obtain

Φ^{n−1}(E_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2})))
= (E_{k,M}^{(1)}|(V_{0,1}(p^{2}), V_{1,0}(p^{2}))) (∏_{l=2}^{n} p^{−k+l+ \frac{1}{2}} ) B_{2,3}(p^{k− \frac{5}{2}} ) ⋯ B_{n,n+1}(p^{k−n− \frac{1}{2}} )
= p^{(−k+ \frac{1}{2} )(n−1)+ \frac{n(n+1)}{2} −1} (E_{k,M}^{(1)}|(V_{0,1}(p^{2}), V_{1,0}(p^{2}))) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} )
= (E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)})  (7.1)

×p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)} (\begin{matrix}0 & a_{0,m,p,k} \\ p^{−2k+2} & a_{1,m,p,k} \\ 0 & a_{2,m,p,k} \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

From Proposition 5.5 there exists a matrix M ∈ ℂ^{(3,n+1)} which satisfies

\begin{matrix}E_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2})) \\ = (E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}) M. \end{matrix} ; (7.2)

Thus

Φ^{n−1}(E_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2})))
= (E_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M}^{(1)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(1)}) M.

From Lemma 7.3 the matrix M is uniquely determined. Therefore, by using the identity (7.1), we have

M = p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)} (\begin{matrix}0 & a_{0,m,p,k} \\ p^{−2k+2} & a_{1,m,p,k} \\ 0 & a_{2,m,p,k} \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

Therefore we conclude that this Proposition follows from the identity (7.2). ⊓ ⊔

We recall that the form e_{k,M}^{(n)} is the M-th Fourier-Jacobi coefficient of Siegel-

Eisenstein series E_{k}^{(n+2)} of weight k of degree n + 2.

Proposition 7.5. We obtain the identity

e_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2}))
= p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)}

× (e_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)})

× (\begin{matrix}0 & p^{−k+1} \\ p^{−2k+2} & p^{−2k+2} (\frac{−m}{p}) \\ 0 & p^{−3k+4} \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

Proof. For any ϕ ∈ J_{k,M}^{(n)} and for any L = ( \begin{matrix}a & 0 \\ b & 1 \end{matrix} ) ∈ ℤ^{(2,2)}, a straightforward calculation gives the identity

(ϕ|U_{L}) |V_{α,n−α}(p^{2}) = (ϕ|V_{α,n−α}(p^{2})) |U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}L( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}.  (7.3)

We recall from Proposition 3.3 the identity

e_{k,M}^{(n)} = ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{k,M[W_{d}^{−1}]}^{(n)}(τ, z^{t}W_{d})

= ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}}

where W_{d} is a matrix such that M [W_{d}^{−1}] ∈ Sym_{2}^{+}. We choose the set of ma-

trices \{W_{d}\}_{d} which satisfies M [W_{d}^{−1} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}] ∈ Sym_{2}^{+}, if d|\frac{f}{p}. In particular, 182 S.Hayashida we choose W_{pd} = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) W_{d} for d such that pd|f. By virtue of Lemma 3.1 and of the identity W_{pd} = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) W_{d}, we have

E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )} = E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )W_{pd}^{−1}]}^{(n)}|U_{W_{pd}}.  (7.4)

For the sake of simplicity we write

\begin{aligned}E_{0}(d) & = E_{k,M[W_{pd}^{−1}]}^{(n)}|U_{W_{pd}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, \\ E_{1}(d) & = E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, \\ E_{2}(d) & = E_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )W_{d}^{−1}]}^{(n)}|U_{W_{d}}. \end{aligned}  (7.5)

We remark E_{0}(d) = E_{1}(pd) and E_{1}(d) = E_{2}(pd) due to the identity (7.4). From Proposition 7.4 and due to identities (7.3) and (7.5) we get

(E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}}) | (V_{0,n}(p^{2}), ⋯, V_{n,0}(p^{2}))

= p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)} (E_{0}(d), E_{1}(d), E_{2}(d)) (\begin{matrix}0 & a_{0, \frac{m}{d^{2}} ,p,k} \\ p^{−2k+2} & a_{1, \frac{m}{d^{2}} ,p,k} \\ 0 & a_{2, \frac{m}{d^{2}} ,p,k} \end{matrix})
× A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

Hence from Proposition 3.3 we have e_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2}))

= ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) (E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}}) |(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2}))
= p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)} ∑_{d|f} g_{k}(\frac{m}{d^{2}} )

× (E_{0}(d), E_{1}(d), E_{2}(d)) (\begin{matrix}0 & a_{0, \frac{m}{d^{2}} ,p,k} \\ p^{−2k+2} & a_{1, \frac{m}{d^{2}} ,p,k} \\ 0 & a_{2, \frac{m}{d^{2}} ,p,k} \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

On the RHS of the above identity we obtain

∑_{d|f} g_{k}(\frac{m}{d^{2}} ) (E_{0}(d), E_{1}(d), E_{2}(d)) (\begin{matrix}0 \\ p^{−2k+2} \\ 0 \end{matrix})

= p^{−2k+2} ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{1}(d)

Documenta Mathematica 21 (2016) 125–196

Lifting to Siegel Modular Forms … 183

= p^{−2k+2} ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{k,M[W_{d}^{−1}]}^{(n)}|U_{W_{d}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}

= p^{−2k+2}e_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}.

By using Lemma 3.2 we now have

∑_{d|f} g_{k}( \frac{m}{d^{2}} ) (E_{0}(d), E_{1}(d), E_{2}(d)) (\begin{matrix}a_{0, \frac{m}{d^{2}} ,p,k} \\ a_{1, \frac{m}{d^{2}} ,p,k} \\ a_{2, \frac{m}{d^{2}} ,p,k} \end{matrix})

= ∑_{d|f} g_{k}( \frac{m}{d^{2}} ) \{δ(p| \frac{f}{d} ) p^{−3k+4}E_{0}(d)

+ (p^{−2k+2} ( \frac{−m/d^{2}}{p} ) + δ(p |̸ \frac{f}{d} ) p^{−3k+4}) E_{1}(d)

+ (p^{−k+1} − p^{−2k+2} ( \frac{−m/d^{2}}{p} )) E_{2}(d)\}

= p^{−3k+4} ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≡0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{0}(d) + p^{−2k+2} ( \frac{−D_{0}}{p} ) ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{1}(d)

+p^{−3k+4} ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{1}(d)

+p^{−3k+4} ∑_{d|f} g_{k}( \frac{m}{d^{2}} ) (p^{2k−3} − p^{k−2} ( \frac{−m/d^{2}}{p} )) E_{2}(d)

= p^{−3k+4}δ(p|f) ∑_{d| \frac{f}{p}} (p^{2k−3} − p^{k−2} ( \frac{−m/(dp)^{2}}{p} )) g_{k}( \frac{m}{d^{2}p^{2}} ) E_{0}(d)

+p^{−2k+2} ( \frac{D_{0}}{p} ) ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{1}(d) + p^{−3k+4} ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{mp^{2}}{(dp)^{2}} ) E_{2}(pd)

+p^{−3k+4} ∑_{d|f} g_{k}( \frac{mp^{2}}{d^{2}} ) E_{2}(d)

= p^{−k+1}δ(p|f) ∑_{d| \frac{f}{p}} g_{k}( \frac{m}{d^{2}p^{2}} ) E_{0}(d) − p^{−2k+2}δ(p|f) ( \frac{D_{0}}{p} )

× ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{0}( \frac{d}{p} )

+p^{−2k+2} ( \frac{D_{0}}{p} ) ∑_{\begin{matrix}d|f \\ \frac{f}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{m}{d^{2}} ) E_{1}(d) + p^{−3k+4} ∑_{\begin{matrix}d|fp \\ \frac{fp}{d} ≢0 (p) \end{matrix}} g_{k}( \frac{mp^{2}}{d^{2}} ) E_{2}(d)

184 S.Hayashida

+p^{−3k+4} ∑_{\begin{matrix}d|fp \\ \frac{fp}{d} ≡0 (p) \end{matrix}} g_{k}( \frac{mp^{2}}{d^{2}} ) E_{2}(d)

= p^{−k+1}δ(p|f) ∑_{d| \frac{f}{p}} g_{k}( \frac{m}{d^{2}p^{2}} ) E_{0}(d) + p^{−2k+2} ( \frac{D_{0}f^{2}}{p} ) ∑_{d|f} g_{k}( \frac{m}{d^{2}} ) E_{1}(d)

+p^{−3k+4} ∑_{d|fp} g_{k}( \frac{mp^{2}}{d^{2}} ) E_{2}(d).

Hence

∑_{d|f} g_{k}(\frac{m}{d^{2}} ) (E_{0}(d), E_{1}(d), E_{2}(d)) (\begin{matrix}a_{0, \frac{m}{d^{2}} ,p,k} \\ a_{1, \frac{m}{d^{2}} ,p,k} \\ a_{2, \frac{m}{d^{2}} ,p,k} \end{matrix})

= (∑_{d| \frac{f}{p}} g_{k}(\frac{m}{d^{2}p^{2}} ) E_{0}(d), ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) E_{1}(d), ∑_{d|fp} g_{k}(\frac{mp^{2}}{d^{2}} ) E_{2}(d))

× (\begin{matrix}p^{−k+1} \\ p^{−2k+2} (\frac{−m}{p}) \\ p^{−3k+4} \end{matrix})

= (e_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)})

× (\begin{matrix}p^{−k+1} \\ p^{−2k+2} (\frac{−m}{p}) \\ p^{−3k+4} \end{matrix}) .

Therefore e_{k,M}^{(n)}|(V_{0,n}(p^{2}), ..., V_{n,0}(p^{2}))

= p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)} ∑_{d|f} g_{k}(\frac{m}{d^{2}} ) (E_{0}(d), E_{1}(d), E_{2}(d))(\begin{matrix}0 & a_{0, \frac{m}{d^{2}} ,p,k} \\ p^{−2k+2} & a_{1, \frac{m}{d^{2}} ,p,k} \\ 0 & a_{2, \frac{m}{d^{2}} ,p,k} \end{matrix})

×A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} )

= p^{(−k+ \frac{1}{2} n+ \frac{3}{2} )(n−1)}

× (e_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}, e_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)})

× (\begin{matrix}0 & p^{−k+1} \\ p^{−2k+2} & p^{−2k+2} (\frac{−m}{p}) \\ 0 & p^{−3k+4} \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ).

Documenta Mathematica 21 (2016) 125–196

⊓ ⊔

Proposition 7.5 is a generalized Maass relation for matrix index of integralweight. The generalized Maass relation for integer index of half-integral weight is as follows.

Theorem 7.6. Let e_{k− \frac{1}{2} ,m}^{(n)} be the m-th Fourier-Jacobi coefficient of generalized

Cohen-Eisenstein series H_{k− \frac{1}{2}}^{(n+1)}. (See (1.1)). Then we obtain

e_{k− \frac{1}{2} ,m}^{(n)}|(Ṽ_{0,n}(p^{2}), Ṽ_{1,n−1}(p^{2}), ..., Ṽ_{n,0}(p^{2}))
= p^{k(n−1)− \frac{1}{2} (n^{2}+5n−5)} (e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(n)} |U_{p^{2}}, e_{k− \frac{1}{2} ,m}^{(n)}|U_{p}, e_{k− \frac{1}{2} ,mp^{2}}^{(n)})

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ) diag(1, p^{1/2}, ⋯, p^{n/2}).

Here A_{2,n+1}^{p}(p^{k− \frac{n+2}{2} − \frac{1}{2}} ) is a 2 × (n + 1) matrix which is introduced in the beginning of §7 and the both side of the above identity are vectors of forms.

Proof. From Lemma 4.2 and from the definitions of e_{k,M}^{(n)} and e_{k− \frac{1}{2} ,m}^{(n)}, we have

ι_{M}(e_{k,M}^{(n)}) = e_{k− \frac{1}{2} ,m}^{(n)}.

By using Proposition 4.4 we have

e_{k− \frac{1}{2} ,m}^{(n)}|Ṽ_{α,n−α}(p^{2}) = ι_{M}(e_{k,M}^{(n)})|Ṽ_{α,n−α}(p^{2})
= p^{k(2n+1)−n(n+ \frac{7}{2} )+ \frac{1}{2} α}ι_{M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}(e_{k,M}^{(n)}|V_{α,n−α}(p^{2}))

From Proposition 4.3 we also have identities

e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(n)} |U_{p^{2}} = ι_{M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]} (e_{k,M[X^{−1}( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1}]}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )X( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}) ,

e_{k− \frac{1}{2} ,m}^{(n)}|U_{p} = ι_{M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]} (e_{k,M}^{(n)}|U_{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}) ,

and

e_{k− \frac{1}{2} ,mp^{2}}^{(n)} = ι_{M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]} (e_{k,M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]}^{(n)}) .

Because the map ι_{M[( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )]} is a linear map, this theorem follows from Propo-

sition 7.5 and from the above identities. ⊓ ⊔ 8 Maass relation for Siegel cusp forms of half-integral weight

and lifts

In this section we shall prove Theorem 8.3. We denote by S_{k}(Γ_{n}) ⊂ M_{k}(Γ_{n}), S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)) ⊂ M_{k− \frac{1}{2}}^{+} (Γ_{0}^{(n)}(4)), J_{k,1}^{(n) cusp} ⊂

J_{k,1}^{(n)} and J_{k− \frac{1}{2} ,m}^{(n)∗ cusp} ⊂ J_{k− \frac{1}{2} ,m}^{(n)∗} the spaces of the cusp forms, respectively (cf §4.3, §4.4, §2.5 and §2.6). Let k be an even integer and f ∈ S_{2(k−n)}(Γ_{1}) be an eigenform for all Hecke operators. Let

h(τ) = ∑_{\begin{matrix}N∈ℤ \\ N≡0,3 mod 4, N>0 \end{matrix}} c(N) e(Nτ) ∈ S_{k−n+ \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4))

be a Hecke eigenform which corresponds to f by the Shimura correspondence. We assume that the Fourier coefficient of f at e^{2πiz} is 1. Let

I_{2n}(h)(τ) = ∑_{T ∈Sym_{2n}^{+}} A(T) e(T τ) ∈ S_{k}(Γ_{2n})

be the Ikeda lift of h. For T ∈ Sym_{2n}^{+} the T -th Fourier coefficient A(T) of I_{2n}(h) is

A(T) = c(|D_{T} |) f_{T}^{k−n− \frac{1}{2}} ∏_{\begin{matrix}q: prime \\ q|f_{T} \end{matrix}} F̃_{q}(T, α_{q}),

where D_{T} is the fundamental discriminant and f_{T} is the natural number which satisfy det(2T) = |D_{T} | f_{T}^{2}, and where {α_{q}^{±}} is the set of Satake parameters of f in the sense of Ikeda [Ik 01], it means that (α_{q} + α_{q}^{−1})q^{k−n−1/2} is the q-th Fourier coefficient of f. Here F̃_{q}(T, X) ∈ ℂ[X + X^{−1}] is a Laurent polynomial. For the detail of the definition of F̃_{q}(T, X) the reader is referred to [Ik 01, page 642]. Let

I_{2n}(h) ((\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{a=1}^{∞} ψ_{a}(τ, z) e(aω)

be the Fourier-Jacobi expansion of I_{2n}(h), where τ ∈ ℌ_{2n−1}, ω ∈ ℌ_{1} and z ∈ ℂ^{(2n−1,1)}. Note that ψ_{a} ∈ J_{k,a}^{(2n−1) cusp} is a Jacobi cusp form of weight k of index a of degree 2n − 1. By the Eichler-Zagier-Ibukiyama correspondence (see §4.3) there exists a Siegel cusp form F ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−1)}(4)) which corresponds to ψ_{1} ∈ J_{k,1}^{(2n−1) cusp}.

For g ∈ S_{k−1/2}^{+}(Γ_{0}^{(1)}(4)) we put

F_{h,g}(τ) := \frac{1}{6} ∫_{Γ_{0}^{(1)}(4)\ℌ_{1}} F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) \overline{g(ω)} Im(ω)^{k− \frac{5}{2}} dω

for τ ∈ ℌ_{2n−2}. It is not difficult to show that the form F_{h,g} belongs to S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−2)}(4)). The above construction of F_{h,g} was suggested by T.Ikeda to the author. To show properties of F_{h,g} we consider the Fourier-Jacobi expansion of F. Let

F ((\begin{matrix}τ & z \\ ^{t}z & ω \end{matrix})) = ∑_{\begin{matrix}m∈ℤ \\ m≡0,3 mod 4 \end{matrix}} ϕ_{m}(τ, z) e(mω)

be the Fourier-Jacobi expansion of F, where τ ∈ ℌ_{2n−2}, ω ∈ ℌ_{1} and z ∈ ℂ^{(2n−2,1)}. Note that ϕ_{m} ∈ J_{k− \frac{1}{2} ,m}^{(2n−2)∗ cusp} is a Jacobi cusp form of weight k − \frac{1}{2} of index m and of degree 2n − 2. Let

ϕ_{m}(τ, z) = ∑_{\begin{matrix}M∈Sym_{2n−2}^{+}, S∈ℤ^{(2n−2,1)} \\ 4Mm−S^{t}S>0 \end{matrix}} C_{m}(M, S) e(Mτ + S^{t}z)

be the Fourier expansion of ϕ_{m}, where τ ∈ ℌ_{2n−2} and z ∈ ℂ^{(2n−2,1)}. We have the diagram

[figure: I_{2n}(h) ∈ S_{k}(Γ_{2n}); 1st F-J; ψ_{1} ∈ J_{k,1}^{(2n−1) cusp}; E-Z-I; F ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−1)}(4)); F-J; \{ϕ_{m}\}_{m} ∈ ⨂_{m≡0,3 mod 4} J_{k− \frac{1}{2} ,m}^{(2n−2)∗ cusp}; h ∈ S_{k−n+ \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)); Ikeda lift; f ∈ S_{2(k−n)}(Γ_{1}).]

Lemma 8.1. The (M, S)-th Fourier coefficient C_{m}(M, S) of ϕ_{m} is

C_{m}(M, S) = c(|D_{T} |) f_{T}^{k−n− \frac{1}{2}} ∏_{q|f_{T}} F̃_{q}(T, α_{q}),

where T ∈ Sym_{2n}^{+} is the matrix which satisfies

T = (\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & 1 \end{matrix})

and N ∈ Sym_{2n−1}^{+} and R ∈ ℤ^{(2n−1,1)} are the matrices which satisfy

4N − R^{t}R = (\begin{matrix}M & \frac{1}{2}S \\ \frac{1}{2}^{t}S & m \end{matrix}) .

Proof. The Fourier expansion of ψ_{1} is

ψ_{1}(τ, z) = ∑_{\begin{matrix}N∈Sym_{2n−1}^{+}, R∈ℤ^{(2n−1,1)} \\ 4N−R^{t}R>0 \end{matrix}} A ((\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & 1 \end{matrix})) e(Nτ + R^{t}z).

And the Fourier expansion of F is

F (τ) = ∑_{4N−R^{t}R>0} A ((\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & 1 \end{matrix})) e((4N − R^{t}R)τ).

Since ϕ_{m} is the m-th Fourier-Jacobi coefficient of F, the (M, S)-th Fourier coefficient C_{m}(M, S) of ϕ_{m} is A(T), where T is in the statement of this lemma.

⊓ ⊔

The following theorem is a generalization of the Maass relation for Siegel cusp forms of half-integral weight.

Theorem 8.2. Let ϕ_{m} be the m-th Fourier-Jacobi coefficient of F as above. Then we obtain

ϕ_{m}|(Ṽ_{0,2n−2}(p^{2}), Ṽ_{1,2n−3}(p^{2}), ..., Ṽ_{2n−2,0}(p^{2}))

= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} (ϕ_{\frac{m}{p^{2}}} |U_{p^{2}}, ϕ_{m}|U_{p}, ϕ_{mp^{2}}) (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix})

×A_{2,2n−1}^{p}(α_{p}) diag(1, p^{\frac{1}{2}} , p, ..., p^{n−1})

for any prime p, where the 2 × (n + 1)-matrix A_{2,2n−1}^{p}(α_{p}) is introduced in the beginning of §7.

Proof. Let

(ϕ_{m}|Ṽ_{α,2n−2−α}(p^{2}))(τ, z) = ∑_{\begin{matrix}M∈Sym_{2n−2}^{+}, S∈ℤ^{(2n−2,1)} \\ 4Mmp^{2}−S^{t}S>0 \end{matrix}} C_{m}(α; M, S) e(Mτ + S^{t}z).

be the Fourier expansion of ϕ_{m}|Ṽ_{α,2n−2−α}(p^{2}). We first calculate the Fourier coefficients C_{m}(α; M, S). There exist matrices N ∈ ℤ^{(2n−1,2n−1)} and R ∈

ℤ^{(2n−1,1)} which satisfy 4N − R^{t}R = (\begin{matrix}M & \frac{1}{2}S \\ \frac{1}{2}^{t}S & mp^{2} \end{matrix}). We put T = (\begin{matrix}N & \frac{1}{2}R \\ \frac{1}{2}^{t}R & 1 \end{matrix}).

Due to Proposition 4.4 and due to the definition of Ṽ_{α,2n−2−α}(4) in §4.7, we can take N and R which satisfy

T = (\begin{matrix}N′ & \frac{1}{2}R′ \\ \frac{1}{2}^{t}R′ & M [( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )] \end{matrix})

with matrices N′ ∈ ℤ^{(2n−2,2n−2)} and R′ ∈ ℤ^{(2n−2,2)}. We assume that p is an odd prime. Let

\{((\begin{matrix}p^{2}^{t}D_{i}^{−1} & B_{i} \\ 0 & D_{i} \end{matrix}) , γ_{i} p^{−n+1} (det D_{i})^{\frac{1}{2}} )\}_{i}

be a complete set of the representatives of Γ_{0}^{(n)}(4)^{∗}\Γ_{0}^{(n)}(4)^{∗}Y Γ_{0}^{(n)}(4)^{∗}, where Y is Y = (diag(1_{α}, p1_{2n−2−α}, p^{2}1_{α}, p1_{2n−2−α}), p^{α/2}) and γ_{i} is a root of unity (see [Zh 83, Prop.7.1] or [Zh 84, Lemma 3.2] for the detail of these representatives). Then by a straightforward calculation and from Lemma 8.1 we obtain

C_{m}(α; M, S) = p^{k(2n−3)+2n^{2}− \frac{1}{2} −4n(n−1)}c(|D_{T} |)f_{T}^{k−n− \frac{1}{2}}  (8.1)

× ∑_{i} γ_{i} (det D_{i})^{−n} e(\frac{1}{p^{2}} N^{t}D_{i}B_{i}) ∏_{q|f_{T[Q_{i}]}} F̃_{q} (T [Q_{i}] , α_{q}) ,

where D_{T} is the fundamental discriminant and f_{T} > 0 is the natural number which satisfy det(2T) = |D_{T} |f_{T}^{2}, and where Q_{i} = diag(p^{−1}^{t}D_{i}, p^{−1}, 1) ∈ ℚ^{(2n,2n)}. The number c(|D_{T} |) is the |D_{T} |-th Fourier coefficient of h. By virtue of the definition of Ṽ_{α,2n−2−α}(4) the identity (8.1) also holds for p = 2. For any prime p the (M, S)-th Fourier coefficients of ϕ_{\frac{m}{p^{2}}} |U_{p^{2}}, ϕ_{m}|U_{p} and ϕ_{mp^{2}} are C_{\frac{m}{p^{2}}} (M, p^{−2}S), C_{m}(M, p^{−1}S) and C_{mp^{2}}(M, S), respectively. These are

C_{\frac{m}{p^{2}}} (M, p^{−2}S) = p^{−2(k−n− \frac{1}{2} )}c(|D_{T} |)f_{T}^{k−n− \frac{1}{2}} ∏_{q|f_{T} p^{−2}} F̃_{q} (T_{0}, α_{q}) ,

C_{m}(M, p^{−1}S) = p^{−(k−n− \frac{1}{2} )}c(|D_{T} |)f_{T}^{k−n− \frac{1}{2}} ∏_{q|f_{T} p^{−1}} F̃_{q} (T_{1}, α_{q})

and

C_{mp^{2}}(M, S) = c(|D_{T} |)f_{T}^{k−n− \frac{1}{2}} ∏_{q|f_{T}} F̃_{q} (T, α_{q}) ,

respectively, where we put T_{0} = T [( \begin{matrix}1_{2n−2} & 0 & 0 \\ 0 & p^{−2} & 0 \\ 0 & 0 & 1 \end{matrix} )] and T_{1} =

T [( \begin{matrix}1_{2n−2} & 0 & 0 \\ 0 & p^{−1} & 0 \\ 0 & 0 & 1 \end{matrix} )]. Note that if p^{−1}S ∈ ℤ^{(2n−2,1)}, then f_{T} is divisible

by p, and if p^{−2}S ∈ ℤ^{(2n−2,1)}, then f_{T} is divisible by p^{2}. Note that the Fourier coefficients of e_{k− \frac{1}{2} ,m}^{(2n−2)}|Ṽ_{α,2n−2−α}(p^{2}), e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(2n−2)} |U_{p^{2}},

e_{k− \frac{1}{2} ,m}^{(2n−2)}|U_{p} and e_{k− \frac{1}{2} ,mp^{2}}^{(2n−2)} have the same form of the above expressions by substituting α_{q} = q^{k−n− \frac{1}{2}} and by replacing c(|D_{T} |) by c h_{k−n+ \frac{1}{2}} (|D_{T} |), where h_{k−n+ \frac{1}{2}} (|D_{T} |) is the |D_{T} |-th Fourier coefficient of the Cohen-Eisenstein series H_{k−n+ \frac{1}{2}}^{(1)} of weight k − n + \frac{1}{2}, and where c := c_{k,2n} = 2^{n}ζ(1 − k)^{−1} ∏_{i=1}^{n} ζ(1 + 2i − 2k)^{−1}. On the other hand, Theorem 7.6 is valid for infinitely many integer k. Therefore Theorem 7.6 deduces not only the relation among the Fourier coefficients of three forms e_{k− \frac{1}{2} , \frac{m}{p^{2}}}^{(2n−2)} , e_{k− \frac{1}{2} ,m}^{(2n−2)} and e_{k− \frac{1}{2} ,mp^{2}}^{(2n−2)}, but also the relation

among the polynomials {F̃_{q}(T, X)}_{T} of X. (cf. [Ik 01, Lemma 10.5 and page 665. line 2]. More precisely, we can conclude that the polynomial

p^{k(2n−3)+2n^{2}− \frac{1}{2} −4n(n−1)} ∑_{i} γ_{i} (det D_{i})^{−n} e(\frac{1}{p^{2}} N^{t}D_{i}B_{i}) ∏_{q|f_{T[Q_{i}]}}F̃_{q} (T [Q_{i}] , X)

of X coincides with the (α + 1)-th component of the vector

p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}}

×(p^{−2(k−n− \frac{1}{2} )} ∏_{q|f_{T} p^{−2}} F̃_{q} (T_{0}, X) , p^{−(k−n− \frac{1}{2} )} ∏_{q|f_{T} p^{−1}} F̃_{q} (T_{1}, X) , ∏_{q|f_{T}}F̃_{q} (T, X))

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,2n−1}^{p}(X) diag(1, p^{1/2}, ..., p^{(2n−2)/2}).

Therefore C_{m}(α; M, S) coincides the (α + 1)-th component of the vector

p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} (C_{\frac{m}{p^{2}}} (M, p^{−2}S), C_{m}(M, p^{−1}S), C_{mp^{2}}(M, S))

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ..., p^{(2n−2)/2}).

Thus we conclude this theorem. ⊓ ⊔

Let T̃_{α,2n−2−α}(p^{2}) be the Hecke operator introduced in §2.8 and let L(s, F) be the L-function for a Hecke eigenform F ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(m)}(4)) introduced in §2.9.

Lifting to Siegel Modular Forms … 191

Theorem 8.3. Let k be an even integer and n be an integer greater than 1. Let h ∈ S_{k−n+ \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)) and g ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)) be eigenforms for all Hecke operators. Then there exists a F_{h,g} ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−2)}). Under the assumption that F_{h,g} is not identically zero, then F_{h,g} is an eigenform with the L-function which satisfies

L(s, F_{h,g}) = L(s, g) ∏_{i=1}^{2n−3} L(s − i, h).

Proof. The construction of F_{h,g} is stated in the above:

F_{h,g}(τ) = \frac{1}{6} ∫_{Γ_{0}^{(1)}(4)\ℌ_{1}} F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) \overline{g(ω)} Im(ω)^{k− \frac{5}{2}} dω,

where F ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−1)}(4)) is constructed from h. By the definition of Ṽ_{α,2n−2−α}(p^{2}) and due to Theorem 8.2 we have

ϕ_{m}(τ, 0)| (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2}))
= (ϕ_{m}| (Ṽ_{0,2n−2}(p^{2}), ..., Ṽ_{2n−2,0}(p^{2}))) (τ, 0)
= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} ((ϕ_{\frac{m}{p^{2}}} |U_{p^{2}}) (τ, 0), (ϕ_{m}|U_{p}) (τ, 0), ϕ_{mp^{2}}(τ, 0))

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2})

= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} (ϕ_{\frac{m}{p^{2}}} (τ, 0), ϕ_{m}(τ, 0), ϕ_{mp^{2}}(τ, 0))

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2}).

We remark

∑_{\begin{matrix}m \\ m≡0,3 mod 4 \end{matrix}} (p^{2k−3}ϕ_{\frac{m}{p^{2}}} (τ, 0) + p^{k−2} (\frac{−m}{p}) ϕ_{m}(τ, 0) + ϕ_{mp^{2}}(τ, 0)) e(mω)

= F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) |T̃_{1,0}(p^{2}).

Documenta Mathematica 21 (2016) 125–196

192 S.Hayashida Thus

F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) | (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2}))

= ∑_{\begin{matrix}m \\ m≡0,3 mod 4 \end{matrix}} \{ϕ_{m}(τ, 0)| (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2}))\} e(mω)
= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} ∑_{\begin{matrix}m \\ m≡0,3 mod 4 \end{matrix}} \{ (ϕ_{\frac{m}{p^{2}}} (τ, 0), ϕ_{m}(τ, 0), ϕ_{mp^{2}}(τ, 0))

× (\begin{matrix}0 & p^{2k−3} \\ p^{k−2} & p^{k−2} (\frac{−m}{p}) \\ 0 & 1 \end{matrix}) e(mω)\} A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2})

= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} (F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) |_{ω} (T̃_{0,1}(p^{2}), T̃_{1,0}(p^{2})))

×A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2}). Hence

F_{h,g}| (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2}))

= ∫_{Γ_{0}^{(1)}(4)\ℌ_{1}} (F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) |_{τ} (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2})))
×\overline{g(ω)} Im(ω)^{k− \frac{5}{2}} dω
= p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}}

× ∫_{Γ_{0}^{(1)}(4)\ℌ_{1}} (F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) |_{ω} (T̃_{0,1}(p^{2}), T̃_{1,0}(p^{2}))) \overline{g(ω)} Im(ω)^{k− \frac{5}{2}} dω

×A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2}). Let b(p) be the eigenvalue of g with respect to T̃_{1,0}(p^{2}). We remark that b(p) is a real number. We have

\begin{matrix}F_{h,g}| (T̃_{0,2n−2}(p^{2}), ..., T̃_{2n−2,0}(p^{2})) \\ = p^{k(2n−3)−2n^{2}−n+ \frac{11}{2}} F_{h,g}(τ) \\ × \{(p^{k−2}, b(p))A_{2,2n−1}^{p}(α_{p}) diag(1, p^{1/2}, ⋯, p^{(2n−2)/2})\} . \end{matrix}  (8.2)

Therefore F_{h,g} is an eigenform for any T̃_{α,2n−2−α}(p^{2}). Let {β_{p}^{±}} be the set of complex numbers which satisfy

1 − b(p)z + p^{2k−3}z^{2} = (1 − β_{p}p^{k−3/2}z)(1 − β_{p}^{−1}p^{k−3/2}z).

Let {μ_{0,p}^{2}, μ_{1,p}^{±}, ...μ_{2n−2,p}^{±}} be the p-parameters of F_{h,g} (see §2.9 for the definition of p-parameters). We remark μ_{0,p}^{2}μ_{1,p} ⋯ μ_{2n−2,p} = p^{2(n−1)(k−n)}.

Documenta Mathematica 21 (2016) 125–196

We now assume that p is an odd prime. Let Ψ_{2n−2}(K_{α}^{(2n−2)}) ∈ R_{2n−2} be the Laurent polynomial of {z_{i}}_{i=0,...,2n−2} introduced in §2.9. The explicit formula of Ψ_{2n−2}(K_{α}^{(2n−2)}) was obtained in Proposition 6.3. The eigenvalue of F_{h,g} for T̃_{α,2n−2−α}(p^{2}) (α = 0, ..., 2n − 2) is obtained by substituting z_{i} = μ_{i} into Ψ_{2n−2}(K_{α}^{(2n−2)}). We remark that the eigenvalue of F_{h,g} for T̃_{0,2n−2}(p^{2}) is p^{(n−1)(2k−4n+1)}. From the identities (8.2) and (6.5), we obtain

\begin{matrix}p^{2n^{2}−6n+5}(p^{−1/2}, μ_{1,p} + μ_{1,p}^{−1}) ∏_{l=2}^{2n−2} B_{l,l+1}(μ_{l,p}) \\ = p^{2n^{2}−6n+5}(p^{−1/2}, β_{p} + β_{p}^{−1}) ∏_{l=2}^{2n−2} B_{l,l+1}(p^{n−l}α_{p}). \end{matrix}  (8.3)

Here the components of the vectors in the above identity (8.3) are eigenvalues of F_{h,g} for

T̃_{0,2n−2}(p^{2})^{−1}T̃_{α,2n−2−α}(p^{2}) (α = 0, ..., 2n − 2).

If we substitute z_{1} = β_{p} and z_{i} = p^{n−i}α_{p} (i = 2, ..., 2n − 2) into the Laurent polynomial (Ψ_{2n−2}(K_{0}^{(2n−2)}))^{−1}Ψ_{2n−2}(K_{α}^{(2n−2)}), then due to (8.3) this value is the eigenvalue of F_{h,g} for T̃_{0,2n−2}(p^{2})^{−1}T̃_{α,2n−2−α}(p^{2}). Because R_{2n−2} is generated by Ψ_{2n−2}(K_{α}^{(2n−2)}) (α = 0, ..., 2n−2) and Ψ_{2n−2}(K_{0}^{(2n−2)})^{−1} and because of the fact that the p-parameters are uniquely determined up to the action of the Weyl group W_{2}, we therefore can take the p-parameters {μ_{1,p}^{±}, ⋯, μ_{2n−2,p}^{±}} of F_{h,g} as

{β_{p}^{±}, p^{n−2}α_{p}^{±}, p^{n−3}α_{p}^{±}, ⋯, p^{−n+2}α_{p}^{±}} .

Hence the Euler p-factor Q_{F_{h,g},p}(z) of F_{h,g} for odd prime p is

\begin{aligned}Q_{F_{h,g},p}(z) & = ∏_{i=1}^{2n−2} \{(1 − μ_{i,p}z) (1 − μ_{i,p}^{−1}z)\} \\ & = (1 − β_{p}z) (1 − β_{p}^{−1}z) ∏_{i=1}^{2n−3} \{(1 − α_{p}p^{−n+i}z) (1 − α_{p}^{−1}p^{−n+i}z)\} . \end{aligned} (8.4)

We now consider the case p = 2. The identity (8.2) is also valid for p = 2. Because γ̃_{j,2} is defined in the same formula as in the case of odd primes, we also obtain the identity (8.4) for p = 2. Thus we conclude

L(s, F_{h,g}) = ∏_{p} ∏_{i=1}^{2n−2} \{(1 − μ_{i,p}p^{−s+k− \frac{3}{2}} ) (1 − μ_{i,p}^{−1}p^{−s+k− \frac{3}{2}} )\}^{−1}

= L(s, g) ∏_{i=1}^{2n−3} L(s − i, h).
⊓ ⊔

9 Examples of non-vanishing

Lemma 9.1. The form F_{h,g} in Theorem 8.3 is not identically zero, if (n, k) = (2, 12), (2, 14), (2, 16), (2, 18), (3, 12), (3, 14), (3, 16), (3, 18), (3, 20), (4, 10), (4, 12), (4, 14), (4, 16), (4, 18), (4, 20), (5, 14), (5, 16), (5, 18), (5, 20), (6, 12), (6, 14), (6, 16), (6, 18) or (6, 20).

Proof. Let h ∈ S_{k−n+ \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)), F ∈ S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−1)}(4)) and F_{h,g} ∈

S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(2n−2)}(4)) be the same symbols in §8. We have

F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) = ∑_{g} \frac{1}{⟨g, g⟩}F_{h,g}(τ)g(ω).  (9.1)

Here in the summation g runs over a basis of S_{k− \frac{1}{2}}^{+} (Γ_{0}^{(1)}(4)) which consists of Hecke eigenforms. On the other hand, we have

F ((\begin{matrix}τ & 0 \\ 0 & ω \end{matrix})) = ∑_{M∈Sym_{2n−2}^{+}, m∈Sym_{1}^{+}} K(M, m)e(Nτ)e(mω), (9.2)

where

K(M, m) = ∑_{\begin{matrix}S∈ℤ^{(2n−2,1)} \\ 4Mm−S^{t}S>0 \end{matrix}} C_{m}(M, S)

and where C_{m}(M, S) is the (\begin{matrix}M & S \\ ^{t}S & m \end{matrix})-th Fourier coefficient of F. By using a

computer algebraic system and Katsurada’s formula for Siegel series [Ka 99], we can compute the explicit values of Fourier coefficients C_{m}(M, S). Hence we can also compute some Fourier coefficients K(M, m). By virtue of the identities (9.1) and (9.2), we obtain

K(M, m) = ∑_{g} \frac{1}{⟨g, g⟩}A(M; F_{h,g})A(m; g),

where A(M; F_{h,g}) is the M-th Fourier coefficient of F_{h,g} and where A(m; g) is the m-th Fourier coefficient of g. Here Fourier coefficients A(m; g) are calculated through the structure theorem of Kohnen plus space [Ko 80]. Therefore we can calculate some Fourier coefficients A(M; F_{h,g}). For example, if (n, k) = (2, 10), then k − 1/2 = 19/2 and k − n + 1/2 = 17/2. We have dim S_{19/2}^{+}(Γ_{0}^{(1)}(4)) = dim S_{17/2}^{+}(Γ_{0}^{(1)}(4)) = 1. Let g ∈ S_{19/2}^{+}(Γ_{0}^{(1)}(4))

and h ∈ S_{17/2}^{+}(Γ_{0}^{(1)}(4)) be Hecke eigenforms such that the Fourier coefficients satisfy A(3; g) = A(1; h) = 1. We remark that all Fourier coefficients of g and h are real numbers. Let K(M, m) be the number defined in (9.2), where F ∈ S_{19/2}^{+}(Γ_{0}^{(3)}(4)) is the Siegel modular form constructed from h. Because

dim S_{19/2}^{+}(Γ_{0}^{(1)}(4)) = 1, we need to check K(M, m) ≠ 0 for a pair (M, m) ∈

Sym_{2n−2}^{+} × Sym_{1}^{+}. We take M = (\begin{matrix}3 & 1 \\ 1 & 3 \end{matrix}) and m = 3, then

K(M, m)

= C_{3}(M, (\begin{matrix}2 \\ 2 \end{matrix})) + C_{3}(M, (\begin{matrix}2 \\ −2 \end{matrix})) + C_{3}(M, (\begin{matrix}−2 \\ 2 \end{matrix})) + C_{3}(M, (\begin{matrix}−2 \\ −2 \end{matrix}))
= −336 − 168 − 168 − 336
≠ 0.

Therefore F_{h,g} ≢ 0 for (n, k) = (2, 10). Similarly, by using a computer algebraic system, we can also check F_{h,g} ≢ 0 for any h and g for other (n, k) in the lemma. ⊓ ⊔

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Shuichi Hayashida

Department of Mathematics Joetsu University of Education 1 Yamayashikimachi Joetsu, Niigata 943-8512 JAPAN e-mail hayasida@juen.ac.jp