Geometry and Arithmetic on the Siegel-Jacobi Space
Jae-Hyun Yang

TL;DR
This paper explores the geometric and arithmetic properties of the Siegel-Jacobi space, a significant non-symmetric homogeneous space, providing insights into its structure and number-theoretic aspects.
Contribution
It develops the theory of geometry and arithmetic on the Siegel-Jacobi space, advancing understanding of its mathematical properties.
Findings
Detailed geometric descriptions of the Siegel-Jacobi space
Arithmetic properties related to number theory
New theoretical frameworks for analysis
Abstract
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Analytic and geometric function theory · Mathematical Dynamics and Fractals
Geometry and arithmetic on the Siegel-Jacobi Space
Jae-Hyun Yang
Department of Mathematics, Inha University, Incheon 402-751, Korea
Abstract.
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
Subject Classification: Primary 11F30, 11F55, 11Fxx, 13A50, 15A72, 32F45, 32M10, 32Wxx
Keywords and phrases: Jacobi group, Siegel-Jacobi space, Invariant metrics, Laplacians, Invariant differential operators, Partial Cayley transform, Siegel-Jacobi disk, Jacobi forms, Siegel-Jacobi operator, Schrödinger-Weil representation, Maass-Jacobi forms, Theta sums.
The author was supported by Basic Science Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (47724-1)
To the memory of my teacher, Professor Shoshichi Kobayashi
Table of Contents
Introduction
-
Invariant Metrics and Laplacians on the Siegel-Jacobi Space
-
Invariant Differential Operators on the Siegel-Jacobi Space
-
The Partial Cayley Transform
-
Invariant Metrics and Laplacians on the Siegel-Jacobi Disk
-
A Fundamental Domain for the Siegel-Jacobi Space
-
Jacobi Forms
-
Singular Jacobi Forms
-
The Siegel-Jacobi Operator
-
Construction of Vector-Valued Modular Forms from Jacobi Forms
-
Maass-Jacobi Forms
12 The Schrödinger-Weil Representation
- Final Remarks and Open Problems
Acknowledgements
References
1. Introduction
For a given fixed positive integer , we let
[TABLE]
be the Siegel upper half plane of degree and let
[TABLE]
be the symplectic group of degree , where denotes the set of all matrices with entries in a commutative ring for two positive integers and , denotes the transposed matrix of a matrix and
[TABLE]
Then acts on transitively by
[TABLE]
where and Let
[TABLE]
be the Siegel modular group of degree . This group acts on properly discontinuously. C. L. Siegel investigated the geometry of and automorphic forms on systematically. Siegel [57] found a fundamental domain for and described it explicitly. Moreover he calculated the volume of We also refer to [23], [38], [58] for some details on
For two positive integers and , we consider the Heisenberg group
[TABLE]
endowed with the following multiplication law
[TABLE]
with \big{(}\lambda,\mu;\kappa\big{)},\big{(}\lambda^{\prime},\mu^{\prime};\kappa^{\prime}\big{)}\in H_{\mathbb{R}}^{(n,m)}. We define the Jacobi group of degree and index that is the semidirect product of and
[TABLE]
endowed with the following multiplication law
[TABLE]
with and . Then acts on transitively by
[TABLE]
where and We note that the Jacobi group is not a reductive Lie group and the homogeneous space is not a symmetric space. From now on, for brevity we write The homogeneous space is called the Siegel-Jacobi space of degree and index .
The aim of this paper is to discuss and survey the geometry and the arithmetic of the Siegel-Jacobi space . This article is organized as follows. In Section 2, we provide Riemannian metrics which are invariant under the action (1.2) of the Jacobi group and their Laplacians. In Section 3, we discuss -invariant differential operators on the Siegel-Jacobi space and give some related results. In Section 4, we describe the partial Cayley transform of the Siegel-Jacobi disk onto the Siegel-Jacobi space which gives a partially bounded realization of the Siegel-Jacobi space. We provide a compatibility result of a partial Cayley transform. In Section 5, we provide Riemannian metrics on the Siegel-Jacobi disk which is invariant under the action (4.8) of the Jacobi group and their Laplacians using the partial Cayley transform. In Section 6, we find a fundamental domain for the Siegel-Jacobi space with respect to the Siegel-Jacobi modular group. In Section 7, we give the canonical automorphic factor for the Jacobi group which is obtained by a geometrical method and review the concept of Jacobi forms. In Section 8, we characterize singular Jacobi forms in terms of a certain differential operator and their weights. In Section 9, we define the notion of the Siegel-Jacobi operator. We give the result about the compatibility with the Hecke-Jacobi operator. In Section 10, we differentiate a given Jacobi form with respect to the toroidal variables by applying a homogeneous pluriharmonic differential operator to a Jacobi form and then obtain a vector-valued modular form of a new weight. As an application, we provide an identity for an Eisenstein series. In Section 11, we discuss the notion of Maass-Jacobi forms. In Section 12, we construct the Schrödinger-Weil representation and give some results on theta sums constructed from the Schrödinger-Weil representation. In Section 13, we give some remarks and propose some open problems about the geometry and the arithmetic of the Siegel-Jacobi space.
Notations: We denote by and the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by and the ring of integers and the set of all positive integers respectively. The symbol “:=” means that the expression on the right is the definition of that on the left. For two positive integers and , denotes the set of all matrices with entries in a commutative ring . For a square matrix of degree , denotes the trace of . For any denotes the transpose of a matrix . denotes the identity matrix of degree . For and , we set For a complex matrix , denotes the complex conjugate of . For and , we use the abbreviation For a number field , we denote by the ring of adeles of . If , the subscript will be omitted.
2. Invariant Metrics and Laplacians on the Siegel-Jacobi Space
For we write with real. We put and . We also put
[TABLE]
C. L. Siegel [57] introduced the symplectic metric on invariant under the action (1.1) of that is given by
[TABLE]
and H. Maass [37] proved that its Laplacian is given by
[TABLE]
And
[TABLE]
is a -invariant volume element on (cf. [59], p. 130).
For a coordinate with and , we put as before and set
[TABLE]
[TABLE]
Yang proved the following theorems in [71].
Theorem 2.1**.**
For any two positive real numbers and ,
[TABLE]
is a Riemannian metric on which is invariant under the action (1.2) of In fact, is a Kähler metric of
Proof. See Theorem 1.1 in [71].
Theorem 2.2**.**
The Laplacian of the -invariant metric is given by
[TABLE]
where
[TABLE]
and
[TABLE]
Furthermore and are differential operators on invariant under the action (1.2) of
Proof. See Theorem 1.2 in [71].
Remark 2.1**.**
Erik Balslev [2] developed the spectral theory of on for certain arithmetic subgroups of the Jacobi modular group to prove that the set of all eigenvalues of satisfies the Weyl law.
Remark 2.2**.**
The sectional curvature of is and hence is independent of the parameter . We refer to [76] for more detail.
Remark 2.3**.**
For an application of the invariant metric we refer to [79].
3. **Invariant Differential Operators on the Siegel-Jacobi Space **
Before we discuss -invariant differential operators on the Siegel-Jacobi space , we review differential operators on the Siegel upper half plane invariant under the action (1.1).
For brevity, we write The isotropy subgroup at for the action (1.1) is a maximal compact subgroup given by
[TABLE]
Let be the Lie algebra of . Then the Lie algebra of has a Cartan decomposition , where
[TABLE]
[TABLE]
[TABLE]
The subspace of may be regarded as the tangent space of at The adjoint representation of on induces the action of on given by
[TABLE]
Let be the vector space of symmetric complex matrices. We let be the map defined by
[TABLE]
We let be the isomorphism defined by
[TABLE]
where denotes the unitary group of degree . We identify (resp. ) with (resp. ) through the map (resp. ). We consider the action of on defined by
[TABLE]
Then the adjoint action (3.1) of on is compatible with the action (3.4) of on through the map Precisely for any and , we get
[TABLE]
The action (3.4) induces the action of on the polynomial algebra and the symmetric algebra respectively. We denote by \Big{(}\textrm{resp.}\ S(T_{n})^{U(n)}\,\Big{)} the subalgebra of \Big{(}\textrm{resp.}\ S(T_{n})\,\Big{)} consisting of -invariants. The following inner product on defined by
[TABLE]
gives an isomorphism as vector spaces
[TABLE]
where denotes the dual space of and is the linear functional on defined by
[TABLE]
It is known that there is a canonical linear bijection of onto the algebra of differential operators on invariant under the action (1.1) of . Identifying with by the above isomorphism (3.6), we get a canonical linear bijection
[TABLE]
of onto . The map is described explicitly as follows. Similarly the action (3.1) induces the action of on the polynomial algebra and the symmetric algebra respectively. Through the map , the subalgebra of consisting of -invariants is isomorphic to . We put . Let be a basis of a real vector space . If , then
[TABLE]
where . We refer to [20, 21] for more detail. In general, it is hard to express explicitly for a polynomial .
According to the work of Harish-Chandra [18, 19], the algebra is generated by algebraically independent generators and is isomorphic to the commutative algebra with indeterminates. We note that is the real rank of . Let be the complexification of . It is known that is isomorphic to the center of the universal enveloping algebra of .
Using a classical invariant theory (cf. [22, 61], we can show that is generated by the following algebraically independent polynomials
[TABLE]
For each with the image of is an invariant differential operator on of degree . The algebra is generated by algebraically independent generators In particular,
[TABLE]
We observe that if we take with real , then q_{1}(\omega)=q_{1}(x,y)=\,\textrm{tr}\big{(}x^{2}+y^{2}\big{)} and
[TABLE]
It is a natural question to express the images explicitly for We hope that the images for are expressed in the form of the trace as .
H. Maass [38] found algebraically independent generators of . We will describe explicitly. For and with real , we set
[TABLE]
We set
[TABLE]
Then it is easily seen that
[TABLE]
[TABLE]
and
[TABLE]
Using Formulas (3.11), (3.12) and (3.13), we can show that
[TABLE]
Therefore we get
[TABLE]
We set
[TABLE]
We define recursively by
[TABLE]
We set
[TABLE]
As mentioned before, Maass proved that are algebraically independent generators of .
In fact, we see that
[TABLE]
is the Laplacian for the invariant metric on .
Example 3.1. We consider the case when The algebra is generated by the polynomial
[TABLE]
Using Formula (3.8), we get
[TABLE]
Therefore \mathbb{D}(\mathbb{H}_{1})=\mathbb{C}\big{[}\Theta_{1}(q)\big{]}=\,\mathbb{C}[H_{1}].
Example 3.2. We consider the case when The algebra is generated by the polynomial
[TABLE]
Using Formula (3.8), we may express and explicitly. is expressed by Formula (3.10). The computation of might be quite tedious. We leave the detail to the reader. In this case, was essentially computed in [11], Proposition 6. Therefore
[TABLE]
In fact, the center of the universal enveloping algebra was computed in [11].
G. Shimura [56] found canonically defined algebraically independent generators of . We will describe his way of constructing those generators roughly. Let denote the complexication of respectively. Then we have the Cartan decomposition
[TABLE]
with the properties
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
For a complex vector space and a nonnegative integer , we denote by the vector space of complex-valued homogeneous polynomial functions on of degree . We put
[TABLE]
denotes the vector space of all -multilinear maps of into . An element of is called symmetric if
[TABLE]
for each permutation of Given , there is a unique element symmetric element of such that
[TABLE]
Moreover the map is a -linear bijection of onto the set of all symmetric elements of . We let denote the subspace consisting of all homogeneous elements of degree in the symmetric algebra . We note that and are dual to each other with respect to the pairing
[TABLE]
Let be the dual space of , that is, Let be a basis of and be the basis of dual to where . We note that and are dual to each other with respect to the pairing
[TABLE]
where and runs over Let be the universal enveloping algebra of and its subspace spanned by the elements of the form with and We recall that there is a -linear bijection of the symmetric algebra of onto which is characterized by the property that for all For each we define an element of by
[TABLE]
where runs over If , then as an element of is defined by
[TABLE]
Hence According to (2.25), we see that if for with a polynomial , then
[TABLE]
Thus is a -linear injection of into independent of the choice of a basis. We observe that \omega\big{(}\textrm{Pol}_{r}({\mathfrak{p}}_{\mathbb{C}}^{*})\big{)}=\,\psi(S_{r}({\mathfrak{p}}_{\mathbb{C}})). It is a well-known fact that if , then
[TABLE]
We have a canonical pairing
[TABLE]
defined by
[TABLE]
where (resp. ) are the unique symmetric elements of (resp. , and and are dual bases of and with respect to the Killing form , and runs over \big{\{}1,\cdots,{\widetilde{N}}\big{\}}^{r}.
The adjoint representation of on induces the representation of on . Given a -irreducible subspace of we can find a unique -irreducible subspace of such that is the direct sum of and the annihilator of . Then and are dual with respect to the pairing (3.26). Take bases of and of that are dual to each other. We set
[TABLE]
It is easily seen that belongs to and is independent of the choice of dual bases and Shimura [56] proved that there exists a canonically defined set with a -irreducible subspace of such that are algebraically independent generators of . We can identify with . We recall that denotes the vector space of symmetric complex matrices. We can take as the subspace of spanned by the functions for all where denotes the determinant of the upper left submatrix of . For every , we let denote the element of represented by . Then is the polynomial ring generated by algebraically independent elements
Now we investigate differential operators on the Siegel-Jacobi space invariant under the action (1.2) of . The stabilizer of at is given by
[TABLE]
Therefore is a homogeneous space which is not symmetric. The Lie algebra of has a decomposition
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
Thus the tangent space of the homogeneous space at is identified with .
If and are elements of , then the Lie bracket of and is given by
[TABLE]
where
[TABLE]
Lemma 3.1**.**
[TABLE]
Proof.
The proof follows immediately from Formula (3.29). ∎
Lemma 3.2**.**
Let
[TABLE]
with and
[TABLE]
with Then the adjoint action of on is given by
[TABLE]
where
[TABLE]
Proof.
We leave the proof to the reader. ∎
We recall that denotes the vector space of all symmetric complex matrices. For brevity, we put We define the real linear isomorphism by
[TABLE]
where and
Let denote the additive group consisting of all real symmetric matrices. Now we define the isomorphism by
[TABLE]
where is the map defined by (3.3). Identifying with , we can identify with .
Theorem 3.1**.**
The adjoint representation of on is compatible with the natural action of on defined by
[TABLE]
through the maps and . Precisely, if and , then we have the following equality
[TABLE]
Here we regard the complex vector space as a real vector space.
Proof.
Let
[TABLE]
with and
[TABLE]
with Then we have
[TABLE]
where and are given by the formulas (3.31), (3.32), (3.33) and (3.34) respectively. ∎
We now study the algebra of all differential operators on invariant under the natural action (1.2) of . The action (3.37) induces the action of on the polynomial algebra We denote by the subalgebra of consisting of all -invariants. Similarly the action (3.30) of induces the action of on the polynomial algebra \textrm{Pol}\big{(}\mathfrak{p}^{J}\big{)}. We see that through the identification of with , the algebra \textrm{Pol}\big{(}\mathfrak{p}^{J}\big{)} is isomorphic to The following -invariant inner product of the complex vector space defined by
[TABLE]
gives a canonical isomorphism
[TABLE]
where is the linear functional on defined by
[TABLE]
According to Helgason ([21], p. 287), one gets a canonical linear bijection of onto . Identifying with by the above isomorphism, one gets a natural linear bijection
[TABLE]
of onto The map is described explicitly as follows. We put . Let \big{\{}\eta_{\alpha}\,|\ 1\leq\alpha\leq N_{\star}\,\big{\}} be a basis of . If P\in\textrm{Pol}\big{(}\mathfrak{p}^{J}\big{)}^{K}=\mathrm{Pol}_{n,m}^{U(n)}, then
[TABLE]
where and . In general, it is hard to express explicitly for a polynomial P\in\textrm{Pol}\big{(}\mathfrak{p}^{J}\big{)}^{K}.
We propose the following natural problems.
Problem 1. Find a complete list of explicit generators of .
Problem 2. Find all the relations among a set of generators of .
Problem 3. Find an easy or effective way to express the images of the above invariant polynomials or generators of under the Helgason map explicitly.
Problem 4. Find a complete list of explicit generators of the algebra . Or construct explicit -invariant differential operators on
Problem 5. Find all the relations among a set of generators of .
Problem 6. Is finitely generated ?
Problem 7. Is finitely generated ?
We will give answers to Problems 1, 2 and 6.
We put . Moreover, for and , we put
[TABLE]
Then we have the following relations:
[TABLE]
Then we have the following theorem:
Theorem 3.2**.**
The algebra is generated by the following polynomials:**
[TABLE]
Here the indices run as follows:
[TABLE]
This is seen from the following theorem by using (3.40):
Theorem 3.3**.**
The algebra is generated by , , , and . Here the indices run as follows:**
[TABLE]
Proof. See Theorem 3.3 in [26].
Problem 2, that is, the second fundamental theorem for is stated as follows. We consider indeterminates and corresponding to and , respectively. For these, we assume the relations
[TABLE]
We denote by the polynomial algebra in the following indeterminates:
[TABLE]
Here the indices run as follows:
[TABLE]
The relations among the generators of are described as follows:
Theorem 3.4**.**
The kernel of the natural map from to is generated by the entries of with
[TABLE]
[TABLE]
Here the notation is as follows. We put
[TABLE]
Here means the Kostka number. Namely, in general, we define by
[TABLE]
where is the Schur polynomial. In other words, is the image of the Schur polynomial under the linear map
[TABLE]
Moreover we replace in by
[TABLE]
Finally and are the following matrices (an alternating matrix of size and a matrix):
[TABLE]
Here we put .
The proof of Theorem 3.4 is complicated, but it is deduced from the second fundamental theorem of invariant theory for vector invariants (this is quite parallel with the fact that Theorem 3.3 follows from the first fundamental theorem of invariant theory for vector invariants). The detail will be given in the forthcoming paper.
Remark 3.1**.**
Itoh, Ochiai and Yang [26] solved all the problems (Problem 1–Problem 7) proposed in this section when
We present some interesting -invariants. For an matrix , we define the following invariant polynomials in :
[TABLE]
where and .
We define the following -invariant polynomials in .
[TABLE]
4. The Partial Cayley Transform
Let
[TABLE]
be the generalized unit disk. We set
[TABLE]
It is easily seen that
[TABLE]
Then acts on transitively by
[TABLE]
It is well known that the action (1.1) is compatible with the action (4.1) through the Cayley transform given by
[TABLE]
In other words, if and then
[TABLE]
where . We refer to [31] for generalized Cayley transforms of bounded symmetric domains.
For brevity, we write This homogeneous space is called the Siegel-Jacobi disk of degree and index . For a coordinate with and we put
[TABLE]
and
[TABLE]
[TABLE]
We can identify an element of with the element
[TABLE]
of We set
[TABLE]
We now consider the group defined by
[TABLE]
If with , then is given by
[TABLE]
where
[TABLE]
[TABLE]
and are given by the formulas
[TABLE]
and
[TABLE]
From now on, we write
[TABLE]
In other words, we have the relation
[TABLE]
Let
[TABLE]
be the complex Heisenberg group endowed with the following multiplication
[TABLE]
We define the semidirect product
[TABLE]
endowed with the following multiplication
[TABLE]
where and
If we identify with the subgroup
[TABLE]
of we have the following inclusion
[TABLE]
We define the mapping by
[TABLE]
where and are given by (4.4) and (4.5). We can see that if , then
According to [69, p. 250], is of the Harish-Chandra type (cf. [53, p. 118]). Let
[TABLE]
be an element of Since the Harish-Chandra decomposition of an element in is given by
[TABLE]
the -component of the following element
[TABLE]
of is given by
[TABLE]
We can identify with the subset
[TABLE]
of the complexification of Indeed, is embedded into given by
[TABLE]
This is a generalization of the Harish-Chandra embedding (cf. [53, p. 119]). Then we get the natural transitive action of on defined by
[TABLE]
where and
The author [72] proved that the action (1.2) of on is compatible with the action (4.8) of on through the partial Cayley transform defined by
[TABLE]
In other words, if and ,
[TABLE]
where . is a biholomorphic mapping of onto which gives the partially bounded realization of by . The inverse of is
[TABLE]
5. Invariant Metrics and Laplacians on the Siegel-Jacobi Disk
For we write and . We put
[TABLE]
Using the Cayley transform , Siegel [57] showed that
[TABLE]
is a -invariant Riemannian metric on and Maass [37] showed that its Laplacian is given by
[TABLE]
Yang [73] proved the following theorems.
Theorem 5.1**.**
For any two positive real numbers and , the following metric defined by
[TABLE]
is a Riemannian metric on which is invariant under the action (4.8) of the Jacobi group .
Proof. See Theorem 1.3 in [73].
Theorem 5.2**.**
The following differential operators and on defined by
[TABLE]
and
[TABLE]
are invariant under the action (4.8) of The following differential operator
[TABLE]
is the Laplacian of the invariant metric on .
Proof. See Theorem 1.4 in [73].
Itoh, Ochiai and Yang [26] proved that the following differential operator on defined by
[TABLE]
is invariant under the action (4.8) of on . Furthermore the authors [26] proved that the following matrix-valued differential operator on defined by
[TABLE]
and each -entry of given by
[TABLE]
are invariant under the action (4.8) of on .
[TABLE]
is an invariant differential operator of degree three on and
[TABLE]
is an invariant differential operator of degree on .
Indeed it is very complicated and difficult at this moment to express the generators of the algebra of all -invariant differential operators on explicitly.
6. A Fundamental Domain for the Siegel-Jacobi Space
Let
[TABLE]
be an open connected cone in with Then the general linear group acts on transitively by
[TABLE]
Thus is a symmetric space diffeomorphic to
The fundamental domain for which was found by H. Minokwski [42] is defined as a subset of consisting of satisfying the following conditions (M.1) and (M.2): (M.1) for every in which are relatively prime for (M.2) for
We say that a point of is Minkowski reduced.
Let be the Siegel modular group of degree . Siegel determined a fundamental domain for We say that with real is Siegel reduced or S-reduced if it has the following three properties : (S.1) for all ; (S.2) is Minkowski reduced, that is, ; (S.3) for , where
is defined as the set of all Siegel reduced points in Using the highest point method, Siegel proved the following (F1)-(F3): (F1) , i.e., ; (F2) is closed in ; (F3) is connected and the boundary of consists of a finite number of hyperplanes.
Let be the matrix with entry 1 where the -th row and the -the column meet, and all other entries 0. For an element , we set for brevity
[TABLE]
For each , we define the subset of by
[TABLE]
For each , we define the subset of by
[TABLE]
Let
[TABLE]
be the Siegel-Jacobi (or simply Jacobi) modular group of degree and index .
Yang found a fundamental domain for using Siegel’s fundamental domain in [70].
Theorem 6.1**.**
The set
[TABLE]
is a fundamental domain for
Proof. See Theorem 3.1 in [70].
7. Jacobi Forms
Let be a rational representation of on a finite dimensional complex vector space Let be a symmetric half-integral semi-positive definite matrix of degree . The canonical automorphic factor
[TABLE]
for on is given as follows :
[TABLE]
where and We refer to [66] for a geometrical construction of
Let be the algebra of all functions on with values in For we define
[TABLE]
where and
Definition 7.1**.**
Let and be as above. Let
[TABLE]
be the discrete subgroup of . A Jacobi form of index with respect to on a subgroup of of finite index is a holomorphic function satisfying the following conditions (A) and (B):
(A) for all .
*(B) For each , has a Fourier expansion of *
* the following form :*
[TABLE]
with and only if .
If the condition (B) is superfluous by Köcher principle ( cf. [82] Lemma 1.6). We denote by the vector space of all Jacobi forms of index with respect to on . Ziegler ( cf. [82] Theorem 1.8 or [12] Theorem 1.1 ) proves that the vector space is finite dimensional. In the special case with and a fixed , we write instead of and call the weight of the corresponding Jacobi forms. For more results about Jacobi forms with and , we refer to [62]-[68] and [82]. Jacobi forms play an important role in lifting elliptic cusp forms to Siegel cusp forms of degree (cf. [24, 25]).
Now we will make brief historical remarks on Jacobi forms. In 1985, the names Jacobi group and Jacobi forms got kind of standard by the classic book [12] by Eichler and Zagier to remind of Jacobi’s “Fundamenta nova theoriae functionum ellipticorum”, which appeared in 1829 (cf. [27]). Before [12] these objects appeared more or less explicitly and under different names in the work of many authors. In 1966 Pyatetski-Shapiro [49] discussed the Fourier-Jacobi expansion of Siegel modular forms and the field of modular abelian functions. He gave the dimension of this field in the higher degree. About the same time Satake [52]-[53] introduced the notion of “groups of Harish-Chandra type” which are non reductive but still behave well enough so that he could determine their canonical automorphic factors and kernel functions. Shimura [54]-[55] gave a new foundation of the theory of complex multiplication of abelian functions using Jacobi theta functions. Kuznetsov [34] constructed functions which are almost Jacobi forms from ordinary elliptic modular functions. Starting 1981, Berndt [3]-[5] published some papers which studied the field of arithmetic Jacobi functions, ending up with a proof of Shimura reciprocity law for the field of these functions with arbitrary level. Furthermore he investigated the discrete series for the Jacobi group and developed the spectral theory for in the case (cf. [6]-[8]). The connection of Jacobi forms to modular forms was given by Maass, Andrianov, Kohnen, Shimura, Eichler and Zagier. This connection is pictured as follows. For even, we have the following isomorphisms
[TABLE]
Here denotes Maass’s Spezialschar or Maass space and denotes the Kohnen plus space. For a precise detail, we refer to [39]-[41], [1], [12], [29, 30] and [80]. In 1982 Tai [60] gave asymptotic dimension formulae for certain spaces of Jacobi forms for arbitrary and and used these ones to show that the moduli of principally polarized abelian varieties of dimension is of general type for Feingold and Frenkel [13] essentially discussed Jacobi forms in the context of Kac-Moody Lie algebras generalizing the Maass correspondence to higher level. Gritsenko [17] studied Fourier-Jacobi expansions and a non-commutative Hecke ring in connection with the Jacobi group. After 1985 the theory of Jacobi forms for had been studied more or less systematically by the Zagier school. A large part of the theory of Jacobi forms of higher degree was investigated by Kramer [32, 33], Runge [51], Yang [62]-[66]and Ziegler [82]. There were several attempts to establish -functions in the context of the Jacobi group by Murase [46, 47] and Sugano [48] using the so-called “Whittaker-Shintani functions”. Kramer [32, 33] developed an arithmetic theory of Jacobi forms of higher degree. Runge [51] discussed some part of the geometry of Jacobi forms for arbitrary and For a good survey on some motivation and background for the study of Jacobi forms, we refer to [9]. The theory of Jacobi forms has been extensively studied by many people until now and has many applications in other areas like geometry and physics.
8. Singular Jacobi Forms
Definition 8.1**.**
A Jacobi form is said to be cuspidal if for any with A Jacobi form is said to be singular if it admits a Fourier expansion such that a Fourier coefficient vanishes unless
Let be the Minkowski-Euclid space, where is the open cone consisting of positive symmetric real matrices. For a variable with and , we put
[TABLE]
[TABLE]
where and
We define the following differential operator
[TABLE]
In [65], Yang characterized singular Jacobi forms in the following way:
Theorem 8.1**.**
Let be a Jacobi form of index with respect to a rational representation of . Then the following conditions are equivalent: (Sing-1) is a singular Jacobi form. (Sing-2) satisfies the differential equation
Proof. See Theorem 4.1 in [65].
Theorem 8.2**.**
Let be a symmetric, positive definite, unimodular even matrix of degree . Assume that is irreducible and satisfies the condition
[TABLE]
Then a nonvanishing Jacobi form in is singular if and only if .
Proof. See Theorem 4.5 in [65].
Remark 8.1**.**
We let
[TABLE]
be the semidirect product of and with multiplication law
[TABLE]
Then we have the natural action of on the Minkowski-Euclid space defined by
[TABLE]
where Without difficulty we see that the differential operator is invariant under the action (8.2) of We refer to [77] for more detail about invariant differential operators on the Minkowski-Euclid space .
9. The Siegel-Jacobi Operator
Let be a rational representation of on a finite dimensional vector space . For a positive integer , we let be a rational representation of defined by
[TABLE]
The Siegel-Jacobi operator is defined by
[TABLE]
where and
In [62], Yang investigated the injectivity, surjectivity and bijectivity of the Siegel-Jacobi operator.
Theorem 9.1**.**
Let be a symmetric, positive definite, unimodular even matrix of degree . Assume that is irreducible and satisfies the condition
[TABLE]
If , then the Siegel-Jacobi operator is injective.
Proof. See Theorem 3.5 in [62].
Theorem 9.2**.**
Let be a symmetric, positive definite, unimodular even matrix of degree . Assume that is irreducible and satisfies the condition
[TABLE]
If , then the Siegel-Jacobi operator is an isomorphism.
Proof. See Theorem 3.6 in [62].
Theorem 9.3**.**
Let be a symmetric, positive definite, unimodular even matrix of degree . Assume that and Then the Siegel-Jacobi operator is an isomorphism.
Proof. See Theorem 3.7 in [62].
Now we review the action of the Hecke operators on Jacobi forms. For a positive integer , we define
[TABLE]
Then is decomposed into finitely many double cosets mod , that is,
[TABLE]
We define
[TABLE]
Let For a Jacobi form , we define
[TABLE]
where (finite disjoint union) and denotes the weight of . We see easily that if and , then
[TABLE]
For a prime , we define
[TABLE]
Let be the -module generated by all left cosets and the -module generated by all double cosets Then is a commutative associative algebra. We associate to a double coset
[TABLE]
the element
[TABLE]
We extend linearly to the Hecke algebra and then we have a monomorphism We now define a bilinear mapping
[TABLE]
by
[TABLE]
This mapping is well defined because the definition does not depend on the choice of representatives.
Let be a Jacobi form. For a left coset with , we put
[TABLE]
We extend this operator (9.2) linearly to If we write
[TABLE]
Obviously we have
[TABLE]
In a left coset we can choose a representative of the form
[TABLE]
[TABLE]
where Then we have
[TABLE]
For an integer , we define
[TABLE]
If (disjoint union), then we define in a natural way
[TABLE]
We extend the above map (9.3) linearly on and then we have an algebra homomorphism
[TABLE]
It is known that the above map (9.4) is a surjective map ([81] Theorem 2).
Let be the modified Siegel-Jacobi operator defined by
[TABLE]
where is a finite dimensional representation of defined by
[TABLE]
In [62], Yang proved that the action of the Hecke operators is compatible with that of the Siegel-Jacobi operator:
Theorem 9.4**.**
Suppose we have (a) a rational finite dimensional representation
[TABLE]
(b) a rational finite dimensional representation
[TABLE]
(c) a linear map satisfying the following properties (1) and (2): (1) for all (2) for some Then for any and , we have
[TABLE]
Proof. See Theorem 4.2 in [62].
Remark 9.1**.**
Freitag [14] introduced the concept of stable modular forms using the Siegel operator and developed the theory of stable modular forms. We can define the concept of stable Jacobi forms using the Siegel-Jacobi operator and develop the theory of stable Jacobi forms.
10. Construction of Vector-Valued Modular Forms from Jacobi Forms
Let and be two positive integers and let be the ring of complex valued polynomials on For any homogeneous polynomial , we put
[TABLE]
Let be a positive definite symmetric rational matrix of degree . Let be the inverse of . For each with we denote by the following differential operator
[TABLE]
A polynomial on is said to be harmonic with respect to if
[TABLE]
A polynomial on is said to be pluriharmonic with respect to if
[TABLE]
If there is no confusion, we just write harmonic or pluriharmonic instead of harmonic or pluriharmonic with respect to . Obviously a pluriharmonic polynomial is harmonic. We denote by the space of all pluriharmonic polynomials on . The ring has a symmetric nondegenerate bilinear form \langle P,Q\rangle:=\big{(}P(\partial_{Z})Q\big{)}(0) for It is easy to check that satisfies
[TABLE]
Lemma 10.1**.**
* is invariant under the action of given by*
[TABLE]
Here O(S):=\big{\{}B\in GL(m,\mathbb{C})\ |\ {}^{t}BSB=S\,\big{\}} denotes the orthogonal group of the quadratic form .
Proof. See Corollary 9.11 in [45].
Remark 10.1**.**
In [28], Kashiwara and Vergne investigated an irreducible decomposition of the space of complex pluriharmonic polynomials defined on under the action (10.1). They showed that each irreducible component occurring in the decomposition of under the action (10.1) has multiplicity one and the irreducible representation of is determined uniquely by the irreducible representation of
Throughout this section we fix a rational representation of on a finite dimensional complex vector space and a positive definite symmetric, half integral matrix of degree once and for all.
Definition 10.1**.**
A holomorphic function is called a modular form of type on if
[TABLE]
for all If the additional cuspidal condition will be added. We denote by the vector space of all modular forms of type on .
Let be the vector space of of all pluriharmonic polynomials on with respect to According to Lemma 10.1, there exists an irreducible subspace invariant under the action of given by (10.1). We denote this representation by . Then we have
[TABLE]
The action of on is defined by
[TABLE]
where and
Definition 10.2**.**
Let be a Jacobi form of index with respect to on . Let be a homogeneous pluriharmonic polynomial. We put
[TABLE]
Now we define the mapping
[TABLE]
by
[TABLE]
Yang proved the following theorem in [66].
Theorem 10.1**.**
Let and be as before. Let be a Jacobi form of index with respect to on . Then is a modular form of type , i.e.,
Proof. See Main Theorem in [66].
We obtain an interesting and important identity by applying Theorem 10.1 to the Eisenstein series. Let be a half integral positive symmetric matrix of degree . We set
[TABLE]
Let be a complete system of representatives of the cosets and be a complete system of representatives of the cosets \mathbb{Z}^{(m,n)}/\big{(}{\rm Ker}({\mathcal{M}})\cap\mathbb{Z}^{(m,n)}\big{)}, where Let be a positive integer. In [82], Ziegler defined the Eisenstein series of Siegel type by
[TABLE]
where Now we assume that and is even. Then according to [82], Theorem 2.1, is a nonvanishing Jacobi form in By Theorem 10.1, \big{(}E_{k,{\mathcal{M}}}^{(n)}\big{)}_{\tau} is a -valued modular form of type We define the automorphic factor by
[TABLE]
Then according to the relation occurring in the process of the proof of Theorem 10.1, for any homogeneous pluriharmonic polynomial with respect to we obtain the following identity
[TABLE]
for all and
For any homogeneous pluriharmonic polynomial with respect to we define the function by
[TABLE]
where and Then according to Formula (10.3), we obtain the following relation
[TABLE]
If is a constant, we see from (10.3) and (10.5) that satisfies the following relation
[TABLE]
for all and . Therefore for any , the function is a Siegel modular form of weight .
11. Maass-Jacobi Forms
Using -invariant differential operators on the Siegel-Jacobi space, we introduce a notion of Maass-Jacobi forms.
Definition 11.1**.**
Let
[TABLE]
be the discrete subgroup of , where
[TABLE]
A smooth function is called a Maass-Jacobi form on if satisfies the following conditions (MJ1)-(MJ3) :(MJ1) is invariant under
(MJ2) is an eigenfunction of the Laplacian (cf. Formula (2.4)).
(MJ3) has a polynomial growth, that is, there exist a constant
and a positive integer such that
[TABLE]
where is a polynomial in
Remark 11.1**.**
We also may define the notion of Maass-Jacobi forms as follows. Let be a commutative subalgebra of containing the Laplacian . We say that a smooth function is a Maass-Jacobi form with respect to if satisfies the conditions and : the condition is given by is an eigenfunction of any invariant differential operator in .
Remark 11.2**.**
Erik Balslev [2] developed the spectral theory of on to prove that the set of all eigenvalues of satisfies the Weyl law.
It is natural to propose the following problems.
Problem A : Find all the eigenfunctions of
Problem B : Construct Maass-Jacobi forms.
If we find a nice eigenfunction of the Laplacian , we can construct a Maass-Jacobi form on in the usual way defined by
[TABLE]
where
[TABLE]
is a subgroup of
We consider the simple case when and . A metric on given by
[TABLE]
is a -invariant Kähler metric on . Its Laplacian is given by
[TABLE]
We provide some examples of eigenfunctions of . with eigenvalue
Here
[TABLE]
where
with eigenvalue
with eigenvalue
with eigenvalue [math].
All Maass wave forms.
Let be a rational representation of on a finite dimensional complex vector space . Let be a symmetric half-integral semi-positive definite matrix of degree . Let be the algebra of all functions on with values in . We define the -slash action of on as follows: If ,
[TABLE]
where and . We recall the Siegel’s notation for suitable matrices and . We define to be the algebra of all differential operators on satisfying the following condition
[TABLE]
for all and for all We denote by the center of .
We define another notion of Maass-Jacobi forms as follows.
Definition 11.2**.**
A vector-valued smooth function is called a Maass-Jacobi form on of type and index if it satisfies the following conditions and : for all
* is an eigenfunction of all differential operators in the center of .*
* has a growth condition*
[TABLE]
as for some
Remark 11.3**.**
In the sense of Definition 11.2, Pitale [50] studied Maass-Jacobi forms on the Siegel-Jacobi space We refer to [74, 75] for more details on Maass-Jacobi forms.
12. The Schrödinger-Weil Representation
Throughout this section we assume that is a positive definite symmetric real matrix. We consider the Schrödinger representation of the Heisenberg group with the central character . Then is expresses explicitly as follows:
[TABLE]
where and For the construction of we refer to [78]. We note that the symplectic group acts on by conjugation inside . For a fixed element , the irreducible unitary representation of defined by
[TABLE]
has the property that
[TABLE]
Here denotes the identity operator on the Hilbert space According to Stone-von Neumann theorem, there exists a unitary operator on with such that
[TABLE]
We observe that is determined uniquely up to a scalar of modulus one.
From now on, for brevity, we put According to Schur’s lemma, we have a map satisfying the relation
[TABLE]
We recall that denotes the multiplicative group of complex numbers of modulus one. Therefore is a projective representation of on and defines the cocycle class in The cocycle yields the central extension of by . The group is a set equipped with the following multiplication
[TABLE]
We see immediately that the map defined by
[TABLE]
is a true representation of As in Section 1.7 in [35], we can define the map satisfying the relation
[TABLE]
Thus we see that
[TABLE]
is the metaplectic group associated with that is a two-fold covering group of . The restriction of to is the Weil representation of associated with .
If we identify (resp. ) with (resp. every element of can be written as with and . In fact,
[TABLE]
Therefore we define the projective representation of the Jacobi group with cocycle by
[TABLE]
Indeed, since is a normal subgroup of , for any and ,
[TABLE]
We let
[TABLE]
be the semidirect product of and with the multiplication law
[TABLE]
where and If we identify (resp. with (resp. we see easily that every element \big{(}(g,t),(\lambda,\mu\,;\kappa)\big{)} of can be expressed as
[TABLE]
Now we can define the true representation of by
[TABLE]
Indeed, since is a normal subgroup of ,
[TABLE]
Here we used the fact that
We recall that the following matrices
[TABLE]
generate the symplectic group (cf. [15, p. 326], [44, p. 210]). Therefore the following elements and of defined by
[TABLE]
generate the group We can show that the representation is realized on the representation H(\chi_{\mathcal{M}})=L^{2}\big{(}\mathbb{R}^{(m,n)}\big{)} as follows: for each f\in L^{2}\big{(}\mathbb{R}^{(m,n)}\big{)} and the actions of on the generators are given by
[TABLE]
[TABLE]
Let
[TABLE]
be the semidirect product of and . Then is a subgroup of which is a two-fold covering group of the Jacobi group The restriction of to is called the Schrödinger-Weil representation of associated with .
We denote by L^{2}_{+}\big{(}\mathbb{R}^{(m,n)}\big{)} \big{(}\textrm{resp.}\,\,L^{2}_{-}\big{(}\mathbb{R}^{(m,n)}\big{)}\big{)} the subspace of L^{2}\big{(}\mathbb{R}^{(m,n)}\big{)} consisting of even (resp. odd) functions in L^{2}\big{(}\mathbb{R}^{(m,n)}\big{)}. According to Formulas (12.11)–(12.13), is decomposed into representations of
[TABLE]
where and are the even Weil representation and the odd Weil representation of that are realized on L^{2}_{+}\big{(}\mathbb{R}^{(m,n)}\big{)} and L^{2}_{-}\big{(}\mathbb{R}^{(m,n)}\big{)} respectively. Obviously the center of is given by
[TABLE]
We note that the restriction of to coincides with and for all
Remark 12.1**.**
In the case is dealt in [10] and [36]. We refer to [16] and [28] for more details about the Weil representation .
Remark 12.2**.**
The Schrödinger-Weil representation is applied to the theory of Maass-Jacobi forms [50].
Let be a positive definite symmetric real matrix of degree . We recall the Schrödinger representation of the Heisenberg group associate with given by Formula (12.1). We note that for an element of , we have the decomposition
[TABLE]
We consider the embedding defined by
[TABLE]
For we put
[TABLE]
According to Formulas (12.11)-(12.13), for any and , we have the following explicit representation
[TABLE]
where
[TABLE]
Indeed, if and , using the decomposition
[TABLE]
and if and , using the decomposition
[TABLE]
we obtain Formula (12.15).
If
[TABLE]
with , the corresponding cocycle is given by
[TABLE]
where
[TABLE]
In the special case when
[TABLE]
we find
[TABLE]
where
[TABLE]
It is well known that every admits the unique Iwasawa decomposition
[TABLE]
where and This parametrization in leads to the natural action of on defined by
[TABLE]
Lemma 12.1**.**
For two elements and in , we let
[TABLE]
and
[TABLE]
be the Iwasawa decompositions of and respectively, where and Let
[TABLE]
be the Iwasawa decomposition of Then we have
[TABLE]
and
[TABLE]
where
[TABLE]
Proof. If has the unique Iwasawa decomposition (12.17), then we get the following
[TABLE]
We set
[TABLE]
Since
[TABLE]
by an easy computation, we obtain the desired results.
Now we use the new coordinates with and in According to Formulas (12.11)-(12.13), the projective representation of reads in these coordinates as follows:
[TABLE]
where and
[TABLE]
Here
[TABLE]
Now we set
[TABLE]
We note that
[TABLE]
for
Remark 12.3**.**
For Schwartz functions we have
[TABLE]
Therefore the projective representation is not continuous at in general. If we set
[TABLE]
* corresponds to a unitary representation of the double cover of (cf. (3.5) and [35]). This means in particular that*
[TABLE]
where parametrises the double cover of
We observe that for any element with and , we have the following decomposition
[TABLE]
Thus acts on naturally by
[TABLE]
Definition 12.1**.**
For any Schwartz function we define the function on the Jacobi group by
[TABLE]
where and . The projective representation of the Jacobi group was already defined by Formula (12.8). More precisely, for and , we have
[TABLE]
Lemma 12.2**.**
We set for . Then for any , there exists a constant such that for all and
[TABLE]
Proof. Following the arguments in the proof of Lemma 4.3 in [36], pp. 428-429, we get the desired result.
Theorem 12.1** (Jacobi 1).**
Let be a positive definite symmetric integral matrix of degree such that Then for any Schwartz function we have
[TABLE]
where
[TABLE]
Proof. See Theorem 6.1 in [78].
Theorem 12.2** (Jacobi 2).**
Let be a positive definite symmetric integral matrix and let be integral. Then we have
[TABLE]
for all and .
Proof. See Theorem 6.2 in [78].
Theorem 12.3** (Jacobi 3).**
Let be a positive definite symmetric integral matrix and let be an integral element of Then we have
[TABLE]
for all and .
Proof. See Theorem 6.3 in [78].
We put . Let
[TABLE]
be the group with the following multiplication law
[TABLE]
where and .
We define
[TABLE]
Then acts on naturally through the multiplication law (12.23).
Lemma 12.3**.**
* is generated by the elements*
[TABLE]
where
[TABLE]
Proof. Since is generated by and , we get the desired result.
We define
[TABLE]
Theorem 12.4**.**
Let be the subgroup of generated by the elements
[TABLE]
where
[TABLE]
Let be a positive definite symmetric unimodular integral matrix such that Then for the function
[TABLE]
is invariant under the action of on .
Proof. See Theorem 6.4 in [78].
13. Final Remarks and Open Problems
The Siegel-Jacobi space is a non-symmetric homogeneous space that is important geometrically and arithmetically. As we see in the formula (7.2), the theory of Jacobi forms is applied in the study of modular forms. The theory of Jacobi forms reduces to that of Siegel modular forms if the index is zero. Unfortunately the theory of the geometry and the arithmetic of the Siegel-Jacobi space has not been well developed so far.
Now we propose open problems related to the geometry and the arithmetic of the Siegel-Jacobi space.
Problem 1. Find the analogue of the Hirzebruch-Mumford Proportionality Theorem. Let us give some remarks for this problem. Before we describe the proportionality theorem for the Siegel modular variety, first of all we review the compact dual of the Siegel upper half plane . We note that is biholomorphic to the generalized unit disk of degree through the Cayley transform. We suppose that is a symplectic lattice with a symplectic form We extend scalars of the lattice to . Let
[TABLE]
be the complex Lagrangian Grassmannian variety parameterizing totally isotropic subspaces of complex dimension . For the present time being, for brevity, we put and The complexification of acts on transitively. If is the isotropy subgroup of fixing the first summand , we can identify with the compact homogeneous space We let
[TABLE]
be an open subset of . We see that acts on transitively. It can be shown that is biholomorphic to A basis of a lattice is given by a unique matrix with . Therefore we can identify with in . In this way, we embed into as an open subset of . The complex projective variety is called the compact dual of
Let be an arithmetic subgroup of . Let be a -equivariant holomorphic vector bundle over of rank . Then is defined by the representation That is, is a homogeneous vector bundle over . We naturally obtain a holomorphic vector bundle over is often called an automorphic or arithmetic vector bundle over . Since is compact, carries a -equivariant Hermitian metric which induces a Hermitian metric on . According to Main Theorem in [43], admits a unique extension to a smooth toroidal compactification of such that is a singular Hermitian metric good on . For the precise definition of a good metric on we refer to [43, p. 242]. According to Hirzebruch-Mumford’s Proportionality Theorem (cf. [43, p. 262]), there is a natural metric on such that the Chern numbers satisfy the following relation
[TABLE]
for all with nonegative integers and where is the -equivariant holomorphic vector bundle on the compact dual of defined by a certain representation of the stabilizer of a point in . Here is the volume of that can be computed (cf. [57]).
As before we consider the Siegel-Jacobi modular group with For an arithmetic subgroup of , we set
[TABLE]
Problem 2. Compute the cohomology of Investigate the intersection cohomology of
Problem 3. Generalize the trace formula on the Siegel modular variety obtained by Sophie Morel to the universal abelian variety. For her result on the trace formula on the Siegel modular variety, we refer to her paper, Cohomologie d’intersection des vari’etés modulaires de Siegel, suite.
Problem 4. Develop the theory of the stability of Jacobi forms using the Siegel-Jacobi operator. The theory of the stability involves in the theory of unitary representations of the infinite dimensional symplectic group and the infinite dimensional unitary group .
Problem 5. Compute the geodesics, the distance between two points and curvatures explicitly in the Siegel-Jacobi space
Siegel proved the following theorem for the Siegel space
Theorem 13.1**.**
(Siegel [57]). (1) There exists exactly one geodesic joining two arbitrary points in . Let be the cross-ratio defined by
[TABLE]
For brevity, we put Then the symplectic length of the geodesic joining and is given by
[TABLE]
where
[TABLE]
(2) For , we set
[TABLE]
Then and have the same eigenvalues.
(3) All geodesics are symplectic images of the special geodesics
[TABLE]
where are arbitrary positive real numbers satisfying the condition
[TABLE]
The proof of the above theorem can be found in [57], pp. 289-293.
Problem 6. Solve Problem 4 and Problem 5 in Section 3. Express the center of the algebra of all -invariant differential operators on explicitly. Describe the center of the universal enveloping algebra of the Lie algebra of the Jacobi group explicitly.
Problem 7. Develop the spectral theory of the Laplacian on for an arithmetic subgroup of Balslev [2] developed the spectral theory of the Laplacian on for certain arithmetic subgroup of
Problem 8. Develop the theory of harmonic analysis on the Siegel-Jacobi disk
Problem 9. Study unitary representations of the Jacobi group . Develop the theory of the orbit method for the Jacobi group
Problem 10. Attach Galois representations to cuspidal Jacobi forms.
Problem 11. Develop the theory of automorphic -function for the Jacobi group .
Problem 12. Find the trace formula for the Jacobi group .
Problem 13. Decompose the Hilbert space L^{2}\big{(}G^{J}(\mathbb{Q})\backslash G^{J}(\mathbb{A})\big{)} into irreducibles explicitly.
Problem 14. Construct Maass-Jacobi forms. Express the Fourier expansion of a Maass-Jacobi form explicitly.
Problem 15. Investigate the relations among Jacobi forms, hyperbolic Kac-Moody algebras, infinite products, the monster group and the Moonshine (cf. [67]).
Problem 16. Provide applications to physics (quantum mechanics, quantum optics, coherent states,), the theory of elliptic genera, singularity theory of K. Saito etc.
Acknowledgements
I would like to give my hearty thanks to Eberhard Freitag and Don Zagier for their advice and their interest in this subject. In particular, it is a pleasure to thank E. Freitag for letting me know the paper [37] of Hans Maass.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 5[5] R. Berndt, Shimuras Reziprozitätsgesetz für den Körper der arithmetischen elliptischen Funktionen beliebiger Stufe , J. reine angew. Math., 343 (1983), 123-145.
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