### Article

Kyungpook Mathematical Journal 2024; 64(3): 407-416

**Published online** September 30, 2024 https://doi.org/10.5666/KMJ.2024.64.3.407

Copyright © Kyungpook Mathematical Journal.

### Problems in the Geometry of the Siegel-Jacobi Space

Jae-Hyun Yang

Department of Mathematics, Inha University, Incheon 22212, Republic of Korea

e-mail : jhyang@inha.ac.kr; jhyang8357@gmail.com

**Received**: September 5, 2023; **Revised**: October 25, 2023; **Accepted**: October 26, 2023

### Abstract

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space.

**Keywords**: Siegel-Jacobi space, Invariant metrics, Laplace operator, Invariant differential operators, Compactification

### 1. Introduction

For a given fixed positive integer *n*, we let

be the Siegel upper half plane of degree *n* and let

be the symplectic group of degree *n*, where *F* for two positive integers *k* and *l*, * ^{t}M* denotes the transposed matrix of a matrix

*M*and

Then

where

be the Siegel modular group of degree *n*. This group acts on

For two positive integers *m* and *n*, we consider the Heisenberg group

endowed with the following multiplication law

with *Jacobi group* *n* and index *m* that is the semidirect product of

endowed with the following multiplication law

with

where *not* a reductive Lie group and the homogeneous space *Siegel-Jacobi space* of degree *n* and index *m*.

In this short article, we propose the basic and natural problems in the geometry of the Siegel-Jacobi space.

**Notations:** We denote by

*k* and *l*, *F*. For a square matrix *k*, *A*. For any *M*. *n*. For a complex matrix *A*, *conjugate* of *A*. For a number field *F*, we denote by *F*. If

### 2. Brief Review on the Geometry of the Siegel Space

We let *K*=*U*(*n*). The stabilizer of the action (1.1) at

Thus we get the biholomorphic map

For

C. L. Siegel [16] introduced the symplectic metric

It is known that the metric

And

is a

Siegel proved the following theorem for the Siegel space

**Theorem 2.1.** (**Siegel[16]**). (1) There exists exactly one geodesic joining two arbitrary points

For brevity, we put

where

(2) For

Then

(3) All geodesics are symplectic images of the special geodesics

where

The proof of the above theorem can be found in [16, pp.289-293].

Let

where *n* algebraically independent invariant differential operators on *G*. It is known that

**Example 2.2.** We consider the simplest case *n*=1 and *A*=1. Let

is a *x*-axis or circular arcs perpendicular to the *x*-axis (half-circles whose origin is on the *x*-axis). The Laplace operator *Δ* of

and

is a

The distance between two points

### 3. Basic Problems in the Geometry of the Siegel-Jacobi Space

For a coordinate

The author proved the following theorems in [18].

**Theorem 3.1.** For any two positive real numbers *A* and *B*,

is a Riemannian metric on

**Theorem 3.2.** The Laplace operator

where

and

Furthermore

**Remark 3.3.** Erik Balslev [2] developed the spectral theory of

**Remark 3.4.** The scalar curvature of *B*. We refer to [21] for more detail.

**Remark 3.5.** Yang and Yin [22] showed that

Now we propose the basic and natural problems.

**Problem 1.** Find all the geodesics of

**Problem 2.** Compute the distance between two points

**Problem 3.** Compute the Ricci curvature tensor and the scalar curvature of

**Problem 4.** Find all the eigenfunctions of the Laplace operator

**Problem 5.** Develop the spectral theory of

**Problem 6.** Describe the algebra of all

**Problem 7.** Find the trace formula for the Jacobi group

**Problem 8.** Discuss the behaviour of the analytic torsion of the Siegel-Jacobi space

We make some remarks on the above problems.

**Remark 3.6.** Problem 1 reduces to trying to solve a system of ordinary differential equations explicitly. If Problem 2 is solved, the distance formula would be a very beautiful one that generalizes the distance formula

**Remark 3.7.** Problem 3 was recently solved in the case that *n*=1 and *m* is arbitrary. Precisely the scalar and Ricci curvatures of the Siegel-Jacobi space

**Remark 3.8.** Concerning Problem 4 and Problem 5, computing eigenfunctions explicitly is a tall order, but if this can be done it will shed a lot of light onto the geometry of this space. And understanding the spectral geometry seems to be a central question which will likely have applications in number theory and other areas.

**Remark 3.9.** The algebra *n*=*m*=1 was completely solved by M. Itoh, H. Ochiai and J.-H. Yang in 2013. They proved that the noncommutative algebra

**Remark 3.10** The solution of Problem 7 will provide lots of arithmetic properties of the Siegel-Jacobi space.

### 4. Final Remarks

Let *n*-dimensional principally polarized abelian varieties with level *N*-structure. The Mumford school [1] found toroidal compactifications of *good singular* Hermitian metric on an automorphic vector bundle on a smooth toroidal compactification of

We set

where

Let

be the universal abelian variety. An arithmetic toroidal compactification of

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