• Kyungpook Mathematical Journal
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• v.64 no.3
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• pp.407-416
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• 2024

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.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.781-794
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• 2008

Let $\mathbb{H}_g$ and $\mathbb{D}_g$ be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let $\mathbb{C}^{(h,g)}$ be the Euclidean space of all $h{\times}g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ onto the Siegel-Jacobi space $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ which gives a partial bounded realization of $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ by $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. and $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$. are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. invariant under the natural action of the Jacobi group $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ explicitly.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.787-799
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• 2013

In this paper, we compute the sectional curvatures and the scalar curvature of the Siegel-Jacobi space $\mathb{H}_1{\times}\mathb{C}$ of degree 1 and index 1 explicitly.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.647-707
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• 2023

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the André-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem.

• Kyungpook Mathematical Journal
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• v.53 no.1
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• pp.49-86
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• 2013

This article is a continuation of the paper [21]. In this paper we deal with Maass-Jacobi forms on the Siegel-Jacobi space $\mathbb{H}{\times}\mathbb{C}^m$, where H denotes the Poincar$\acute{e}$ upper half plane and $m$ is any positive integer.