arXiv subjects
Publications and source records attributed to Jae-Hyun Yang.
We consider a special abelian surface $A_\Omega$ deduced from the work of Tianze Wang, Tianqin Wang and Hongwen Lu \cite{WWL}. We study holomorphic line bundles over a special abelian surface explicitly.
We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We conjecture that a simple abelian variety over the rational number field is modular.
In this paper, we introduce the notion of automorphic forms for $GL(n,\BZ)\ltimes \BZ^{(m,n)}$ and discuss invariant differential operators on the Minkowski-Euclid space. The group $GL{n,\BR}\ltimes \BR^{(m,n)}$ is the semidirect product of $GL(n,\BR)$ and the additive group $\BR^{(m,n)}$ and is {\it not} a reductive group. The Minkowski-Euclid space is the quotient space of $GL(n,\BR)\ltimes \BR^{(m,n)}$ by $O(n,\BR)$. The Minkowski-Euclid space is an important non-symmetric homogeneous space geometrically and number theoretically. We present some open problems to be solved in the future.
In this paper, we introduce the notion of the stability of automorphic forms for the general linear group and relate the stability of automorphic forms to the moduli space of real tori and the Jacobian real locus.
In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the Andr{é}-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem.
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.
In this paper, we investigate the derivatives of L-functions, in particular, the Riemann zeta function, the Hasse-Weil L-function, the Rankin L-function and the Artin L-function, and survey the relations between the derivatives of L-functions and the geometry and arithmetic of the associated Shimura varieties.
The relatively-new subject, symplectic geometry, has been studied the past five decades. Symplectic geometry is the mathematical subject studying the geometry of symplectic manifolds. A symplectic manifold is an even-dimensional smooth manifold equipped with a closed non-degenerate two form. In this paper, we survey the progress on the study of symplectic geometry the past five decades. We deal with the brief history of symplectic geometry and symplectic topology, Arnold's conjecture, pseudoholomorphic curves, the convexity properties of a moment map, recent results on Hamiltonian and non-Hamiltonian symplectic group actions, the classification of symplectic group actions, recent results on the symplectic embedding problems, the theory of Gromov-Witten invariants, Lagrangian Floer homology, the Fukaya category and mirror symmetry.
In this paper, we introduce the concept of stable automorphic forms for semisimple algebraic groups and use the stability of automorphic forms to study the geometry of infinite dimensional arithmetic quotients.
In this article, we study adjoint orbits of the Jacobi group, and in particular describe nilpotent orbits explicitely.
We give a brief description of the Birch-Swinnerton-Dyer conjecture and present related conjectures. We describe the relation between the nilpotent orbits of SL(2,R) and CM points.
In this article, we discuss the relation between Kac-Moody algebras, the monstrous moonshine, Jacobi forms and infinite products. We also review Borcherds' solution of the Moonshine Conjecture and his work of constructing automorphic forms on the orthogonal group which can be written as infinite products.
In this article, we prove that there do not exist stable Schottky-Jacobi forms for the universal Jacobian locus and also prove that there exist non-trivial stable Schottky-Jacobi forms for the universal hyperelliptic locus.
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.
In this paper, we obtain some behaviours of theta sums of higher index for the Schroedinger-Weil representation of the Jacobi group associated with a positive definite symmetric real matrix of degree m.
In this paper, we define the concept of Jacobi forms of half-integral weight using Takase's automorohic factor of weight 1/2 for a two-fold covering group of the symplectic group on the Siegel upper half plane and find covariant maps for the Schroedinger-Weil representation. Using these covariant maps, we construct Jacobi forms of half-integral weight with respect to an arithmetic subgroup of the Jacobi group.
In this manuscript, we develope the theory of harmonic analysis on the Heisenberg group G of high dimension. We investigate the theta functions and the Weil representation related to this Heisenberg group and describe the connection among them.
For a fixed positive integer $g$, we let ${\mathcal P}_g = \big\{Y\in {\mathbb R}^{(g,g)} | Y= {}^tY>0 \big\}$ be the open convex cone in the Euclidean space ${\mathbb R}^{g(g+1)/2}$. Then the general linear group $GL(g,{\mathbb R})$ acts naturally on ${\mathcal P}_g$ by $A\star Y= AY {}^tA$ ($A\in GL(g,{\mathbb R}), Y\in {\mathcal P}_g$). We introduce a notion of polarized real tori. We show that the open cone ${\mathcal P}_g$ parametrizes principally polarized real tori of dimension $g$ and that the Minkowski domain ${\mathfrak R}_g= GL(g,{\mathbb Z})\backslash {\mathcal P}_g$ may be regarded as a moduli space of principally polarized real tori of dimension $g$. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.