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A study of Siegel modular forms of half integral weight by a method of algebraic geometry

半积分权Siegel模形式的代数几何研究

基本信息

批准号：
10640044
负责人：
INATOMI Akira
金额：
$ 1.98万
依托单位：
Meiji University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

Siegel modular forms of half integral weight are identified with holomorphic sections of a certain holomorphic line bundle over a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the spaces of Siegel modular forms of degree two and half integral weight by applying the formula of Riemann-Roch (holomorphic Lefschetz fixed point theorem) and Kodaira vanishing theorem to this line bundie. We classified fixed points by using computer.The space of Siegel modular forms of half integral weight has a subspace called plus space. This subspace is a very important subspace concerning the lifting theory of modular forms. There exists an isomorphism between this plus space and the space of Jacobi forms of index one. We computed the dimension of the spaces of Jacobi forms of degree two to know the dimension of the plus space by this isomorphism. In this way we knew the dimension of the plus space and its structure was determined (Ibukiyama and Hayashida).Jac … More obi forms are holomorphic functions on the product space of Siegel upper half plane and complex vector space which behave like modular forms with respect to the variables of Siegel upper half plane and behave like theta functions with respect to the variables of complex vector space. Since Jacobi forms of index m behave like theta functions of degree 2m with respect to the variables of complex vector space, they are represented by a linear combination of theta series which consist of a basis of theta functions of degree 2m. The coefficients of this combination are holomorphic functions on Siegel upper half plane. The vector consisting of these coefficients becomes a vector valued modular form with respect to a certain automorphic factor on Siegel upper half plane. Therefore Jacobi forms are identified with holomorphic sections of a certain holomorphic vector bundle on a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the space of holomorphic sections which is the dimension of the space of Jacobi forms by applying the formula of Riemann-Roch and the vanishing theorem of Kodaira-Nakano. Less

半整权的Siegel模形式与Siegel上半平面的商空间上的某个全纯线丛的全纯截面通过离散群来标识。利用Riemann-Roch公式（全纯Lefschetz不动点定理）和科代拉消失定理，计算了二次半整权Siegel模形式空间的维数.利用计算机对不动点进行了分类，半整权Siegel模形式空间存在一个子空间，称为正空间。这个子空间是模形式提升理论中一个非常重要的子空间。在这个正空间和指数为1的Jacobi形式空间之间存在同构。我们计算了二次Jacobi形式空间的维数，从而知道了这个同构的正空间的维数。通过这种方式，我们知道了正空间的维数，并确定了它的结构（Ibukiyama和Hayashida）。 ...更多信息 obi形式是Siegel上半平面和复向量空间的乘积空间上的全纯函数，其对于Siegel上半平面的变量表现为模形式，并且对于复向量空间的变量表现为θ函数。由于指数m的Jacobi形式表现为关于复向量空间的变量的2 m次θ函数，因此它们由θ级数的线性组合表示，该线性组合由2 m次θ函数的基组成。这种组合的系数是Siegel上半平面上的全纯函数。由这些系数组成的向量在Siegel上半平面上关于某个自守因子成为一个向量值模形式。因此，Jacobi型通过一个离散群与Siegel上半平面的商空间上的某个全纯向量丛的全纯截面相一致。应用Riemann-Roch公式和Kodaira-Nakano消失定理，计算了全纯截面空间的维数，即Jacobi形式空间的维数。少