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

用离散群将半积分权的Siegel模形式与Siegel上半平面商空间上的某全纯线束的全纯截面进行了识别。将Riemann-Roch（全纯Lefschetz不动点定理）公式和Kodaira消失定理应用于该直线模型，计算了二阶半积分权的Siegel模形式空间的维数。我们用计算机对不动点进行分类。半积分权的西格尔模形式的空间有一个子空间叫做正空间。这个子空间是模形式提升理论中一个非常重要的子空间。在这个加空间和指标1的雅可比形式的空间之间存在同构。我们计算了二阶雅可比形式的空间的维数通过这个同构知道了正空间的维数。通过这种方式，我们知道了正空间的维度，并确定了它的结构（Ibukiyama和Hayashida）。更多的obi形式是西格尔上半平面和复向量空间积空间上的全纯函数，它们对于西格尔上半平面的变量表现得像模形式，而对于复向量空间的变量表现得像函数。由于指标m的雅可比形式对复向量空间的变量表现得像2m次的函数，它们由由2m次的函数的基组成的级数的线性组合来表示。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西格尔上半平面上对某自同构因子形成了一个向量值模形式。因此，在Siegel上半平面的商空间上，用离散群将Jacobi形式与某全纯向量束的全纯截面进行了识别。应用Riemann-Roch公式和Kodaira-Nakano的消失定理，计算了全纯截面空间的维数即Jacobi形式空间的维数。少

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