论文标题
矢量值正交模块形式
Vector-valued orthogonal modular forms
论文作者
论文摘要
该专着专门介绍了签名正交组的矢量值模块形式的理论（2，n）。我们的目的是多层的：（1）为矢量值正交模块化形式的理论奠定基础； （2）以更深入的方式开发理论的某些方面，例如西格尔操作员的几何形状，与一维尖端相关的过滤，矢量值雅各比形式的分解，平方的集成性等； （3）应用程序得出了几种类型的消失定理的矢量值模块化形式。我们消失的定理尤其暗示着在正交模块化品种上<n/2-1的全体形态张量消失，这是一般结合。 该理论的基本要素是两个霍奇束。第一个是霍奇线束，它已经出现在标量值模块化形式的理论中。第二霍奇束出现在矢量值理论中，并发挥着核心作用。它对应于O（2，n）的最大紧凑型亚组的非亚伯式o（n，r）。该专着的主要重点是围绕第二霍奇束在矢量值正交模块形式的特性和作用。
This monograph is devoted to the theory of vector-valued modular forms for orthogonal groups of signature (2,n). Our purpose is multi-layered: (1) to lay a foundation of the theory of vector-valued orthogonal modular forms; (2) to develop some aspects of the theory in more depth such as geometry of the Siegel operators, filtrations associated to 1-dimensional cusps, decomposition of vector-valued Jacobi forms, square integrability etc; and (3) as applications derive several types of vanishing theorems for vector-valued modular forms of small weight. Our vanishing theorems imply in particular vanishing of holomorphic tensors of degree <n/2-1 on orthogonal modular varieties, which is optimal as a general bound. The fundamental ingredients of the theory are the two Hodge bundles. The first is the Hodge line bundle which already appears in the theory of scalar-valued modular forms. The second Hodge bundle emerges in the vector-valued theory and plays a central role. It corresponds to the non-abelian part O(n,R) of the maximal compact subgroup of O(2,n). The main focus of this monograph is centered around the properties and the role of the second Hodge bundle in the theory of vector-valued orthogonal modular forms.
