Arithmetic geometry of moduli spaces and applications

模空间的算术几何及其应用

• 批准号: 227040-2009
• 负责人: Goren, Eyal
• 金额: $ 2.48万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=503704
• 关键词: Arithmetic; geometry; moduli; spaces; applications

Number fields are obtained when one wishes to define solutions to polynomial equations. They are the primary object of study of number theory. Abelian varieties are a generalization of elliptic curves and are one of the primary objects of study of algebraic geometry. The spaces that parameterize abelian varieties are a generalization of modular curves, called Shimura varieties. Shimura varieties play a central role in modern number theory. They are intimately related to Galois representations (projections of the absolute Galois group, whose study, arguably, is the chief purpose of algebraic number theory), explicit class field theory (that attempts to give an explicit handle on number fields), modular forms (examples of which are theta functions which encode the number of times a quadratic form represents an integer, and have many applications to physics and combinatorics), nonabelian harmonic analysis and more.

当希望将解决方案定义为多项式方程时，将获得数字字段。它们是研究数理论的主要对象。阿贝尔（Abelian）品种是椭圆曲线的概括，是代数几何学研究的主要对象之一。参数化Abelian品种的空间是模块化曲线的概括，称为Shimura品种。 Shimura品种在现代数字理论中起着核心作用。它们与Galois的表示密切相关（绝对Galois集团的预测，他们的研究可以说是代数数理论的主要目的），显式的类领域理论（试图对数字字段进行明确处理），模块化形式（其中的示例是theta函数，编码二次形式的次数代表整数，并且在物理和组合学上有许多应用），nonabelian谐波分析等。