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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=523885
• 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.

当一个人想要定义多项式方程的解时，可以得到数域。它们是数论研究的主要对象。阿贝尔簇是椭圆曲线的推广，是代数几何的主要研究对象之一。阿贝尔簇的参数空间是模曲线的推广，称为下村簇。下村变种在现代数论中起着核心作用。它们与伽罗瓦表示(绝对伽罗瓦群的投影，可以说其研究是代数数论的主要目的)、显式类场理论(试图对数域进行显式处理)、模形式(其示例是编码二次型表示整数的次数的theta函数，在物理学和组合学中有许多应用)、非阿贝尔调和分析等密切相关。