Motivic fundamental groups, multiple polylogarithms, and Diophantine geometry

动机基本群、多重多对数和丢番图几何

• 批准号: 0753012
• 负责人: Minhyong Kim
• 金额: $ 3.39万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0753012&HistoricalAwards=false
• 关键词: Motivic; fundamental; groups; multiple; polylogarithms

The theory of mixed motives provides one of the most fertile grounds for investigations in number theory via its connection to the theory of L-functions of algebraic varieties. Although the study of motives is essentially homological in nature, Deligne has defined a motivic fundamental group attached to varieties over number fields. Coordinate functions on such fundamental groups are related to special functions like multiple polylogarithms and, thereby, also to special values of L-functions. On the other hand, the proposer has discovered a direct connection betweenmotivic fundamental groups and Diophantine geometry, somewhat along the lines suggested by Gronthendieck's `anabelian' philosophy. The technical tools involve p-adic integration, p-adic Hodge theory, and the global study of Galois cohomology. He proposes to continue research into this connection, aiming towards homotopy-theoretic proofs of well-known theorems, such as those of Faltings or Wiles, and eventually new higher-dimensional results in the line of the conjectures of Serge Lang on hyperbolic varieties.The deep relationship between geometry and arithmetic is a venerable topic of study going back at least to ruler and compass constructions of special numbers in ancient Greece. The modern manifestation of this tradition is the subject of arithmetic geometry, an area of mathematics that has yielded some of the most profound mathematical results of the previous century.This proposal describes several related ideas for obtaining new results in the theory of algebraicequations using ideas at the interface of linear and non-linear geometry.

混合动机理论通过与代数簇的 L 函数理论的联系，为数论研究提供了最肥沃的基础之一。尽管对动机的研究本质上是同源的，但德利涅定义了一个与数域上的变体相关的动机基本群。这些基本群上的坐标函数与特殊函数（如多重多对数）相关，因此也与 L 函数的特殊值相关。另一方面，提议者发现了动机基本群和丢番图几何之间的直接联系，有点类似于格隆腾迪克的“阿贝尔”哲学所暗示的路线。技术工具包括 p-adic 积分、p-adic Hodge 理论和伽罗瓦上同调的全局研究。他建议继续研究这种联系，旨在对著名定理（例如法尔廷斯或怀尔斯的定理）进行同伦理论证明，并最终在塞尔日·朗（Serge Lang）关于双曲簇的猜想中获得新的高维结果。几何和算术之间的深厚关系是一个古老的研究课题，至少可以追溯到古希腊特殊数字的尺子和指南针的构造。这一传统的现代表现形式是算术几何学科，这是一个数学领域，它产生了上个世纪一些最深刻的数学成果。该提案描述了使用以下思想在代数方程理论中获得新结果的几个相关思想：线性和非线性几何的接口。