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.

混合动机理论通过其与代数簇的L函数理论的联系，为数论的研究提供了最肥沃的基础之一。尽管动机的研究本质上是同调的，但德利涅定义了一个依附于数域上的变体的动机基本群。在这样的基本群上的坐标函数与特殊函数（如多重多项式）有关，因此也与L-函数的特殊值有关。另一方面，提议者发现了motivic基本群和丢番图几何之间的直接联系，有点沿着Grontendieck的“anabelian”哲学所建议的路线。技术工具包括p-adic积分，p-adic Hodge理论，以及Galois上同调的整体研究。他建议继续研究这方面，旨在同伦理论证明著名的定理，如那些Faltings或怀尔斯，并最终新的高维结果线的prostitutures的塞尔日朗双曲品种。深刻的关系几何和算术是一个古老的研究课题至少可以追溯到统治者和指南针建设的特殊号码在古希腊。这一传统的现代表现是算术几何的主题，这是一个数学领域，产生了上个世纪一些最深刻的数学结果。这一建议描述了几个相关的想法，以获得新的结果，在代数方程理论中使用的思想在线性和非线性几何的接口。