Motivic fundamental groups, multiple polylogarithms, and Diophantine geometry

混合动机理论通过与代数变量的l -函数理论的联系，为数论的研究提供了最肥沃的土壤之一。虽然动机的研究本质上是同调的，但德列涅已经定义了一个与数域上的变量相关的动机基本群。这些基群上的坐标函数与一些特殊函数有关，比如多个多对数，因此也与l函数的特殊值有关。另一方面，提出者发现了动机基本群和丢番图几何之间的直接联系，这在某种程度上与格隆腾狄克的“安娜贝尔”哲学相一致。技术工具包括p进积分、p进霍奇理论和伽罗瓦上同调的全局研究。他建议继续研究这一联系，以同伦理论证明著名的定理为目标，如Faltings或Wiles的定理，并最终在Serge Lang关于双曲变的猜想中获得新的高维结果。几何和算术之间的深刻关系是一个值得尊敬的研究主题，至少可以追溯到古希腊的尺子和指南针构造特殊数字。这种传统的现代表现形式是算术几何，这是一个数学领域，在上个世纪产生了一些最深刻的数学结果。本文描述了利用线性和非线性几何界面的思想在代数方程理论中获得新结果的几个相关思想。

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.