Mathematical Sciences: Rational Points on Algebraic Varieties

数学科学：代数簇上的有理点

• 批准号: 9501057
• 负责人: James Milne
• 金额: $ 3.8万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501057&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Rational; Points; Algebraic

The main goal of this research is to study a conjecture of Mazur concerning the topology of rational points on a variety over the rational number field. Questions related to the Hasse Principle, weak approximation and the Brauer-Manin obstruction play a key role in this study. The approach depends on a detailed study of special properties of each variety under consideration and the construction of canonical height functions when applicable. This research is in the general mathematical area of Arithmetic Geometry. This ultramodern research area combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.

本研究的主要目的是研究Mazur关于有理数域上簇上有理点的拓扑的一个猜想。与Hasse原理、弱逼近和Brauer-Manin阻塞有关的问题在本研究中起着关键作用。该方法依赖于详细研究的特殊性质，每个品种的考虑和建设的典范高度函数时，适用。 本研究是在算术几何的一般数学领域。这个超现代的研究领域结合了数学的两个最古老的分支：数论和几何。这种结合产生的新见解正在产生越来越强大的工具来解决像费马大定理这样的长期存在的问题，这些问题已经抵制了三个多世纪数学家的最强努力。此外，虽然算术几何有时被认为是最纯粹的纯数学之一，但它也一直在开发有见地的新技术，从而在纠错码和密码学等应用领域取得了巨大进展。