Rational Points on Algebraic Varieties

代数簇上的有理点

• 批准号: 9700781
• 负责人: Lan Wang
• 金额: $ 6.61万
• 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: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700781&HistoricalAwards=false
• 关键词: Rational; Points; Algebraic; Varieties

Wang 9700781 This award funds research into the following two distinct projects. (Project 1) In 1974, S. Lang made a conjecture which connects the geometry (hyperbolicity) of an algebraic variety defined over a number field with the arithmetic (Mordellicity) of the variety. The conjecture is true for curves and subvarieties of Abelian varieties. Little is known for other varieties. Professors Sarnak and Wang have shown that some hypersurfaces yield either a violation of the Hasse principle which is not accounted for by the Brauer--Manin obstruction or a violation of the above Lang's conjecture. Professor Wang will continue to investigate this problem for 0-cycles of degree 1. (Project 2) The densities of rational points on algebraic varieties have been studied extensively. Professor Wang plans to study weak approximation, Brauer--Manin obstruction and the modified Mazur's conjecture on the topology of rational points on the Hurwitz families, and more generally, on Hurwitz spaces. Since every variety is uniformized by Hurwitz spaces, this research will shed light on these problems in the general case. This research falls into the general mathematical field of Number Theory. It concerns the solutions to algebraic equations in whole numbers and generalizations. Number theory is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century, it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

Wang 9700781 该奖项资助以下两个不同项目的研究。 （项目 1） 1974 年，S. Lang 提出了一个猜想，它将在数域上定义的代数簇的几何（双曲性）与该簇的算​​术（Mordelliity）联系起来。该猜想对于阿贝尔簇的曲线和子簇是正确的。其他品种知之甚少。 Sarnak 和 Wang 教授表明，某些超曲面要么违反了布劳尔-马宁障碍无法解释的哈斯原理，要么违反了上述朗猜想。王教授将继续研究1次0循环的问题。（项目2）代数簇上有理点的密度已经得到了广泛的研究。王教授计划研究弱逼近、Brauer-Manin 阻塞和修正的 Mazur 猜想关于 Hurwitz 族上的有理点拓扑，更一般地说，是 Hurwitz 空间上的有理点拓扑。由于每种变体都由赫尔维茨空间统一，因此这项研究将在一般情况下阐明这些问题。 这项研究属于数论的一般数学领域。它涉及整数代数方程的解和概括。数论是数学最古老的分支之一，几个世纪以来纯粹出于美学原因而被人们研究。然而，在过去的半个世纪里，它已成为数据传输和处理、通信系统等多种应用中不可或缺的工具。