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; 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该奖项将研究资助以下两个不同的项目。 （项目1）在1974年，S。Lang做出了一个猜想，该猜想连接了在数字字段上定义的代数变体的几何（双曲线），并具有该品种的算术（mordellicity）。猜想对于阿贝尔品种的曲线和亚地区是正确的。以其他品种而闻名。萨尔纳克（Sarnak）和王（Wang）教授表明，某些高空产生了违反哈萨（Hasse）原则的行为，这是brauer-manin阻塞或违反上述朗的猜想所无法解释的。 Wang教授将继续研究0学位的0个学位的问题。（项目2）已广泛研究了代数品种上理性点的密度。 Wang教授计划研究弱近似，Brauer-Manin障碍物以及修改后的Mazur对Hurwitz家族的理性观点拓扑的猜想，更普遍地是在Hurwitz空间上。由于每个品种都被Hurwitz的空间统一，因此在一般情况下，这项研究将阐明这些问题。 这项研究属于数量理论的一般数学领域。它涉及整体和概括的代数方程的解决方案。数字理论是数学最古老的分支之一，由于纯粹的审美原因而被追捕了许多世纪。但是，在过去的半个世纪中，它已成为数据传输和处理以及通信系统等领域的不同应用中必不可少的工具。