Topics in Arithmetic Algebraic Geometry

算术代数几何专题

• 批准号: 0201691
• 负责人: Shou-wu Zhang
• 金额: $ 51.33万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201691&HistoricalAwards=false
• 关键词: Topics; Arithmetic; Algebraic; Geometry

The investigator and his collaborators study the following topics in arithmetic algebraic geometry:the uniformity of small points for family of heights,the arithmetic Kodaira-Spencer class for arithmetic surfaceswith applications to the effective Mordell conjecture, the Gross-Zagier type formula for Shimura varietieswith applications to the Birch and Swinnerton-Dyer conjecture andthe Andre-Oort conjecture.This is a project in a subfield of mathematics known asnumber theory. Many of these questions are motivated by the philosophy that algebraic information can be obtained by geometric methods. At the center of this projectis the use of a symmetric group, or algebraic group,which is an object that is both algebraic and geometricin nature. These symmetric groups arise naturally in physics and chemistry. It is not too ambitious to say that the solution to the problems in this proposal will one day affect research in cryptography,theoretical physics, and quantum computing.

研究者和他的合作者研究算术代数几何中的以下主题：高度族小点的均匀性、算术曲面的算术 Kodaira-Spencer 类及其在有效 Mordell 猜想中的应用、Shimura 簇的 Gross-Zagier 型公式及其应用伯奇和斯温纳顿-戴尔猜想以及安德烈-奥尔特猜想。这是数学子领域“数论”中的一个项目。其中许多问题都是由这样的哲学引发的：代数信息可以通过几何方法获得。该项目的核心是使用对称群或代数群，它是一个本质上既是代数又是几何的对象。这些对称基团在物理和化学中自然出现。毫不夸张地说，该提案中问题的解决方案有一天会影响密码学、理论物理和量子计算的研究。