Analytic Number Theory on Groups

群的解析数论

• 批准号: 0098633
• 负责人: Dorian Goldfeld
• 金额: $ 19.2万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098633&HistoricalAwards=false
• 关键词: Analytic; Number; Theory; Groups

This award supports the research of Dorian Goldfeld in thestudy of discrete groups with applications to analytic numbertheory. The proposed research concerns various open problemsrelated to: the distribution of additive characters onFuchsian groups, the ABC-conjecture, higher rank Eisensteinseries twisted by Ash-Borel modular symbols, the Waldspurgercorrespondence for cubic covers of GL(2), and the geometricperiods associated to derivatives of L-series.Number theory has its historical roots in the study of wholenumbers, and is among the oldest branches of mathematics. Diophantus, of the third century, proposed many problems inhis arithmetic, requiring the solutions to be whole numbers.The study of integer solutions to equations is now referredto as Diophantine analysis, and has many applications tocomputational complexity, data transmission, signalprocessing, cryptography, etc. Professor Goldfeld introducesand studies new types of geometrical L-series (infinite sumsof geometric objects depending on a complex parameter) inorder to develop a novel general method to attack a largeclass of hitherto still unsolved Diophantine problems.

该奖项支持Dorian Goldfeld在离散群研究及其在分析数论中的应用方面的研究。本文研究了fuchsian群上加性特征的分布、abc猜想、Ash-Borel模符号扭曲的高阶Eisensteinseries、GL(2)的三次覆盖的Waldspurgercorrespondence、l级数导数的几何周期等开放问题。数论在对整数的研究中有其历史根源，是数学中最古老的分支之一。公元三世纪的丢番图在他的算术中提出了许多问题，要求解必须是整数。对方程整数解的研究现在被称为丢芬图分析，在计算复杂性、数据传输、信号处理、密码学等方面有许多应用。戈德菲尔德教授介绍和研究了几何l系列的新类型（依赖于复杂参数的几何对象的无限和），以开发一种新的通用方法来解决迄今为止仍未解决的丢芬图问题。