FRG: Collaborative

FRG：协作

• 批准号: 0354382
• 负责人: Stephen Kudla
• 金额: --
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354382&HistoricalAwards=false
• 关键词: FRG; Collaborative

FRG collaborative Award with lead DMS-0354353 of Ono, Zhang and Kudla with Co-PI Yang. This project involves arithmetic geometry and number theory andfocuses on a systematic study of cycles on Shimura varieties and applications. The fascination of diophantine problems -- the study of whole numbersolutions of polynomial equations-- goes back to ancient times.The mathematical techniques developed in the last 50 year toattack such questions have lead to significant advances in our knowledgeof this subject, for example the proof of Fermat's last theorem.These same tools have, meanwhile, proved to be of great importancein cryptography, the construction of new algorithms for computer scienceand new error correcting codes for electronics.Shimura varieties are the geometric objectsassociated to systems of diophantine equations with a great degree of symmetry. Their diophantine properties have deep connection with many important parts of mathematics.An extensive study will be made of the arithmetic geometry ofcycles on Shimura varieties, with an emphasis on the interactionof their heights, arithmetic intersections and density propertieswith modular forms and special values of L-functions and theirderivatives. Applications will be made to Gauss's class numberproblem, equidistribution problems for cycles on Shimuravarieties, and the Andre-Oort conjecture, and the Tate and Bloch-Beilinsonconjectures. This collaborative project will takeadvantage of recent developments including nonvanishing propertiesof Fourier coefficients of modular forms,the theory of Borcherds forms and their connections with theta functions, and integral representations of Langlands L-functions. A long range goal of the project is to establish relationsbetween the height pairings, periods, and algebraic cycles and thederivatives of L-functions.

FRG合作奖与Ono，Zhang和Kudla的Lead DMS-0354353与Co-Pi Yang的合作奖。该项目涉及对Shimura品种和应用的系统研究的算术几何学和数量理论和集合。 The fascination of diophantine problems -- the study of whole numbersolutions of polynomial equations-- goes back to ancient times.The mathematical techniques developed in the last 50 year toattack such questions have lead to significant advances in our knowledgeof this subject, for example the proof of Fermat's last theorem.These same tools have, meanwhile, proved to be of great importancein cryptography, the construction of new algorithms for computer scienceand new error校正电子设备的代码。Shimura品种是与具有很高对称性的Diophantine方程系统相关的几何对象。他们的养生特性与数学的许多重要部分具有密切的联系。将对Shimura品种的算术几何形状进行广泛的研究，重点是其高度，算术相互作用和密度性质的相互作用，并具有指示性和特殊价值的Loctuntions and untunctions and untunctions and untunctions and'sdertiment and esterntiment and esterntiment and derdunciptions。将在高斯的类杂志问题上提出应用程序，Shimuravarieties周期的等分分配问题以及Andre-oort猜想，以及Tate和Bloch-BeilinsonConjoctures。 这个协作项目将对最新的发展（包括模块化形式的傅立叶系数，Borcherds形式的理论及其与Theta函数的联系以及Langlands L功能的整体表示）的最新发展。该项目的一个远距离目标是在高度配对，周期和代数周期和L功能的theverations之间建立关系。