Collaborative Research / FRG: Arakelov Theory and Modular Forms

合作研究/FRG：阿拉克洛夫理论和模块化形式

• 批准号: 0354436
• 负责人: Shou-wu Zhang
• 金额: $ 18.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354436&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Arakelov; Theory

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 合作奖由 Ono、Zhang 和 Kudla 的领导 DMS-0354353 以及联合 PI Yang 颁发。该项目涉及算术几何和数论，重点对志村品种的周期及其应用进行系统研究。丢番图问题（对多项式方程的整数解的研究）的魅力可以追溯到远古时代。过去 50 年为解决此类问题而开发的数学技术使我们对这一学科的认识取得了重大进展，例如证明费马最后定理。同时，这些相同的工具已被证明在密码学、计算机科学新算法的构建和电子学新纠错码中非常重要。志村簇是相关的几何对象具有高度对称性的丢番图方程组。它们的丢番图性质与数学的许多重要部分有着深刻的联系。将对 Shimura 簇上的圈的算术几何进行广泛的研究，重点是它们的高度、算术交集和密度性质与模形式和 L- 的特殊值的相互作用。函数及其导数。将应用于高斯的类数问题、Shimuravarieties 循环的等分布问题、Andre-Oort 猜想、Tate 和 Bloch-Beilinson 猜想。 该合作项目将利用最新的进展，包括模形式的傅立叶系数的非零特性、Borcherds 形式的理论及其与 theta 函数的联系，以及 Langlands L 函数的积分表示。该项目的长期目标是建立高度配对、周期、代数环与 L 函数的导数之间的关系。