Mathematical Sciences: Arithmetic of L-Values and Iwasawa Theory

数学科学：L 值算术和岩泽理论

• 批准号: 9525833
• 负责人: Li Guo
• 金额: $ 4万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9525833&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Values; Iwasawa

This award supports research centered on Iwasawa theory and special values of Hecke L-functions. This project involves work on three questions regarding these special values, the first two of which are concerned with the conjectures made by Beilinson and Bloch-Kato. These conjectures reveal an intrinsic connection between the special values of a motivic L-function and the arithmetic properties of the motive. The first part of this research deals with the Bloch-Kato conjecture at critical values of Hecke L-functions from CM elliptic curves. The second part addresses the Beilinson conjecture at the central critical values of Hecke L-functions. The third part explores a possible duality property in Iwasawa theory induced by the generalized Tate duality for p-adic Galois representations. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful, having recently solved problems that have withstood generations. Among its many consequences are new error-correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

该奖项支持以岩泽理论和Hecke L函数的特殊值为中心的研究。这个项目涉及三个问题的工作，这些特殊的价值观，其中前两个是关于贝林森和布洛赫-加藤所作的说明。这些结构揭示了动机L-函数的特殊值与动机的算术性质之间的内在联系。本文的第一部分研究了CM椭圆曲线上Hecke L-函数在临界值处的Bloch-Kato猜想。第二部分讨论了Hecke L-函数中心临界值处的Beilinson猜想。第三部分探讨了由p-adic Galois表示的广义Tate对偶引起的岩泽理论中的一个可能的对偶性质。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。事实证明，这种结合非常富有成效，最近解决了几代人都无法解决的问题。它的许多结果之一是新的纠错码。这种代码对于现代计算机（硬盘）和光盘都是必不可少的。