Mathematical Sciences: Arithmetic of L-Values and Iwasawa Theory

该奖项支持以岩泽理论和赫克·L函数的特殊价值为中心的研究。这个项目涉及关于这些特殊值的三个问题，前两个问题与Beilinson和Bloch-Kato提出的猜想有关。这些猜想揭示了动机L函数的特殊值与动机的算术性质之间的内在联系。本研究的第一部分讨论CM椭圆曲线上的Hecke L函数在临界值的Bloch-Kato猜想。第二部分讨论了贝林森猜想在黑克-L函数的中心临界值上的问题。第三部分研究了岩泽理论中的一个可能的对偶性质，它是由p-进Galois表示的广义Tate对偶引出的。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效，最近解决了几代人经受住的问题。在其众多后果中，有一种是新的纠错码。这种代码对于现代计算机(硬盘)和光盘都是必不可少的。

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.