Mathematical Sciences: Arithmetic of L-Values and Iwasawa Theory

该奖项支持以Iwasawa理论和Hecke l函数的特殊值为中心的研究。这个项目涉及关于这些特殊值的三个问题，其中前两个与Beilinson和Bloch-Kato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了CM椭圆曲线上Hecke l函数的临界值处的Bloch-Kato猜想。第二部分讨论了在Hecke l -函数中心临界值处的Beilinson猜想。第三部分探讨了由p进伽罗瓦表示的广义Tate对偶引起的Iwasawa理论中可能的对偶性质。这个项目属于算术几何的一般领域，它融合了数学中两个最古老的领域：数论和几何。事实证明，这种结合非常富有成效，最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.