Iwasawa Theory

华盛顿大学Ralph Greenberg教授98 00829格林伯格教授打算研究关于局部Galois上同调群、Selmer群和椭圆曲线的各种问题。根据他的计划，Galois上同调将提供关于Selmer群的信息。Selmer群是我们理解椭圆曲线算术的关键因素。格林伯格教授最终感兴趣的是椭圆曲线上那些可以用代数数字精确定位的点。代数数是可以从我们在学校里学到的整数有限地求出的数。塞尔默小组就是专门为研究这些问题而设计的。在一种观点中，椭圆曲线的形状就像一个甜甜圈的表面。在这个观点中，简单的曲线就像一个球体的曲面，一个没有洞的甜甜圈，而更复杂的曲线就像一个有几个洞的甜甜圈的曲面。简单曲线的算术性质自古以来就被广泛研究。更复杂的曲线具有如此严格的算术结构，以至于简单地计算一条曲线上的点是一个困难的研究问题。然而，单孔甜甜圈恰到好处。它们有一个丰富而神秘的结构，本项目将对其进行探索。其优美的数学结构使得椭圆曲线在信息的数字编码和通信安全等实际问题中非常有用。

ABSTRACT Ralph Greenberg University of Washington 98 00829 Professor Greenberg intends to study a variety of questions concerning local Galois cohomology groups, Selmer Groups and elliptic curves. Under his plan Galois cohomology will provide information about Selmer groups. Selmer groups are key ingredients in our understanding of the arithmetic of elliptic curves. Professor Greenberg is ultimately interested in those points on elliptic curves that can be pinpointed using algebraic numbers. Algebraic numbers are numbers that can be developed in a finite way from the integers we all learn in school. The Selmer Group is specifically designed to study these points. In one view, elliptic curves are shaped like the surface of a donut. In this view, simpler curves are like the surface of a sphere, a donut without a hole, and more complicated curves are like the surface of a donut with several holes. The arithmetical properties of the simple curves have been extensively studied since ancient times. The more complicated curves have such a restrictive arithmetic structure that simply counting the points on one is a difficult research problem. The one hole donuts are, however, just right. They have a rich and mysterious structure which this project will explore. Their beautiful mathematical structure makes elliptic curves very useful in such practical problems as the digital encoding of information and the security of communications.