Iwasawa Theory

摘要 拉尔夫·格林伯格华盛顿大学98 00829 格林伯格教授打算研究各种问题，有关当地伽罗瓦上同调群，塞尔默群和椭圆曲线。根据他的计划伽罗瓦上同调将提供有关塞尔默群体的信息。 塞尔默群是我们理解椭圆曲线算术的关键要素。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精确定位的点。 代数数是可以从我们在学校所学的整数中以有限的方式发展出来的数字。 塞尔默小组就是专门研究这些问题的。 在一个视图中，椭圆曲线的形状像甜甜圈的表面。 在这种观点中，简单的曲线就像一个球体的表面，一个没有孔的甜甜圈，而更复杂的曲线就像一个有几个孔的甜甜圈的表面。 简单曲线的算术性质自古以来就被广泛研究。 更复杂的曲线有这样一个限制性的算法结构，简单地计算一个点是一个困难的研究问题。 但是，一个洞的甜甜圈是正确的。他们有一个丰富而神秘的结构，这个项目将探索。椭圆曲线优美的数学结构使得它在诸如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.