Euler Systems

* * 卡尔·鲁宾斯坦福大学98 00881 这个项目涉及椭圆曲线算术性质的研究。椭圆曲线在数学的许多部分中起着核心作用，包括其最应用的领域。 研究人员感兴趣的是某些点如何出现在这条曲线上，特别是那些可以用简单分数精确定位的点。 分数涉及到我们在学校里学过的整数运算，所以这些点体现了曲线的算术方面。 为了得到这个算法，鲁宾教授将使用所有关于椭圆曲线的几何和分析信息。 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。 从这个项目中获得的洞察力将在很长一段时间内指导数学家。

************************************************************************************ ABSTRACT Karl Rubin Stanford 98 00881 This project involves the study of the arithmetic properties of elliptic curves. Elliptic curves play a central role in many parts of mathematics including its most applied areas. The investigator is interested in how certain points appear on this curve, in particular those points that can be pinpointed using simple fractions. Fractions involve the integer arithmetic we all learn in school, so these points embody the arithmetic aspect of the curve. To get at this arithmetic, Professor Rubin will use all the geometric and analytic information about elliptic curves available. 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. The insight gained from this project will guide mathematicians for a long time to come.