Euler Systems

************************************************************************************ 抽象的卡尔·鲁宾斯坦福98 00881这个项目涉及到椭圆曲线的算术性质的研究。椭圆曲线在数学的许多领域，包括应用最广泛的领域都起着中心作用。研究者感兴趣的是曲线上的某些点是如何出现的，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都学过的整数运算，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解析信息。在一种观点中，椭圆曲线的形状就像甜甜圈的表面。在这个观点中，简单的曲线就像一个球体的表面，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表面。简单曲线的算术性质自古以来就被广泛研究。更为复杂的曲线具有如此严格的算法结构，以至于简单地计算一个点是一个困难的研究问题。然而，单孔甜甜圈是刚刚好。它们有着丰富而神秘的结构，本项目将对其进行探索。从这个项目中获得的洞察力将在未来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.