Selmer Groups, Euler Systems, and Rational Points on Curves

该项目涉及数论的一般领域的工作，在本提案中使用几何方法进行研究。代数变体--由方程定义的几何对象--在数学的许多部分，包括最应用的领域中起着核心作用。例如，称为椭圆曲线的特殊代数簇用于算法中，以加密数据进行传输和有效的数字签名。椭圆曲线的最基本形式是由某种二元多项式方程定义的曲线。从历史上看，数论学家一直对这些方程的解感兴趣，其中变量的值要么是整数，要么是分数。研究者将研究一些关于代数簇上的点的新问题，以及这些点与其他数学对象和概念之间的联系。数论中一些最基本和最重要的问题是关于簇上的有理点。这些问题包括与L-函数的联系，如伯奇和斯温纳顿-戴尔猜想。在这个项目中，研究人员计划使用许多不同的技术，包括代数，p-adic和分析工具，来研究这些问题的各个方面。一组要研究的问题包括细化类数公式数域和更高的秩Kolyvagin系统。在以前的研究工作中，Kolyvagin系统已被证明是一个非常有用的工具，用于关联L值和算术。 在另一个方向，研究人员计划使用塞尔默组的扭曲阿贝尔品种，研究如何设置合理的点在曲线上的变化时，基场增加。

This project concerns work in the general area of number theory, which is studied in this proposal using methods from geometry. Algebraic varieties -- geometric objects defined by equations -- play a central role in many parts of mathematics, including its most applied areas. For example, special algebraic varieties called elliptic curves are used in algorithms to encrypt data for transmission and for efficient digital signatures. In its most basic form, an elliptic curve is a curve defined by a certain type of polynomial equation in two variables. Historically number theorists have been interested in finding solutions of these equations in which the variables take values that are either whole numbers or fractions. The investigator will study some new questions about points on algebraic varieties, and the connections between these points and other mathematical objects and concepts.Some of the most basic and important questions in number theory are about rational points on varieties. These questions include connections with L-functions, such as the Birch and Swinnerton-Dyer conjecture. In this project the investigator plans to use many different techniques, including algebraic, p-adic, and analytic tools, to study various aspects of these questions. One set of questions to be studied includes refined class number formulas over number fields and higher rank Kolyvagin systems. In previous work of the investigator, Kolyvagin systems have proved to be a very useful tool for relating L-values and arithmetic. In another direction, the investigator plans to use Selmer groups of twists of abelian varieties to study how the set of rational points on a curve changes when the base field is increased.