Selmer Groups, Euler Systems, and Rational Points on Curves

这个项目涉及数论的一般领域的工作，在这项建议中使用几何方法进行研究。代数变体--由方程式定义的几何对象--在数学的许多部分中扮演着中心角色，包括它最适用的领域。例如，在算法中使用称为椭圆曲线的特殊代数变体来加密要传输的数据和进行有效的数字签名。在其最基本的形式中，椭圆曲线是由特定类型的两个变量的多项式方程定义的曲线。从历史上看，数论家一直对寻找这些方程的解感兴趣，在这些方程中，变量的值要么是整数，要么是分数。本文将研究关于代数簇上的点的一些新问题，以及这些点与其他数学对象和概念之间的联系。数论中一些最基本也是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Birch和Swinnerton-Dyer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-进和分析工具，来研究这些问题的各个方面。要研究的一组问题包括数域上的精化班数公式和更高等级的Kolyvan系统。在这位研究人员之前的工作中，科利瓦金系统已被证明是将L值与算术联系起来的一个非常有用的工具。在另一个方向上，研究人员计划使用阿贝尔变种的Selmer扭曲群来研究曲线上的有理点集合如何随着基场的增加而变化。

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.