Variation of Selmer groups

数论和算术代数几何中一些最有趣和最重要的开放问题涉及L函数的特殊值及其与算术的关系。这些问题包括在数域的L函数的情况下的斯塔克猜想，以及在椭圆曲线和阿贝尔簇的情况下的Birch和Swinnerton-Dyer猜想。在这个项目中，研究人员和他的同事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。要研究的特殊问题包括椭圆曲线的二次扭曲族中Selmer阶数的分布，数域上的精化类数公式，以及高阶Kolyvan系统。在这位研究者以前的工作中，柯利威林系已被证明是将L值与算术联系起来的一个非常有用的工具。椭圆曲线和阿贝尔变元在数学的许多部分，包括它最应用的领域，发挥着核心作用。例如，椭圆曲线在算法中用于加密要传输的数据，并用于高效的数字签名。在其最基本的形式中，椭圆曲线是一种特殊的二元多项式方程。从历史上看，数学家感兴趣的是找到这些方程的解，在这些方程中，变量的值要么是整数，要么是分数。椭圆曲线的秩是衡量解集合大小的基本不变量。这位研究人员和他的同事研究了椭圆曲线的等级以及它们与其他数学对象和概念的相互关系。

Some of the most interesting and important open questions in number theory and arithmetic algebraic geometry involve special values of L-functions and their connections with arithmetic. These questions include Stark's conjecture in the case of L-functions of number fields, and the Birch and Swinnerton-Dyer conjecture in the case of elliptic curves and abelian varieties. In this project the investigator and his colleagues plan to use many different techniques, including algebraic, p-adic, and analytic tools, to study various aspects of these questions. Particular questions to be studied include the distribution of Selmer ranks in families of quadratic twists of elliptic curves, 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.Elliptic curves and abelian varieties play a central role in many parts of mathematics including its most applied areas. For example, 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 special kind of polynomial equation in two variables. Historically number theorists are interested in finding solutions of these equations in which the variables take values which are either whole numbers, or fractions. The rank of an elliptic curve is a basic invariant which measures the size of the set of solutions. The investigator and his coworkers study ranks of elliptic curves and their interrelations with other mathematical objects and concepts.