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联系。这些问题包括斯塔克猜想的情况下，L-函数的数域，和伯奇和Swinnerton-Dyer猜想的情况下，椭圆曲线和阿贝尔品种。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-adic和分析工具，来研究这些问题的各个方面。特别要研究的问题包括分布的塞尔默行列在家庭的二次扭曲的椭圆曲线，完善类数公式在一些领域，以及更高的排名Kolyvagin系统。在以前的工作调查Kolyvagin系统已被证明是一个非常有用的工具，有关L-值和算术。椭圆曲线和阿贝尔品种发挥了核心作用，在许多部分的数学，包括其最应用领域。例如，椭圆曲线用于算法中以加密数据以进行传输，以及用于有效的数字签名。椭圆曲线的最基本形式是一种特殊的二元多项式方程。从历史上看，数论家对这些方程的解很感兴趣，其中变量的值要么是整数，要么是分数。椭圆曲线的秩是一个基本不变量，它度量了解集的大小。研究者和他的同事们研究椭圆曲线的行列及其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.