Variation of Selmer Groups of Elliptic Curves

椭圆曲线不仅在数论和算术代数几何中越来越重要，而且在密码学和相关应用中也越来越重要。关于椭圆曲线的一些最有趣和最重要的公开问题是Birch和Swinnerton-Dyer猜想以及关于秩、塞尔默群和L-函数的其他问题。在这个项目中，研究人员和他的同事计划研究椭圆曲线的秩在某些数域家族中如何变化，特别是岩泽塔的子域和所有二次域家族。研究将使用许多不同的技术，包括代数，p-adic和分析工具。椭圆曲线在数学的许多部分，包括其最应用的领域中发挥着核心作用。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。这些问题与椭圆曲线的密码学应用有关，椭圆曲线的密码学应用是通过考虑变量在有限域中取值的解来实现的。

Elliptic curves are increasingly important not only in number theory and arithmetic algebraic geometry, but also in cryptography and related applications. Some of the most interesting and important open questions about elliptic curves are the Birch and Swinnerton-Dyer conjecture and other questions about ranks, Selmer groups, and L-functions. In this project the investigator and his colleagues plan to study how the rank of an elliptic curve varies in certain families of number fields, especially the subfields of Iwasawa towers and the family of all quadratic fields. The investigation will make use of many different techniques, including algebraic, p-adic, and analytic tools.Elliptic curves 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. These questions are related to the cryptographic applications of elliptic curves, which come about by considering solutions in which the variables take values in finite fields.