Variation of Selmer Groups of Elliptic Curves

椭圆曲线 Selmer 群的变分

• 批准号: 0457481
• 负责人: Karl Rubin
• 金额: --
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457481&HistoricalAwards=false
• 关键词: Variation; Selmer; Groups; Elliptic; Curves

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.

椭圆曲线不仅在数论和算术代数几何中占有越来越重要的地位，而且在密码学及其相关应用中也占有越来越重要的地位。关于椭圆曲线的一些最有趣和最重要的开放问题是Birch和Swinnerton-Dyer猜想以及其他关于秩、Selmer群和l函数的问题。在这个项目中，研究者和他的同事计划研究椭圆曲线的秩在某些数域族中是如何变化的，特别是Iwasawa塔的子域和所有二次域族。调查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领域，包括应用最广泛的领域都起着中心作用。例如，椭圆曲线被用于加密传输数据的算法，以及高效的数字签名。椭圆曲线最基本的形式是一类特殊的双变量多项式方程。从历史上看，数论学家感兴趣的是找到这些方程的解，其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大小的基本不变量。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，它是通过考虑变量在有限域中取值的解来实现的。