Arithmetic of elliptic curves and abelian varieties

椭圆曲线和阿贝尔簇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。关于椭圆曲线和交换簇的一些最有趣和最重要的开放理论问题是Birch和Swinnerton-Dyer猜想以及其他关于Mordell-Weil群、Selmer群和L函数的秩的问题。在应用方面，一个基本问题是有限域上椭圆曲线上离散对数问题的难度。在这个项目中，研究人员和他的同事们计划使用许多不同的技术，包括代数、p进位和分析工具，来研究这些问题的各个方面。椭圆曲线和阿贝尔变异体在数学的许多部分，包括它最应用的领域中发挥着核心作用。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。他们还研究与椭圆曲线的密码应用直接相关的其他问题，这些问题是通过考虑变量在有限域中取值的解而产生的。

Elliptic curves and abelian varieties 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 theoretical questions about elliptic curves and abelian varieties are the Birch and Swinnerton-Dyer conjecture and other questions about ranks of Mordell-Weil groups, Selmer groups, and L-functions. On the applied side, a fundamental question is the difficulty of the discrete logarithm problem on elliptic curves over finite fields. 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.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. They also study other questions directly related to the cryptographic applications of elliptic curves, which come about by considering solutions in which the variables take values in finite fields.