Euler Systems and Elliptic Curves

研究人员和他的同事研究椭圆曲线、L函数的特殊值以及相关课题。算术代数几何中一些最有趣的公开问题和猜想与L函数的特殊值有关。岩泽理论和欧拉系统已被证明是解决这些问题的最成功的工具之一，它们充当了代数世界和分析世界之间的桥梁。研究人员继续采用这种方法，从欧拉系统及其“衍生”的Kolyvan in系统中提取更多信息。在另一个方向，研究人员和他的同事研究椭圆曲线的阶数，特别是二次扭曲族中阶数的渐近分布。椭圆曲线在数学的许多部分，包括它最应用的领域中扮演着中心角色。例如，椭圆曲线被用在算法中来加密数据以供传输，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，数论者感兴趣的是寻找这些方程的解，在这些方程中，变量的值要么是整数，要么是分数。椭圆曲线的矩是衡量解集合大小的一个基本不变量。这位研究人员和他的同事研究了椭圆曲线的阶数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，椭圆曲线是通过考虑变量在有限域中取值的解而产生的。

The investigator and his coworkers study elliptic curves, special valuesof L-functions, and related topics. Some of the most interesting openquestions and conjectures in arithmetic algebraic geometry relatearithmetic invariants with special values of L-functions. Iwasawa theoryand Euler systems have proved to be among the most successful tools forattacking these problems, by serving as a bridge between the algebraicworld and the analytic world. The investigator continues to pursue thisapproach, by extracting additional information from Euler systems andtheir "derivative" Kolyvagin systems. In another direction, theinvestigator and his colleagues study ranks of elliptic curves,especially the asymptotic distribution of ranks in families of quadratictwists.Elliptic curves play a central role in many parts of mathematicsincluding its most applied areas. For example, elliptic curves are usedin algorithms to encrypt data for transmission, and for efficient digitalsignatures. In its most basic form, an elliptic curve is a special kindof polynomial equation in two variables. Historically number theoristsare interested in finding solutions of these equations in which thevariables take values which are either whole numbers, or fractions. Therank of an elliptic curve is a basic invariant which measures the size ofthe set of solutions. The investigator and his coworkers study ranks ofelliptic curves and their interrelations with other mathematical objectsand concepts. These questions are related to the cryptographicapplications of elliptic curves, which come about by consideringsolutions in which the variables take values in finite fields.