Mathematical Sciences: Regulators, Class Groups and Iwasawa Theory

本研究将集中于三个重要问题 在数论中。 第一个将是锐化下限 以数字字段的调节器而闻名。 第二个将是 研究了Frobenius映射作用于 有限域上超椭圆曲线的雅可比行列式， 研究这种除数类群的分布， 曲线. 这与中的科恩-伦斯特拉猜想有关 数字字段。 第三个问题是纯环的研究 可能决定岩泽消失的理论属性 μ不变 本研究将从数量上考虑各种问题 理论 数论是研究自然界中 整数 提议者带来了各种各样的工具， 进行这项研究。 利用分析和几何学研究 代数数中一个重要不变量的下界 理论，调节器，将进行研究。 的研究 除数类群的分布涉及对 关于这些几何物体的概率。 上的问题 岩泽号大量使用了代数技巧。 的 原则对这些不同的工具和意志有着坚定的控制力 对这个理论有很大的贡献。

This research will concentrate on three important problems in number theory. The first will be to sharpen the lower bounds known for regulators of number fields. The second will be the study of the distribution of the Frobenius map acting on the Jacobians of hyperelliptic curves over a finite field and thereby study the distribution of the divisor class groups of such curves. This is connected with the Cohen-Lenstra conjecture in number fields. The third problem is the study of the purely ring theoretic properties that may govern the vanishing of the Iwasawa mu-invariant. This research will consider various problems in number theory. Number theory is the study of the properties of the integers. The proposer brings to bear a wide variety of tools to perform this research. Using analysis and geometry the study of lower bounds of one of the crucial invariants in algebraic number theory, the regulator, will be studied. The study of the distribution of divisor class groups involves a study in probability concerning these geometric objects. The problem on the Iwasawa involves a heavy use of algebra techniques. The principle has a firm command of these diverse tools and will contribute a great deal to the theory.