Arithmetic Geometry and Galois Module Theory

调查员将研究一些不同的问题有关的算术交换品种和特殊价值的L-功能。他的方法将涉及开发新的方法，其起源于算术伽罗瓦模块理论。调查的主题包括：岩泽理论的阿贝尔品种，椭圆单位和anticyclotomic欧拉系统，和玉川数代数所产生的某些L-功能。他还将研究一些问题有关的等变代数几何和K-理论的品种在有限领域。研究中所描述的建议在于该领域的算术几何。这是一门融合了两个最古老的数学分支--数论和几何--的学科，它已经发展到解决了几个世纪以来一直存在的问题。它在物理学、机器人技术、数据处理和信息理论等不同的科学领域都有应用。

The investigator will study a number of different questions concerning the arithmetic of abelian varieties and special values of L-functions. His methods will involve developing new approaches that have their origins in arithmetic Galois module theory. The topics of investigation include: Iwasawa theory of abelian varieties, elliptic units and anticyclotomic Euler systems, and Tamagawa number conjectures arising via certain L-functions. He will also study some problems concerned with equivariant algebraic geometry and the K-theory of varieties over finite fields.The research described in this proposal lies in the field of arithmetic geometry. This is a subject that blends two of the oldest branches of mathematics--number theory and geometry-- and which has blossomed to the point where it has solved problems that have stood for centuries. It finds applications in fields of science as diverse as physics, robotics, data processing and information theory.