Arithmetic Geometry and Galois Module Theory

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

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.