Shimura Varieties, Galois Modules and L-functions

点击翻译按钮获取中文摘要

This project has two parts: The first part concerns the construction of integral models of Shimura varieties over primes of non-smooth reduction. The second part focuses on the study of the integral representations that appear in the cohomology of arithmetic varieties with an action by a finite group; there are connections with special values of L-functions, with the theory of class groups of cyclotomic fields and applications to determining the geometric Galois module structure of lattices of modular forms. This research falls into a modern area of number theory known as arithmetic geometry in which geometric methods are used to solve problems in number theory. Number theory has its historical roots in the study of the natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.