Shimura Varieties, Galois Modules and L-functions

本课题分为两部分：第一部分是关于素数上非光滑约化的Shimura簇的积分模型的构造。第二部分着重于研究的积分表示，出现在上同调的算术品种的行动，由一个有限的群体;有连接的特殊价值的L-功能，理论的类群的分圆领域和应用，以确定几何伽罗瓦模块结构格的模块形式。 这项研究福尔斯属于现代数论领域称为算术几何，其中几何方法用于解决数论问题。 数论的历史根源在于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.