Shimura Varieties, Galois Modules and L-functions

本课题分为两部分：第一部分是关于非光滑约简素数上的Shimura变量的积分模型的构造。第二部分研究了有限群作用下算术变量上同调中出现的积分表示；与l函数的特殊值有联系，与环切场的类群理论有联系，并应用于确定模形式格的几何伽罗瓦模结构。这项研究属于数论的一个现代领域，称为算术几何，其中几何方法被用来解决数论中的问题。数论的历史根源在于对自然数的研究。它是数学最古老的分支之一。在过去的半个世纪中，它已成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

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.