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化素数上Shimura变种的积分模型的构造。第二部分主要研究了具有有限群作用的算术簇的上同调中的积分表示，它与L函数的特定值、分圆域的类群理论以及在确定模格格的几何Galois模结构中的应用有关。这项研究属于被称为算术几何的现代数论领域，在该领域中，几何方法被用于解决数论中的问题。数论起源于对自然数的研究。它是最古老的数学分支之一。在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

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.