Shimura Varieties and Galois representations

PI将继续他的志村变种算法和应用的工作。一个目标是将PI最近的结果应用于这些变量上的mod p点，以计算它们的zeta函数。另一个目的是将其中一些结果扩展到坏约简的情况。一个可能的应用是关于阿贝尔变体上的de Rham-Tate环结构的结果。志村变量是一类几何对象，在过去50年数论的许多进展中发挥了非常重要的作用。它们可以被认为是阿贝尔变量的参数（或模）空间。本课题旨在研究志村变数的数论性质及其在阿贝尔变数算法中的应用。

The PI will continue his work on the arithmetic of Shimura varieties and applications. One goal is to apply the PI's recent results on the mod p points on these varieties to compute their zeta functions. Another aim is to extend some of these results to cases of bad reduction. One possible application is a result about the structure of de Rham-Tate cycles on abelian varieties.Shimura varieties are a class of geometric objects which have played a very important role in many advances in number theory during the past 50 years. They can be thought of as parameter (or moduli) spaces for abelian varieties. The project aims to study the number theoretic properties of Shimura varieties, and their applications to the arithmetic of abelian varieties.