Arithmetic Geometry: Shimura Varieties, Galois Modules, and Iwasawa Theory

算术几何：志村簇、伽罗瓦模和岩泽理论

基本信息

批准号：
1701619
负责人：
Georgios Pappas
金额：
$ 8.4万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701619&HistoricalAwards=false
关键词：
Arithmetic Geometry Shimura Varieties Galois

项目摘要

This project concerns research in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest areas of mathematics: the geometry of shapes that can be described by the simplest equations, namely polynomials, and the study of numbers. This combination of disciplines has proved extraordinarily fruitful, having solved challenges (such as the recent proof of Fermat's last theorem) that had withstood generations of effort. The field has connections with physics, and has found important applications to the construction of error-correcting codes and cryptography. This research focuses on the study of specific polynomial equations that have many symmetries. This project aims to describe integral models for Shimura varieties at primes of non-smooth reduction and will study related p-adic spaces. In particular, the work continues investigation of the singularities of Shimura varieties of abelian type at such primes. The project aims to characterize these integral models via a suitable Neron extension property and, in the case of orthogonal Shimura varieties, explicitly study the local structure of their reductions. The project also intends to give a general construction and a group theoretic definition of integral models of certain Rapoport-Zink p-adic spaces and, in some cases, fully describe their special fibers. The project additionally studies the representations that appear in the cohomology of varieties over the integers with a finite group action and aims to develop very general fixed point formulae that can be used to calculate equivariant Euler characteristics. Finally, the project explores extending constructions of Iwasawa theory by employing K-theoretic methods; more specifically, obtaining information about the higher codimension primes of the Iwasawa algebra that lie on the support of an Iwasawa module.

该项目涉及算术代数几何领域的研究。这是一门融合了两个最古老的数学领域的学科：可以用最简单的方程（即多项式）描述的形状几何学和数字研究。事实证明，这种学科的结合是卓有成效的，解决了几代人努力所面临的挑战（例如费马大定理的最新证明）。该领域与物理学有联系，并在纠错码和密码学的构建中找到了重要的应用。本研究重点研究具有多种对称性的特定多项式方程。该项目旨在描述志村簇在非平滑归约素数处的积分模型，并将研究相关的 p-adic 空间。特别是，这项工作继续研究此类质数的阿贝尔型志村簇的奇点。该项目旨在通过合适的 Neron 扩展属性来表征这些积分模型，并且在正交 Shimura 簇的情况下，明确研究其还原的局部结构。该项目还打算给出某些 Rapoport-Zink p-adic 空间的积分模型的一般构造和群论定义，并在某些情况下充分描述其特殊纤维。该项目还研究了具有有限群作用的整数上的簇同调中出现的表示，旨在开发非常通用的定点公式，可用于计算等变欧拉特征。最后，该项目探索了利用K理论方法扩展岩泽理论的构造；更具体地说，获取有关 Iwasawa 模支持的 Iwasawa 代数的较高余维素数的信息。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Cup products in the etale cohomology of number fields

数域 etale 上同调中的杯积

DOI：
发表时间：
2018-01
期刊：
New York journal of mathematics
影响因子：
0
作者：
Bleher, F;Chinburg, T;Greenberg, G;Kakde, K: Pappas;Taylor, M.
通讯作者：
Taylor, M.

Arithmetic models for Shimura varieties

志村品种的算术模型

DOI：
10.1142/9789813272880_0059
发表时间：
2018-01
期刊：
Rio de Janeiro
影响因子：
0
作者：
Pappas; Georgios
通讯作者：
Georgios

Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers

阿贝尔算术 Chern-Simons 理论和算术连接数

DOI：
10.1093/imrn/rnx271
发表时间：
2017-11
期刊：
International Mathematics Research Notices
影响因子：
1
作者：
Chung, Hee;Kim, Dohyeong;Kim, Minhyong;Pappas, Georgios;Park, Jeehoon;Yoo, Hwajong
通讯作者：
Yoo, Hwajong

Good and semi-stable reductions of Shimura varieties

志村品种的良好和半稳定还原