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

• 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

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.

本项目涉及算术代数几何领域的研究。这是一个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)描述的形状的几何，以及对数字的研究。这些学科的结合已经被证明是非常有成效的，解决了历经几代人努力的挑战(例如最近费马大定理的证明)。该领域与物理有关，并在构造纠错码和密码学方面发现了重要的应用。本研究主要研究具有多种对称性的特定多项式方程。本项目旨在描述Shimura簇在非光滑约简的素数下的积分模型，并将研究相关的p-进空间。特别地，这项工作继续研究在这样的素数上的阿贝尔型下村变种的奇性。该项目旨在通过适当的Nelon扩张性质来刻画这些积分模型，并且在正交Shimura簇的情况下，显式地研究它们的约化的局部结构。本文还给出了某些Rapoport-Zink p-adi空间的积分模型的一般构造和群论定义，并在某些情况下完全刻画了它们的特殊纤维。此外，该项目还研究了具有有限群作用的整数上的簇的上同调中出现的表示，并旨在发展可用于计算等变欧拉特征的非常一般的不动点公式。最后，本项目利用K-理论的方法研究了岩泽理论的扩展构造；更具体地说，得到了岩泽代数的依赖于岩泽模的高阶余维素数的信息。

项目成果

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
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Chung, Hee-Joong;Kim, Dohyeong;Kim, Minhyong;Pappas, Georgios;Park, Jeehoon;Yoo, Hwajong
• 通讯作者: Yoo, Hwajong

Higher Chern classes in Iwasawa theory

• DOI: 10.1353/ajm.2020.0017
• 发表时间: 2015-12
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: F. Bleher;T. Chinburg;Richard Greenberg;M. Kakde;G. Pappas;R. Sharifi;M. Taylor

Integral models of Shimura varieties with parahoric level structure

• DOI: 10.1007/s10240-018-0100-0
• 发表时间: 2015-12
• 期刊: Publications mathématiques de l'IHÉS
• 作者: M. Kisin;G. Pappas

ARITHMETIC MODELS FOR SHIMURA VARIETIES

• DOI: 10.1142/9789813272880_0059
• 发表时间: 2018-04
• 期刊: Proceedings of the International Congress of Mathematicians (ICM 2018)
• 作者: G. Pappas

Cup products in the etale cohomology of number fields

数域 etale 上同调中的杯积

• 发表时间: 2018
• 期刊: New York journal of mathematics
• 影响因子: 0.6
• 作者: Bleher, F;Chinburg, T;Greenberg, G;Kakde, K: Pappas;Taylor, M.
• 通讯作者: Taylor, M.