Shimura varieties, Galois representations and Riemann-Roch theorems

Shimura 簇、Galois 表示和 Riemann-Roch 定理

• 批准号: 0802686
• 负责人: Georgios Pappas
• 金额: $ 15万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0802686&HistoricalAwards=false
• 关键词: Shimura; varieties; Galois; representations; Riemann

The principal investigator is working on the following three problems:(A) He is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction. In particular, he studies ``local models" for Shimura varieties and their relation with affine flag varieties for infinite dimensional groups and with deformation spaces of Galois representations. The motivation is to obtain information that can be used in the calculation of the Hasse-Weil zeta function of these varieties and in other number theoretic applications.(B) He is developing refined and functorial versions of the Grothendieck-Riemann-Roch theorem that would allow for the calculation of torsion information.(C) He is studying the representations that appear in the cohomology of arithmetic varieties with a finite group action.In particular, he continues his work on developing fixed point formulas for calculating invariants of such (integral) representations using two interconnected themes: the theory of cubic structures and the theory of central extensions of algebraic loop groups.The investigator's research is in the field of arithmetic algebraic geometry, a subject that blends two of the oldest areas of mathematics: the geometry of figures that can be defined by the simplest equations, namely polynomials, and the study of numbers. This combination has proved extraordinarily fruitful - having solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of certain polynomial equations that have many symmetries. There are connections with physics, the construction of error correcting codes and cryptography.

首席研究人员正在研究以下三个问题：（a）他试图以非平滑降低的数量来描述Shimura品种不可或缺的模型。特别是，他研究了Shimura品种的``本地模型''及其与无限尺寸群体和Galois表示的变形空间的仿射旗品种的关系。动机是为了获得可用于计算这些品种的Zeta Zeta功能的信息，并在其他数字的应用中进行了改进。 （c）他正在研究出现在算术品种的共同体中出现的表示形式，尤其是，他继续在开发有限的固定点公式来开发固定点的公式（以这种互构成的构造构建中，及其构造的理论：代数循环组。研究者的研究是在算术代数几何的领域，该主题融合了两个最古老的数学领域：数字的几何形状可以由最简单的方程式来定义，即最简单的方程，即，即多项式和数字的研究。事实证明，这种组合非常富有成果 - 解决了经验丰富的几代问题（例如``Fermat的最后一个定理''）。研究人员的工作主要集中于对某些具有许多对称性的某些多项式方程的研究。与物理学有联系，错误纠正码和密码术的构建。