Shimura Varieties, Galois Modules and the Determinant of Cohomology

Shimura 簇、伽罗瓦模和上同调行列式

• 批准号: 0201140
• 负责人: Georgios Pappas
• 金额: $ 10.59万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201140&HistoricalAwards=false
• 关键词: Shimura; Varieties; Galois; Modules; Determinant

The investigator is working on the following two 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 PEL Shimura varieties and tries to formulate exact predictions about the non-smooth reduction of a general Shimura variety. 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 arithmetic applications. (B) He is studying the representations that appear in the cohomology of arithmetic varieties with a group action. He will continue his work on describing relations between invariants describing such representations and L-functions and develop a theory of ``non-abelian" cubic structures for the determinant of cohomology of general bundles over curves.This is research 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 provedextraordinarily fruitful - having recently solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of specific polynomial equations that have many symmetries. The general field has connections with physics, the construction of error correcting codes and cryptography.

研究人员正在研究以下两个问题：（a）他试图在非平滑降低的数量中描述Shimura品种的整体模型。 In particular, he studies ``local models"for PEL Shimura varieties and tries to formulate exact predictions about the non-smooth reduction of a general Shimura variety. 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 arithmetic applications. (B) He is studying the representations that appear in the cohomology of arithmetic varieties with a group动作。最简单的方程式，即多项式和数字研究。 事实证明，这种组合已被证明是富有成果的 - 最近解决了经验丰富的世代的问题（例如``Fermat的最后一个定理''）。研究者的工作主要集中于研究具有许多对称性的特定多项式方程。该一般领域与物理学有联系，纠正错误的编码和cryperpecraphy和crypecraphys的构建。