Geometric Langlands Program and Arithmetic Algebraic Geometry

几何朗兰兹纲领和算术代数几何

• 批准号: 1602092
• 负责人: Xinwen Zhu
• 金额: $ 39.98万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1602092&HistoricalAwards=false
• 关键词: Geometric; Langlands; Program; Arithmetic; Algebraic

The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and linear algebra. The traditional arithmetic Langlands program has been studied for more than fifty years, and this research has resulted in significant applications to solving classical Diophantine equations, for example, the proof of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development thanks to powerful tools from algebraic geometry. In this project, the investigator will explore connections between these two different facets of the Langlands program by applying geometric methods to study arithmetic problems. In more detail, this is a project to study the geometric Langlands program and its applications to arithmetic geometry. The investigator will apply his results on the geometric Satake correspondence for p-adic groups and the p-adic Riemann-Hilbert correspondence to investigate the mod p and p-adic geometry of Shimura varieties. The investigator will also explore the relations between residues that appear in the theory of automorphic forms and the topology of spaces of rational maps.

朗兰兹纲领是一个数学框架，它统一了许多不同数学领域的问题，特别是数论和线性代数。传统的算术朗兰兹程序已经研究了50多年，这一研究在求解经典丢番图方程中有着重要的应用，例如费马最后定理的证明。几何朗兰兹计划，这是相对较新的，正在迅速发展，由于强大的工具，从代数几何。在这个项目中，研究人员将探索朗兰兹程序的这两个不同方面之间的联系，通过应用几何方法来研究算术问题。更详细地说，这是一个研究几何朗兰兹程序及其在算术几何中的应用的项目。调查员将应用他的结果对几何佐竹对应的p-adic组和p-adic黎曼-希尔伯特对应调查模p和p-adic几何志村品种。调查员也将探讨出现在理论的自守形式和拓扑空间的理性地图之间的关系。