On the p-adic Langlands program

关于 p-adic 朗兰兹纲领

• 批准号: RGPIN-2018-05741
• 负责人: Herzig, Florian
• 金额: $ 2.99万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688331
• 关键词: adic; Langlands; program

Number theory is the study of whole numbers and the solvability of equations in whole numbers. It is one of the oldest branches of mathematics. One particularly famous problem is Fermat's Last Theorem from 1637, which says that the sum of two n-th powers of non-zero whole numbers cannot be the n-th power of a non-zero whole number, as soon as n is at least 3. It was only solved around 1994 by Wiles and Taylor who established a deep connection between elliptic curves (objects of geometry) and modular forms (objects of analysis and the theory of symmetries). This deep connection is a special instance of the Langlands Program, which consists of a number of very general and interlinked conjectures that, in particular, help to explain many phenomena in number theory.******My work concerns a p-adic generalisation of the Langlands program which has been attracting a lot of interest in recent years. So far, it has only been properly understood in the simplest possible case (dimension 2). This alone has led to the solution of problems in number theory that previously seemed out of reach. In the proposed work I aim to shed light on the n-dimensional case of the p-adic Langlands program for n > 2. In particular, I propose to study the global candidate representations that arise in the cohomology of Shimura varieties, as well as their locally analytic vectors.

数论是研究整数和整数方程的可解性的学科。它是数学最古老的分支之一。一个特别著名的问题是1637年的费马大定理，它说只要n至少是3，两个非零整数的n次幂之和就不可能是一个非零整数的n次幂。直到1994年左右，怀尔斯和泰勒才解决了这个问题，他们在椭圆曲线（几何对象）和模形式（分析对象和对称性理论）之间建立了深刻的联系。这种深层联系是朗兰兹纲领的一个特殊例子，朗兰兹纲领由许多非常普遍和相互关联的理论组成，特别有助于解释数论中的许多现象。我的工作涉及一个P-进概括的朗兰兹计划已吸引了大量的兴趣，在最近几年。到目前为止，它只在最简单的可能情况下（维度2）得到了正确的理解。仅这一点就导致了以前似乎遥不可及的数论问题的解决。在拟议的工作中，我的目标是阐明n维情况下的p进朗兰兹计划n > 2。特别是，我建议研究的全球候选代表志村品种的上同调，以及他们的本地解析向量。