The local Langlands correspondence in l-adic families

l-adic 家族中当地朗兰兹的对应

• 批准号: 1161582
• 负责人: David Ben-Zvi
• 金额: $ 13.6万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161582&HistoricalAwards=false
• 关键词: local; Langlands; correspondence; adic; families

The PI proposes several research projects related to the local Langlands correspondence, and in particular to the modular representation theory of general linear groups over p-adic fields. One goal of these projects is to address the question of extending the local Langlands correspondence to a correspondence that works on the level of families of Galois representations. This builds on previous work of the PI and Matthew Emerton that has had applications to the Langlands program for global fields; in particular it plays a role in Emerton's proof of the Fontaine-Mazur conjecture and is likely to be important for generalizations of this result. A long term goal of the project is essentially a "local theory of congruences" for automorphic representations of general linear groups.The proposed project has its roots in the Langlands program, a series of far-reaching conjectures that connect topics in harmonic analysis and representation theory to long-standing problems in number theory. The Langlands program is expect to shed light on questions in theoretical physics, and also has implications for the theory of elliptic curves. Advances in elliptic curves in particular could have considerable practical applications in cryptography and coding theory.

PI提出了几个与局部朗兰兹对应相关的研究项目，特别是p进域上一般线性群的模表示理论。这些项目的一个目标是解决将局部朗兰对应关系扩展到伽罗瓦表示的家族层面上的对应关系的问题。这建立在PI和Matthew Emerton之前的工作基础上，这些工作已经应用到Langlands项目的全球领域；特别是它在Emerton对Fontaine-Mazur猜想的证明中发挥了作用，并且可能对该结果的推广很重要。该项目的长期目标本质上是一般线性群的自同构表示的“局部同余理论”。拟议的项目植根于朗兰兹计划，这是一系列深远的猜想，将谐波分析和表示理论的主题与数论中的长期问题联系起来。朗兰兹计划有望阐明理论物理学中的问题，并对椭圆曲线理论产生影响。特别是椭圆曲线的研究进展在密码学和编码理论方面具有相当大的实际应用价值。