The local Langlands correspondence in l-adic families

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问题的扩展本地朗兰兹对应的对应工作的伽罗瓦表示的家庭的水平。 这是建立在以前的工作PI和马修埃默顿已应用到朗兰兹计划的全球领域，特别是它发挥了作用，埃默顿的证明方丹-马祖尔猜想，并可能是重要的推广这一结果。 该项目的一个长期目标本质上是一般线性群的自守表示的“局部同余理论”。拟议的项目源于朗兰兹计划，这是一系列意义深远的计划，将调和分析和表示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，也对椭圆曲线理论有影响。特别是椭圆曲线的进展可能在密码学和编码理论中有相当大的实际应用。

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.