Mod p local Langlands program for p-adic reductive groups and representations of Hecke algebras

p-进约简群的 Mod p 局部 Langlands 程序和 Hecke 代数的表示

• 批准号: 1201376
• 负责人: Rachel Ollivier
• 金额: $ 15万
• 依托单位: Columbia University
• 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: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201376&HistoricalAwards=false
• 关键词: Mod; local; Langlands; program; adic

The Langlands program, initiated in the 1960s and successfully developed in the last 15 years, is a set of conjectures predicting a unification of number theory and of representation theory of groups: the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa. A decade ago, the question of a p-adic/mod p component of this program was raised, motivated by natural questions of p-adic arithmetic geometry. As of now, only very few cases are understood, but they have already had spectacular consequences such as the proof of most cases of the Fontaine-Mazur conjecture. Because of unexpected and poorly understood phenomena, statements of a general p-adic/mod p local Langlands conjecture remain elusive. The proposal aims to study the mod p representations of p-adic reductive groups in order to understand the right terms of a mod p Langlands correspondence. The PI's previous work has shown that studying mod p representations of Hecke algebras is a promising approach, because it suggests that a mod p Langlands correspondence does exist. The proposal outlines strategies to study families and complexes of mod p representations of p-adic reductive groups and their associated Hecke algebras. At the heart of this proposal is the wish to go beyond the (so far most investigated) focus on irreducible objects and give a homological approach to the representations. It goes along with exploring the possibility of a mod p principle of functoriality and shedding a geometric light on the potential mod p Langlands correspondence. The p-adic/mod p Langlands program holds profound prospects for modern number theory with deep ramifications in arithmetic algebraic geometry. It is a new, fertile area involving tools and objects that were still completely abstruse a few years ago. The proposal aims to give a geometric incarnation to the objects that appear naturally in the mod p framework. The outcome of this work will most likely be naturally connected to other areas of mathematics influenced by the Langlands conjectures, such as the geometric Langlands program, and its links with geometric representation theory. The proposal describes strategies towards such connections.

朗兰兹程序始于20世纪60年代，在过去15年中得到了成功的发展，它是一组预测数论和群的表示理论统一的猜想：朗兰兹对应提供了一种用群论解释数论结果的方法，反之亦然。十年前，这个程序的p进/模p分量的问题被提出，是由p进算术几何的自然问题引起的。到目前为止，只有很少的案例被理解，但它们已经产生了惊人的结果，比如方丹-马祖尔猜想的大多数案例的证明。由于一些意想不到的和难以理解的现象，一般p进/模p局部朗兰兹猜想的陈述仍然是难以捉摸的。本文旨在研究p进约群的模p表示，以理解模p朗兰兹对应的正确项。PI之前的工作表明，研究Hecke代数的模p表示是一种很有前途的方法，因为它表明模p朗兰兹对应确实存在。该提案概述了研究p进约群及其相关Hecke代数的模p表示的族和复合体的策略。这个提议的核心是希望超越（迄今为止研究最多的）对不可约对象的关注，并给出表征的同调方法。同时探讨了模p泛函原理的可能性，并揭示了模p朗兰兹对应的潜在几何意义。p进/模p朗兰兹程序对现代数论具有深远的前景，在算术代数几何中具有深远的影响。这是一个新的、肥沃的领域，涉及到几年前仍然完全晦涩难懂的工具和物体。该提案旨在为在mod p框架中自然出现的物体赋予几何化身。这项工作的结果很可能会自然地与受朗兰兹猜想影响的其他数学领域联系起来，比如几何朗兰兹程序，以及它与几何表示理论的联系。该提案描述了实现这种联系的策略。