Mod p Langlands program for p-adic groups and Hecke algebras

p-adic 群和 Hecke 代数的 Mod p Langlands 程序

• 批准号: RGPIN-2014-04005
• 负责人: Ollivier, Rachel
• 金额: $ 1.02万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566233
• 关键词: Mod; Langlands; program; adic; groups

The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and of representation theory of groups. The numerous developments of this program under its classical form in the last 20 years have had remarkable consequences such as the proof of Fermat's last theorem, and the one of Serre's modularity conjecture. Around 2000, the question of a p-adic/mod p version of these conjectures was raised, motivated by natural questions of p-adic arithmetic geometry. Because of unexpected and poorly understood phenomena involving groups other than GL_2(Q_p), statements of a general p-adic/mod p local Langlands conjecture remain elusive. For GL_2(Q_p) however, a correspondence has been established by P. Colmez and V. Paskunas (and based on the work of many others) with spectacular consequences such as the proof of most cases of the Fontaine-Mazur conjecture (by M. Kisin, M. Emerton). Progress remains to be made for more general groups and exploring the mod p aspect of the local Langlands program is a promising approach. In previous work, the PI highlighted the role of a certain category of Hecke modules and proved the "numerical mod p Langlands correspondence for Hecke modules" for p-adic GL_n conjectured by M.-F. Vigneras. This was the first result with a mod p Langlands flavor involving at once all p-adic general linear groups without any restriction on the rank. The proposal aims at studying the mod p representation theory of p-adic reductive groups and associated Hecke algebras. At the heart of this proposal is the wish to shed a geometric light on the right terms of a potential mod p Langlands correspondence. There are 3 main proposed directions of research: 1 • Explore the possibility of a mod p Langlands correspondence for Hecke modules for a general p-adic reductive group G over a p-adic field F. This project is motivated by a/ recent progress by E. Grosse-Klönne who constructed a functor from Hecke modules to a category of objects that should encode information about certain representations of the absolute Galois group of F b/ the classification by the PI of all supersingular Hecke modules c/the work by K. Koziol, under the supervision of the PI, establishing a Langlands correspondence for packets of Hecke modules for SL(n,F), which is compatible with the one for GL(n,F) and with Grosse-Klönne's functor. Based on this, it is now possible to explore a form of mod p principle of functoriality, in the context of a Langlands correspondence for Hecke modules to start with. 2 • Formalize the representation theory of affine Hecke algebras with parameter zero. This is motivated by previous work of the PI in collaboration with P. Schneider that explores the homological properties (cohomological dimensions, duality functor) of pro-p Iwahori Hecke algebras. Because of the model of GL(2,Q_p) and of Grosse-Klönne's work, it is expected that the mod p Langlands correspondence will be given by a functor though it is still unclear what are the relevant categories to consider. Introducing tools from noncommutative geometry in the study of Hecke modules will contribute to clarifying this crucial point. 3 • Translate the work on Hecke modules in 1 • and 2 • in terms of the mod p representation theory of the p-adic group G and its link to Galois representations. The link between Hecke modules and representations of G is more subtle than in the setting of complex representations and should involve derived categories. Strategies towards such investigations are outlined in the proposal. This far reaching question could eventually be related to a mod p version of a geometric Satake isomorphism.

朗兰兹纲领发起于 20 世纪 60 年代，是一组预测数论和群表示论统一的猜想。过去 20 年来，该程序在其经典形式下的众多发展产生了显着的后果，例如费马大定理的证明和塞尔模性猜想的证明。 2000 年左右，受 p-adic 算术几何的自然问题的推动，提出了这些猜想的 p-adic/mod p 版本的问题。由于涉及 GL_2(Q_p) 以外的群的意外且知之甚少的现象，一般 p-adic/mod p 局部朗兰兹猜想的陈述仍然难以捉摸。然而，对于 GL_2(Q_p)，P. Colmez 和 V. Paskunas（并基于许多其他人的工作）建立了对应关系，并产生了惊人的结果，例如 Fontaine-Mazur 猜想的大多数情况的证明（由 M. Kisin、M. Emerton）。对于更一般的群体来说，仍有待取得进展，探索当地朗兰兹计划的 mod p 方面是一个有前途的方法。在之前的工作中，PI 强调了某一类 Hecke 模的作用，并证明了 M.-F 猜想的 p-adic GL_n 的“Hecke 模的数值 mod p Langlands 对应关系”。维涅拉斯。这是第一个具有 mod p Langlands 风格的结果，同时涉及所有 p-adic 一般线性群，对等级没有任何限制。该提案旨在研究 p-adic 约简群的 mod p 表示理论以及相关的 Hecke 代数。该提案的核心是希望阐明潜在 mod p Langlands 对应关系的正确项的几何意义。主要提出了 3 个研究方向： 1 • 探索 p 进场 F 上的一般 p 进约化群 G 的 Hecke 模的模 p 朗兰兹对应的可能性。该项目的动机是 E. Grosse-Klönne 的最新进展，他从 Hecke 模到一类对象构建了一个函子，该对象应编码有关 F b/ 的绝对伽罗瓦群的某些表示的信息。 PI 对所有超奇异 Hecke 模块的分类 c/K. Koziol 的工作，在 PI 的监督下，为 SL(n,F) 的 Hecke 模块包建立朗兰兹对应关系，该对应关系与 GL(n,F) 的对应关系以及 Grosse-Klönne 函子兼容。基于此，现在可以在 Hecke 模的 Langlands 对应关系的背景下探索一种形式的 mod p 函子性原理。 2 • 形式化参数为零的仿射Hecke 代数的表示论。这是由 PI 之前与 P. Schneider 合作的工作推动的，该工作探索了 pro-p Iwahori Hecke 代数的同调性质（上同调维数、对偶函子）。由于 GL(2,Q_p) 模型和 Grosse-Klönne 的工作，预计 mod p Langlands 对应关系将由函子给出，尽管尚不清楚要考虑的相关类别是什么。在赫克模的研究中引入非交换几何的工具将有助于阐明这一关键点。 3 • 根据p 进群G 的mod p 表示理论及其与Galois 表示的联系，翻译1 • 和2 • 中关于Hecke 模的工作。 Hecke 模块和 G 的表示之间的联系比复杂表示的设置更微妙，并且应该涉及派生类别。提案中概述了此类调查的策略。这个影响深远的问题最终可能与几何 Satake 同构的 mod p 版本有关。