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.

Langlands计划于1960年代发起，是一组猜想，预测了数字理论和群体代表理论的统一。在过去20年中，该计划以其经典形式的众多发展带来了显着的后果，例如Fermat的最后定理证明，以及Serre的模块化猜想之一。在2000年左右，这些猜想的P-ADIC/MOD P版本的问题是由P-Adic算术几何形状的自然问题提出的。由于出乎意料且了解不足的现象涉及GL_2（Q_P）以外的其他群体，因此一般P-Adic/Mod P局部Langlands猜想的陈述仍然难以捉摸。但是，对于GL_2（Q_P），P。Colmez和V. paskunas（并基于许多其他工作）建立了对应关系，例如大多数Fontaine-Mazur猜想的证明（由M. Kisin，M。Kisin，M。Emerton）证明。对于更多的一般群体，仍将取得进展，并探索本地兰兰兹计划的模式方面是一种有前途的方法。在先前的工作中，PI强调了特定类别的Hecke模块的作用，并证明了M.-F.猜想的P-ADIC GL_N的“数值mod p langlands对应”。小葡萄酒。这是Mod P Langlands风味的第一个结果，它立即涉及所有P-ADIC通用线性基团，而无需任何级别。该提案旨在研究P-ADIC还原组和相关Hecke代数的MOD P表示理论。该提议的核心是希望在潜在的Mod P Langlands对应的正确条款上发出几何灯光。 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 reduced group G over a p-adic field F. This project is fused 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在PI的监督下，通过PI对所有超词的Hecke模块C/K. Koziol进行的工作，建立了SL（N，F）的Hecke模块包的Langlands通信，与GL（N，F）以及Grosse-klönne的futlotor兼容。基于此，现在可以在Hecke模块开始的兰格兰通信中探索功能性的模式P原理的形式。 2•以参数为零的仿射Hecke代数的表示理论。这是由PI与P. Schneider合作的先前工作激励的，该工作探讨了Pro-p Iwahori Hecke代数的同源特性（同源尺寸，双重函数）。由于GL（2，Q_P）和Grosse-Klönne的作品的模型，预计函数将由函数提供Mod P Langlands的信件，尽管仍不清楚要考虑的相关类别是什么。在Hecke模块研究中引入非交通性几何形状的工具将有助于阐明这一关键点。 3•在1•和2中翻译有关Hecke模块的工作，以P-ADIC G组G的Mod P表示理论及其与Galois表示的链接。 Hecke模块和G表示之间的联系比在复杂表示的设置中更微妙，应涉及派生类别。提案中概述了这种投资的策略。这个遥不可及的问题最终可能与几何satake同构的mod P版本有关。