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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587546
• 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（英语：Langlands program），始于1960年代，是一套预言数论和群表示论统一的理论。在过去的20年里，这个程序在其经典形式下的许多发展产生了显着的后果，例如费马最后定理的证明和塞尔的模块性猜想。 在2000年左右，这些代数的p-adic/mod p版本的问题被提出，其动机是p-adic算术几何的自然问题。由于GL_2（Q_p）以外的群中存在着意想不到的现象，而且人们对这些现象的理解还很有限，因此对一般的p-adic/modp局部Langlands猜想的陈述仍然是难以捉摸的。 然而，对于GL_2（Q_p），P. Colmez和V. Paskunas（并基于许多其他人的工作）已经建立了一个对应关系，并产生了惊人的结果，如Fontaine-Mazur猜想的大多数情况的证明（M. Kisin，M. Emerton）。 对于更一般的群体，仍有待取得进展，探索当地朗兰兹计划的mod p方面是一种有前途的方法。 在以前的工作中，PI强调了某类Hecke模的作用，并证明了M. F.维格纳斯这是第一个结果与模p朗兰兹风味涉及一次所有p进一般线性群没有任何限制的排名。 该提案旨在研究p-adic约化群和相关Hecke代数的mod p表示理论。在这个建议的核心是希望摆脱一个潜在的模p朗兰兹对应的权利条款的几何光。 有三个主要的研究方向： 1 ·研究了p-adic域F上一般p-adic约化群G的Hecke模的模p Langlands对应的可能性。这个项目的动机是由E.格罗斯-Klönne谁建造了一个函子从Hecke模块的一类对象，应编码信息的某些表示的绝对伽罗瓦群的F B/分类的PI的所有超奇异Hecke模块c/工作的K。Koziol在PI的监督下，建立了SL（n，F）的Hecke模包的Langlands对应，它与GL（n，F）的对应和Grosse-Klönne函子兼容。在此基础上，现在可以在Hecke模的朗兰兹对应的背景下探索函数性的mod p原理的一种形式。 2 ·形式化了参数为零的仿射Hecke代数的表示理论。这是由PI与P. Schneider合作的先前工作激发的，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）的模型和Grosse-Klönne的工作，期望mod p朗兰兹对应将由函子给出，尽管仍然不清楚要考虑的相关范畴是什么。在Hecke模的研究中引入非交换几何的工具将有助于澄清这一关键点。 3.根据p-adic群G的mod p表示理论及其与伽罗瓦表示的联系，翻译1 ·和2 ·中关于Hecke模的工作。 Hecke模和G的表示之间的联系比在复表示的设置中更微妙，并且应该涉及派生范畴。提案中概述了开展此类调查的战略。这个意义深远的问题最终可能与几何佐竹同构的mod p版本有关。