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.

朗兰兹计划始于20世纪60年代，是一组预测数论和群体表征理论统一的猜想。在过去的20年里，这个程序在其经典形式下的许多发展都产生了显著的结果，如费马大定理的证明，以及塞尔的模块化猜想之一。在2000年左右，这些猜想的p进/模p版本的问题被提出，这是由p进算术几何的自然问题引起的。由于涉及GL_2（Q_p）以外的群的一些意想不到的和难以理解的现象，一般p-进/模p局部朗兰兹猜想的表述仍然是难以捉摸的。然而，对于GL_2(Q_p)， P. Colmez和V. Paskunas（并以许多其他人的工作为基础）建立了对应关系，产生了惊人的结果，例如证明了Fontaine-Mazur猜想的大多数情况（由M. Kisin和M. Emerton提出）。对于更一般的群体，仍需取得进展，探索当地朗兰兹计划的现代方面是一种有希望的方法。在之前的工作中，PI强调了一类Hecke模的作用，并证明了m - f推测的p进GL_n的“Hecke模的数值模p Langlands对应”。Vigneras。这是第一个具有模p朗兰兹风格的结果，同时涉及所有p进一般线性群，而不受秩的限制。本文研究了p进约群及其相关Hecke代数的模p表示理论。这个提议的核心是希望对潜在的模p朗兰兹对应的正确条件进行几何解释。主要提出了3个研究方向：1•探索的可能性国防部p Langlands Hecke模块的通信一般p进还原组G / F p进字段这个项目是出于a / e . Grosse-Klonne最新进展,他们建造了一个函子从Hecke模块类别的对象编码信息的某些表示绝对的伽罗瓦群F b /分类的πsupersingular Hecke模块c / k . Koziol工作,在PI的监督下，建立了SL（n，F）的Hecke模块包的Langlands对应关系，该对应关系与GL（n，F）的Langlands对应关系兼容，并与Grosse-Klönne的函子兼容。在此基础上，现在有可能探索一种形式的模p功能原理，在朗兰兹对应的背景下，赫克模块开始。形式化了参数为0的仿射Hecke代数的表示理论。这是由PI先前与P. Schneider合作的工作所激发的，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，我们预计mod p朗兰兹对应将由一个函子给出，尽管目前还不清楚需要考虑哪些相关的类别。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进群G的模p表示理论及其与伽罗瓦表示的联系，翻译1•和2•中的Hecke模的工作。赫克模块和G的表示之间的联系比复杂表示的设置更微妙，应该涉及派生范畴。建议中概述了进行这种调查的战略。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。