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

朗兰兹计划始创于20世纪60年代，是一组预测数论和群表示论统一的猜想。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和Serre的模性猜想的证明。 大约在2000年左右，这些猜想的p进位/mod p版本的问题被提出，其动机是p进位算术几何的自然问题。由于涉及GL_2(Q_P)以外的群的意想不到的和鲜为人知的现象，一般的p-进/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)建立了一个对应关系，得到了惊人的结果，例如证明了Fontaine-Mazur猜想的大多数情况(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现代方面是一个有希望的方法。 在前人的工作中，PI突出了某一范畴的Hecke模的作用，并证明了M.-F.Vigneras猜想的p-进GL_n的“Hecke模的数值模p-朗兰兹对应”。这是第一个具有mod p-朗兰兹性质的结果，同时涉及所有p-进的一般线性群而不对其秩作任何限制。 该方案旨在研究p-进还原群及其相关的Hecke代数的mod p表示理论。这项提议的核心是希望对潜在的现代朗兰兹通信的正确条款进行几何解释。 提出了三个主要的研究方向： 这个项目是由E.Grosse-Klönne最近的进展所推动的，他构造了一个从Hecke模到一类对象的函子，该函子应该编码关于Fb的绝对Galois群的某些表示的信息/在PI的监督下，K.Koziol对所有超奇异Hecke模c/的分类，建立了SL(n，F)上的Hecke模包的Lang land对应，它与GL(n，F)的对应相容F)和Grosse-Klönne的函子。在此基础上，现在可以在Hecke模的朗兰兹对应的背景下，探索一种形式的mod p函数性原理。 2.形式化参数为零的仿射Hecke代数的表示理论。这是由PI与P.Schneider合作研究Pro-p Iwahori Hecke代数的同调性质(上同调维，对偶函子)而得到的。由于GL(2，q_p)的模型和Grosse-Klönne的工作，尽管仍不清楚要考虑的相关范畴是什么，但预计mod p朗兰兹对应将由函数者给出。在Hecke模的研究中引入非对易几何的工具将有助于澄清这一关键问题。 3·根据p-进群群G的mod p表示理论及其到Galois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复杂表示的设置更微妙，应该涉及派生范畴。提案中概述了开展此类调查的战略。这个影响深远的问题最终可能与几何Satake同构的mod p版本有关。

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.