Hecke algebras in the mod p Langlands program

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30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（并以许多其他人的工作为基础）建立了一个对应关系，并产生了惊人的结果，如Fontaine-Mazur猜想的证明（M. Kisin，M. Emerton）。在更一般的群体方面仍有待取得进展。支配朗兰兹图的一般原则是对应应该有一个自然的几何实现。 这个建议建议提出了一种方法，模p朗兰兹计划通过表示理论（导出）Hecke代数。这种方法适合于几何化，已经在申请人的先前工作中证明是有启发性的。我们将研究（与P. Schneider联合）一个特定的“导出”pro-p Iwahori Hecke代数及其与给定p-adic约化群的光滑mod p表示的导出范畴的联系。我们将调查派生版本的逆模p佐竹同构和可能的解释，我们的结果在相干层的仿射旗品种。最终，我们的目标将是把模p朗兰兹对应的这一面与伽罗瓦性质的对象联系起来，例如科尔梅兹、格罗斯-克洛内、施奈德-维涅拉斯的工作中所研究的对象。该提案有几个组成部分，是适合HQP谁将被介绍给一个主题，在数论和表示论的几个非常动态的领域的分支。

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 30 years have had remarkable consequences such as the proof of Fermat's last theorem, and 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 the Fontaine-Mazur conjecture (by M. Kisin, M. Emerton). Progress remains to be made for more general groups. The general principle that governs the Langlands conjectures is that the correspondences should have a natural, geometric realization.  This proposal suggests an approach to the mod p Langlands program via the representation theory of (derived) Hecke algebras. This approach, which is amenable to geometrization, has already proved enlightening in previous work of the applicant. We will study (joint with P. Schneider) a certain "derived" pro-p Iwahori Hecke algebra and its connections to the derived category of smooth mod p representations of a given p-adic reductive group. We will investigate a derived version of the inverse mod p Satake isomorphism and the possible interpretations of our results in terms of coherent sheaves on the affine flag variety. Ultimately, the goal will be to relate this side of the mod p Langlands correspondence to object of Galois nature such as the ones studied in the work of Colmez, Grosse-Klönne, Schneider-Vignéras. The proposal has several components that are suitable for HQP who will be introduced to a subject with ramifications in several very dynamic areas of number theory and representation theory.