Hecke algebras in the mod p Langlands program

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30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，例如费马最后定理的证明，以及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，得到了壮观的结果，例如证明了Fontaine-Mazur猜想(M.Kisin，M.Emerton)。对于更普遍的群体，仍有待取得进展。支配朗兰兹猜想的一般原则是，对应关系应该有一个自然的几何实现。这一建议提出了一种通过(导出的)Hecke代数的表示理论来实现mod p朗兰兹程序的方法。这种方法适用于几何化，在申请人以前的工作中已经被证明是有启发性的。我们将(与P·Schneider一起)研究一种“导出的”Pro-p Iwahori Hecke代数，以及它与给定p-进还原群的光滑mod p表示的导出范畴之间的联系。我们将研究逆mod p Satake同构的一个派生版本，并用仿射旗簇上的相干层来解释我们的结果。最终，我们的目标将是将现代朗兰德通信的这一方面与伽罗瓦性质的对象联系起来，例如科尔梅兹、格罗塞-克隆尼、施奈德-维涅拉的工作中研究的对象。该提案有几个适合HQP的组成部分，HQP将被介绍到一个在数论和表象理论的几个非常动态的领域产生分支的主题。

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.