Hecke algebras in the mod p Langlands program

mod p Langlands 纲领中的赫克代数

• 批准号: RGPIN-2019-03963
• 负责人: Ollivier, Rachel
• 金额: $ 1.38万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758555
• 关键词: Hecke; algebras; mod; Langlands; program

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.

Langlands计划于1960年代启动，是一组猜想，预测了数字理论和群体代表理论的统一。在过去30年中，该计划以其经典形式的众多发展带来了显着的后果，例如Fermat的最后定理证明以及Serre的模块化猜想。在2000年左右，这些猜想的P-ADIC/MOD P版本的问题是由P-Adic算术几何形状的自然问题提出的。由于出乎意料且了解不足的现象涉及GL_2（Q_P）以外的其他群体，因此一般P-Adic/Mod P局部Langlands猜想的陈述仍然具有弹性。但是，对于GL_2（Q_P），P. Colmez和V. paskunas（并基于许多其他工作）建立了对应关系，例如壮观的后果，例如Fontaine-Mazur猜想的证明（M. Kisin，M。Kisin，M。Emerton）。对于更多的一般群体，仍有进步。控制兰兰兹猜想的一般原则是，对应关系应具有自然的几何认识。该提案提出了通过（派生的）Hecke代数的表示理论来实现Mod P Langlands计划的方法。这种方法是适合几何的方法，已经证明了申请人先前工作的启发性。我们将研究（与P. Schneider的联合）特定的“衍生” pro-p iwahori Hecke代数及其与给定P-ADIC还原组的平滑Mod P表示的派生类别的连接。我们将根据仿射标志品种的连贯滑轮来研究逆Mod P satake同构的派生版本，以及我们结果对结果的可能解释。最终，目标是将Mod P Langlands的这一方面与Galois自然的对象联系起来，例如Colmez，Grosse-Klönne，Schneider-Vignéras的作品中所研究的一面。该提案的几个组成部分适用于HQP，这些组件将被介绍给几个非常动态的数字理论和表示理论的主题。