Mod p local Langlands program for p-adic reductive groups and representations of Hecke algebras

朗兰兹纲领（英语：Langlands program），始于1960年代，并在过去15年中得到成功发展，是一套预言数论与群的表示论统一的理论：朗兰兹对应提供了一种用群论解释数论结果的方法，反之亦然。十年前，这个程序的p-adic/mod p组件的问题被提出，其动机是p-adic算术几何的自然问题。到目前为止，只有很少的情况下被理解，但他们已经有了壮观的后果，如证明大多数情况下的丰泰-马祖尔猜想。由于意想不到的和知之甚少的现象，一般的p-adic/mod p局部朗兰兹猜想的陈述仍然难以捉摸。该提案旨在研究p进约化群的mod p表示，以理解mod p朗兰兹对应的右项。PI以前的工作表明，研究Hecke代数的mod p表示是一种很有前途的方法，因为它表明mod p Langlands对应确实存在。该提案概述了研究p进约化群及其相关Hecke代数的mod p表示的家族和复体的策略。在这个建议的核心是希望超越（迄今为止最调查）的重点是不可约对象，并给予同调的方法表示。它沿着探索的可能性，一个模p原则的泛函和脱落的几何光的潜在模p朗兰兹对应。p-adic/mod p Langlands程序对现代数论具有深远的前景，在算术代数几何中具有深刻的分支。这是一个新的，肥沃的领域，涉及工具和物体，几年前仍然完全深奥。该提案旨在为mod p框架中自然出现的对象提供几何化身。这项工作的成果很可能会自然地与受朗兰兹纲领影响的其他数学领域联系起来，例如几何朗兰兹纲领及其与几何表示论的联系。该提案阐述了实现这种联系的战略。

The Langlands program, initiated in the 1960s and successfully developed in the last 15 years, is a set of conjectures predicting a unification of number theory and of representation theory of groups: the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa. A decade ago, the question of a p-adic/mod p component of this program was raised, motivated by natural questions of p-adic arithmetic geometry. As of now, only very few cases are understood, but they have already had spectacular consequences such as the proof of most cases of the Fontaine-Mazur conjecture. Because of unexpected and poorly understood phenomena, statements of a general p-adic/mod p local Langlands conjecture remain elusive. The proposal aims to study the mod p representations of p-adic reductive groups in order to understand the right terms of a mod p Langlands correspondence. The PI's previous work has shown that studying mod p representations of Hecke algebras is a promising approach, because it suggests that a mod p Langlands correspondence does exist. The proposal outlines strategies to study families and complexes of mod p representations of p-adic reductive groups and their associated Hecke algebras. At the heart of this proposal is the wish to go beyond the (so far most investigated) focus on irreducible objects and give a homological approach to the representations. It goes along with exploring the possibility of a mod p principle of functoriality and shedding a geometric light on the potential mod p Langlands correspondence. The p-adic/mod p Langlands program holds profound prospects for modern number theory with deep ramifications in arithmetic algebraic geometry. It is a new, fertile area involving tools and objects that were still completely abstruse a few years ago. The proposal aims to give a geometric incarnation to the objects that appear naturally in the mod p framework. The outcome of this work will most likely be naturally connected to other areas of mathematics influenced by the Langlands conjectures, such as the geometric Langlands program, and its links with geometric representation theory. The proposal describes strategies towards such connections.