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

朗兰兹计划始创于20世纪60年代，在过去的15年里得到了成功的发展，它是一组预测数论和群的表示理论统一的猜想：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进位/mod p分量的问题被提出，其动机是p进位算术几何的自然问题。到目前为止，只有很少的案例被理解，但它们已经产生了惊人的后果，例如证明了方丹-马祖尔猜想的大多数案例。由于意想不到且知之甚少的现象，一般的p进/mod p局域朗兰兹猜想的表述仍然难以捉摸。本文的目的是研究p-进约群的mod p表示，以便理解mod p朗兰兹对应的正确项。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-朗兰兹对应确实存在。该建议概述了研究p-adine还原群及其相关Hecke代数的mod-p表示的族和复形的策略。这一提议的核心是希望超越(迄今为止研究最多的)聚焦于不可约对象，并给出一种表示的同源方法。它伴随着探索mod p函数性原理的可能性，并揭示了潜在的mod p朗兰兹对应的几何意义。P-adi/mod p朗兰兹程序对现代数论有着深刻的前景，在算术代数几何中有着深刻的影响。这是一个新的、肥沃的领域，涉及几年前还完全神秘的工具和物品。该提案旨在为mod p框架中自然出现的对象赋予几何化身。这项工作的结果很可能自然地与受朗兰兹猜想影响的其他数学领域联系在一起，例如几何朗兰兹程序，以及它与几何表示理论的联系。该提案描述了建立这种联系的战略。

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.