Galois moduli and modular Hecke algebras

总体目标是在算术几何中的两个相关程序和算法上取得进展：模p局部朗兰兹对应和广义塞尔模猜想。证明了模p局部Langlands对应将p-adic数域F的绝对Galois群Gal_F的模p表示理论与定义在F上的p-adic约化群（如GL_n（F））的模p表示理论联系起来。朗兰兹猜想和塞尔猜想的中心对象是最近在埃默顿-吉的工作中引入的某个伽罗瓦模栈。这是一个正式的代数堆栈，对于一个固定的自然数n，参数化Gal_F的n维p-adic表示。栈的特殊纤维的不可约分量可以用Serre权（GL_n（F_q）的不可约mod p表示，其中F_q表示F的剩余域）来标记，但其局部mod p几何仍然是神秘的。一方面，它被认为是朗兰兹对应的伽罗瓦侧的L参数的正确堆叠。另一方面，它提供了一个非常有用的几何化的推广的重量部分的塞尔猜想。人们普遍认为，精确理解Emerton-Gee栈的mod p几何可能会导致数论中这两个中心理论的突破。具体目标是分析Emerton-Gee叠加的模p几何。主要的新奇在于使用模块化Hecke代数（如Iwahori-Hecke代数，Hecke DGA或模块化U_p算子）来描述堆栈部分的局部模型（与p-adic Hodge理论数据相关）或与来自几何表示理论的熟悉对象（Vinberg monoid，Satake参数，Springer纤维）产生比较态射。这样的比较态射允许在栈上构造感兴趣的层类（或其复合体），其不变量（支撑、上同调等）可能包含重要的算术信息。代数和几何表示理论的必要工具和方法将部分在项目中开发，并可能在数论的其他研究领域中具有潜在的应用，例如自守形式，岩泽理论和志村变种的研究。

The overall objective is to make progress on two related programs and conjectures in arithmetic geometry: the mod p local Langlands correspondence and the generalized Serre modularity conjecture. The conjectured mod p local Langlands correspondence should relate the mod p representation theory of the absolute Galois group Gal_F of a p-adic number field F to the mod p representation theory of p-adic reductive groups defined over F (like GL_n(F)). A central object in both the Langlands and the Serre conjecture is a certain Galois moduli stack, recently introduced in work of Emerton-Gee. This is a formal algebraic stack which, for a fixed natural number n, parametrizes the n-dimensional p-adic representations of Gal_F. The irreducible components of the special fibre of the stack can be labelled by Serre weights (irreducible mod p representations of GL_n(F_q), where F_q denotes the residue field of F), but its local mod p geometry remains mysterious. On the one hand, it is expected to be the correct stack of L-parameters on the Galois side of the Langlands correspondence. On the other hand, it provides an extremely useful geometrization of the generalizations of the weight part of Serre's conjecture. It is generally believed that a precise understanding of the mod p geometry of the Emerton-Gee stack may lead to a breakthrough in these two central conjectures in number theory. The concrete objective is to analyze the mod p geometry of the Emerton-Gee stack. The main novelty hereby is the use of modular Hecke algebras (such as Iwahori-Hecke algebras, Hecke DGA's or modular U_p-operators) to describe local models for portions of the stack (related to p-adic Hodge theoretic data) or to produce comparison morphisms with familiar objects from geometric representation theory (Vinberg monoids, Satake parameters, Springer fibres). Such comparison morphisms allow for a functorial construction of interesting classes of sheaves (or complexes thereof) on the stack, whose invariants (support, cohomology etc.) may contain important arithmetic information. The necessary tools and methods from algebra and geometric representation theory will partly be developed within the project and may have potential applications to other research fields in number theory, e.g. to automorphic forms, Iwasawa theory and the study of Shimura varieties.