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猜想上取得进展：mod p局部朗兰兹对应和广义Serre模性猜想。猜想的mod p局部朗兰兹对应应将p进数域F的绝对Galois群GalF的mod p表示理论与定义在F上的p进还原群(如GL n(F))的mod p表示理论联系起来.朗兰兹猜想和Serre猜想中的一个中心对象是最近在Emerton-Gee的工作中引入的某个Galois模堆。这是一个形式的代数堆栈，对于固定的自然数n，它将GaL_F的n维p维表示参数化。堆叠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，但它的局部mod_p几何仍然是神秘的。一方面，它有望是朗兰兹通信伽罗瓦侧的L参数的正确堆叠。另一方面，它为Serre猜想的权重部分的推广提供了一个非常有用的几何化。人们普遍认为，对Emerton-Gee堆栈的mod p几何的精确理解可能会导致数论中这两个中心猜想的突破。具体目标是分析Emerton-Gee堆栈的mod p几何结构。因此，主要的创新是使用模Hecke代数(如Iwahori-Hecke代数、Hecke DGA或模U_p-算子)来描述堆栈部分的局部模型(与p-进Hodge理论数据有关)或产生与几何表示理论(Vinberg么半群、Satake参数、Springer纤维)中熟悉的对象的比较态射。这样的比较态射允许堆栈上有趣的片层类(或其复合体)的函数式构造，其不变量(支集、上同调等)。可能包含重要的算术信息。来自代数和几何表示理论的必要工具和方法将在该项目中部分开发，并可能在数论的其他研究领域具有潜在的应用，例如自同构形、岩泽理论和下村变种的研究。

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.