Galois moduli and modular Hecke algebras

伽罗瓦模和模赫克代数

• 批准号: 505496389
• 负责人: Professor Dr. Elmar Große-Klönne
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/505496389?language=en
• 关键词: Galois; moduli; modular; Hecke; algebras

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.

总体目标是在算术几何中两个相关的程序和猜想：模p局部朗兰兹对应和广义Serre模块化猜想上取得进展。推测的模p局部朗兰兹对应将p进数域F的绝对伽罗瓦群Gal_F的模p表示理论与F上定义的p进约群（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都是某个伽罗瓦模堆栈，这是埃默顿-吉最近在工作中引入的。这是一个形式化的代数堆栈，对于一个固定的自然数n，它参数化了Gal_F的n维p进表示。堆栈特殊纤维的不可约分量可以用Serre权（GL_n（F_q）的不可约模p表示，其中F_q表示F的剩余域）来标记，但其局部模p几何仍然是神秘的。一方面，期望它是朗兰兹对应的伽罗瓦侧l参数的正确堆栈。另一方面，它对Serre猜想的权值部分的推广提供了一个非常有用的几何化。人们普遍认为，对埃默顿-吉叠模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。具体目的是分析emton - gee叠层的模态几何。这里的主要新颖之处是使用模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-adic Hodge理论数据相关）或与几何表示理论中熟悉的对象（Vinberg monoids， Satake参数，施普林格纤维）产生比较态射。这样的比较态射允许在堆栈上构造有趣的层（或其复合体）类，其不变量（支持、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该项目中开发，并可能在数论的其他研究领域中有潜在的应用，例如自同构形式，Iwasawa理论和志村变种的研究。