Local models of moduli spaces of shtukas

shtukas 模块化空间的局部模型

• 批准号: 394587809
• 负责人: Professor Dr. Timo Richarz
• 金额: --
• 依托单位: Institut de Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586)
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/394587809?language=en
• 关键词: Local; models; moduli; spaces; shtukas

Arithmetic Algebraic Geometry aims at a better understanding of integer solutions of systems of polynomial equations with integer coefficients. One approach to the study of the solution space X is by studying the solutions modulo a fixed prime number p. The number of these solutions is used to define a function in one complex variable ζ(X, s) where s ∈ C, Re(s) >> 0. These ζ-functions admit conjecturally a meromorphic continuation to the whole complex plane. Favorable properties of these functions carry information on the solution space X, and are closely related with its geometry. By an idea of Langlands, one should be able to express these ζ-functions for certain classes of polynomial equations, the Shimura varieties X, explicitly in terms of automorphic L-functions - thereby proving many favorable properties of these functions. The methods led for example to a proof of Fermat’s Last Theorem by Wiles. In this sense, the following project lies at the interface of Algebraic Number theory, Algebraic Geometry and the theory of Automorphic Forms. Concretely, the project focuses on the local geometry of certain moduli spaces of Drinfeld shtukas, the function field analogs of Shimura varieties, at places of bad reduction and their applications to the local factors of their ζ-functions.Moduli spaces of shtukas were introduced by Drinfeld for the general linear group to study Landlands’ correspondence for function fields. They were generalized by Varshavsky to reductive groups and by Arasteh Rad and Hartl to general smooth affine groups. In a first step of the project, Beilinson-Drinfeld local models are used to study the local geometry of these moduli spaces for Bruhat-Tits groups. In the Shimura case, these models were used by Rapoport in order to apply the Landlands-Kottwitz method for parahoric level, and first result on the local geometry were obtained by Görtz. The novelty of the project is the study of deeper level structures. In a second step of the project, the function given by the semisimple trace of Frobenius on the sheaf of nearby cycles of these models is studied. This function is explicitly determined by the Kottwitz conjecture for parahoric level, and can be used in the Langlands-Kottwitz method. In future projects, these results shall on the one hand contribute to the calculation of local factors of ζ-functions for moduli spaces of shtukas, and on the other hand to a better understanding of analogies with the case of Shimura varieties.

算术代数几何的目的是更好地理解具有整数系数的多项式方程组的整数解。研究解空间X的一种方法是研究模为固定素数p的解，这些解的个数被用来定义一个复变量ζ(X，S)中的函数，其中S∈C，Re(S)&gt；&gt；0.这些ζ函数猜想地允许整个复平面上的亚纯延拓。这些函数的良好性质在解空间X上携带信息，并且与其几何密切相关。根据朗兰兹的思想，人们应该能够用自同构的L函数来显式地表示某些多项式方程的ζ函数，即下村族X，从而证明了这些函数的许多良好性质。例如，这些方法导致了Wiles对费马大定理的证明。在这个意义上，以下项目位于代数论、代数几何和自同构形理论的交界处。具体地说，该项目主要研究了某些模空间的局部几何性质，即Shimura变种的函数域模拟在不良约化位置的局部几何及其在其ζ-函数的局部因子中的应用。它们被Varshavsky推广到约化群，由Arasteh Rad和Hartl推广到一般光滑仿射群。在该项目的第一步，使用Beilinson-Drinfeld局部模型来研究Bruhat-Tits群的这些模空间的局部几何。在Shimura的例子中，Rapoport使用这些模型来将Landland-Kottwitz方法应用于意合水平，Görtz首先获得了关于局部几何的结果。该项目的新奇之处在于对更深层次结构的研究。在项目的第二步中，研究了Frobenius的半单迹在这些模型的邻近循环的束上所给出的函数。该函数由Kottwitz猜想明确确定，可用于朗兰兹-Kottwitz方法。在未来的项目中，这些结果一方面有助于计算Shtukas模空间的ζ函数的局部因子，另一方面有助于更好地理解与Shimura簇的类比。