Integral models and endoscopy for Shimura varieties with deeper level structure

朗兰兹计划试图阐明当前神秘的不同类型对称之间的关系，从而将算术、代数几何和李群上的调和分析的数学领域联系起来。在过去的几十年里，这个程序是回答数论中许多长期存在的问题的基础；举个例子，它在20世纪90年代安德鲁·怀尔斯的费马大定理的解决中发挥了关键作用。经典朗兰兹程序的变体，即由Drinfeld和Laumon发起的几何朗兰兹程序，对理论物理的发展一直是鼓舞人心的，例如在Kapustin和Witten关于量子场论的工作中。这个项目涉及下村簇，在伽罗华群的表示和自同构表示之间建立局部和全局朗兰兹对应所必需的算术对象。PI试图在下村簇的研究和几何朗兰兹程序之间建立新的联系。感兴趣的Shimura簇是具有附加结构的交换簇的模空间，当方程以不同的素数为模时，这种附加结构可以引起奇点。当结构为Prop Iwahori水平时，PI将引入新的局部模型来研究这些奇点；局部模型将是Rapoport-Zink局部模型的推广，而Rapoport-Zink局部模型对于研究意合水平结构是如此有效。他将把这些局部模型与‘Pro-p’仿射旗帜品种及其变形联系起来，将几何朗兰兹程序中使用的想法引入驯服分支的设置。这种联系将允许理解附近的循环和奇点，以及内窥镜和Hasse-Weil Zeta函数，以获得有趣的下村变种的新类别。为了将这些建筑放在一个非常广泛的背景下，PI将制定一个关于Bruhat-Tits建筑的Tannakian理论。

The Langlands program seeks to illuminate currently-mysterious relations between different kinds of symmetries, thereby connecting the mathematical realms of arithmetic, algebraic geometry, and harmonic analysis on Lie groups. This program has been fundamental in answering many longstanding problems in number theory over the last few decades; to cite one example, it played a key role in the solution of Fermat's Last Theorem by Andrew Wiles in the 1990s. Variants of the classical Langlands program, namely the geometric Langlands program initiated by Drinfeld and Laumon, have been inspirational for developments in theoretical physics, for instance in the work of Kapustin and Witten on Quantum Field Theory. This project relates to Shimura varieties, arithmetical objects which have been indispensable for establishing local and global Langlands correspondences between representations of Galois groups on the one hand, and automorphic representations on the other hand.The PI seeks to build new relations between the study of Shimura varieties and the geometric Langlands program. The Shimura varieties of interest are moduli spaces of abelian varieties with additional structure, and that additional structure can give rise to singularities when the equations are reduced modulo various prime numbers. The PI will introduce new local models to study those singularities when the structure is pro-p Iwahori level; the local models will be generalizations of the Rapoport-Zink local models which have been so effective for studying parahoric level structure. He will relate these local models to 'pro-p' affine flag varieties and their deformations, bringing in ideas used in the geometric Langlands program in the setting of tame ramification. This connection will permit an understanding of nearby cycles and singularities, and also of endoscopy and Hasse-Weil zeta functions, for interesting new classes of Shimura varieties. In order to put these constructions into a very general context, the PI will work out a Tannakian theory of Bruhat-Tits buildings.