Measures, orbital integrals, and counting points.

从广义上讲，这一建议旨在促进我们对朗兰兹纲领的理解，更具体地说，是阿瑟迹公式的几何方面，使用统一的几何方法来测量p进流形。直到最近，我的研究在很大程度上受到一个长期项目的推动，该项目是将动机整合应用于p- adic群体的表征理论。动机积分是一种最初建立在代数几何基础上的理论，最近又建立在形式逻辑和模型理论的基础上，它允许人们以一种一致的、p- adic域（更一般地说，是p- adic域上的各种点的集合）独立的方式在p- adic域上进行积分。其根源是观察到在局部域上的各种点的集合上的积分可以简化为在剩余域上的点计数和以1/p为底的几何级数的求和。同样的观察结果为数论中的另一个经典思想提供了动力——局部密度的计算，如闵可夫斯基-西格尔质量公式。在最近与Jeff Achter， Ali Altug和Luis Garcia一起完成的工作中，我们利用这一观察结果重新表达了Langlands和Kottwitz用局部密度积表示有限域上主偏振普通阿贝尔变体的等同系类的基数的公式，西格尔式（Langlands- Kottwitz公式将这种基数表示为adelic轨道积分）。令人惊讶的是，我们必须实施的一些技术步骤（例如，仔细跟踪辛群中半简单元素轨道上度量的归一化）与遵循Langlands- Frenkel- Ngo方法对Langlands的“超越内镜”建议所采取的第一步非常相似。我目前的建议有三个互补的方向，源于这些想法：1。1999年，我的博士导师T.C.黑尔斯开始了对p进群“动机”进行谐波分析的项目，解决了这个项目中一些持续存在的开放性问题。2. 2 .与J. Achter在西格尔式公式和轨道积分与某些局部密度关系方面的进一步进展；试图用伊古萨的思想来理解所谓的基本函数和它们的轨道积分。这个方向仍处于推测阶段，主要是通过与W. Casselman的对话得知的。

Broadly speaking, this proposal aims to contribute to our understanding of the Langlands programme, and more specifically, the geometric side of Arthur's Trace Formula,  using a uniform geometric approach to measures that arise on p-adic manifolds.  Until recently, my research has been largely motivated by a long-term project of applications of motivic integration to the representation theory of p--adic groups. Motivic integration is a theory based initially on algebraic geometry and, more recently, on formal logic and model theory, that allows one to do integration on p--adic fields (and more generally, on the set of points of a variety over a p--adic field) in a uniform, p--independent, way. At its source is the observation that integration over the set of points of a variety over a local field can be reduced to point--counting over the residue field and summation of geometric series with base 1/p. The same observation powers another classical idea in number theory - computation of local densities, as in the Minkowski--Siegel mass formula. In a recently completed work with Jeff Achter, Ali Altug and Luis Garcia, we have used this observation to re-express the formula by Langlands and Kottwitz for the cardinality of the isogeny class of a principally polarized ordinary abelian variety over a finite field in terms of a product of local densities, Siegel--style (the Langlands--Kottwitz formula expresses this cardinality as an adelic orbital integral). Surprisingly, some of the technical steps we had to implement (e.g., careful tracking of the normalization of measures on orbits of semisimple elements in the symplectic group) turned out to be very similar to the first steps one has to take to follow the Langlands--Frenkel--Ngo approach to the `Beyond endoscopy' proposal of Langlands. My current proposal has three complementary directions that stem from these ideas: 1. Resolving  some persistent open questions remaining  in the program of making harmonic analysis on p-adic groups `motivic', that was started by my Ph.D. advisor, T.C. Hales, in 1999. 2. Further advances in the project with J. Achter on Siegel--style formulas and relationships between orbital integrals and certain local densities, and 3. Trying to understand the so--called basic functions and their orbital integrals using the ideas of Igusa. This direction is still in a speculative stage, and is largely informed by conversations with W. Casselman.