Measures, orbital integrals, and counting points.

从广义上讲，这个建议的目的是有助于我们理解朗兰兹计划，更具体地说，几何方面的亚瑟的迹公式，使用统一的几何方法来衡量出现在p-adic流形。 直到最近，我的研究一直在很大程度上是由一个长期项目的动机整合的应用程序的p-进群的表示理论。动机整合是一种理论基础上最初代数几何，最近，在形式逻辑和模型理论，它允许一个做整合的p-进领域（更一般地说，对一组点的各种超过一个p-进领域）在一个统一的，p-独立的方式。在其来源是观察到的积分点的集合的各种在当地的领域可以减少到点-计数的剩余领域和总和的几何级数与基地1/p。同样的观察权力的另一个经典思想在数论-计算局部密度，如在闵可夫斯基-西格尔质量公式。在最近完成的工作与杰夫Achter，阿里Altug和路易斯加西亚，我们已经使用这个观察重新表达公式朗兰兹和Kottwitz的基数的islan-class的主要极化普通阿贝尔品种在有限领域的产品的局部密度，西格尔-风格（朗兰兹-Kottwitz公式表示这种基数作为一个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，仔细跟踪辛群中半单元素轨道上的度量的归一化）被证明非常类似于遵循Langlands-Frenkel-Ngo方法以实现Langlands的“超越内窥镜”建议所必须采取的第一步。我目前的建议有三个互补的方向，源于这些想法：1。解决了一些持续存在的悬而未决的问题，在程序中，使调和分析对p-adic组的“动机”，这是由我的博士开始。顾问，TC 1999年，黑尔斯。2.进一步的进展项目与J. Achter在西格尔-风格的公式和轨道积分和某些局部密度之间的关系，和3。试图理解所谓的基本函数和它们的轨道积分，用Igusa的思想。这一方向仍处于推测阶段，主要是通过与W。卡塞尔曼。

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.