Measures, orbital integrals, and counting points.

测量、轨道积分和计数点。

• 批准号: RGPIN-2020-04351
• 负责人: Gordon, Julia
• 金额: $ 2.26万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758516
• 关键词: Measures; orbital; integrals; counting; points.

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.

从广义上讲，这项提议旨在帮助我们理解朗兰兹方案，更具体地说，通过使用统一的几何方法来理解p-进流形上出现的度量，从而有助于我们理解朗兰兹方案，更具体地说，帮助我们理解Arthur迹公式的几何方面。直到最近，我的研究在很大程度上是由一个长期项目推动的，该项目将动机积分应用于p-进群的表示理论。Motivic积分是一种最初基于代数几何，最近又基于形式逻辑和模型理论的理论，它允许人们以一种统一的、p独立的方式在p-adad场上(更广泛地说，在p-adad场上的各种点的集合上)进行积分。它的起源是这样一种观察，即在局部域上各种点的集合上的积分可以归结为点--在剩余域上的计数和基数为1/p的几何级数的求和。同样的观察为数论中的另一个经典思想-局部密度的计算提供了动力，如在Minkowski-Siegel质量公式中。在最近与Jeff Achter、Ali Altug和Luis Garcia共同完成的一项工作中，我们利用这一观察结果重新表达了Langland和Kottwitz关于有限域上主要极化的普通阿贝尔变种的同源类的基数的公式，其形式是局部密度的乘积，Siegel式(Langland-Kottwitz公式将这种基数表示为adelic轨道积分)。令人惊讶的是，我们必须实现的一些技术步骤(例如，对辛群中半简单元素轨道上的测量的归一化的仔细跟踪)被证明与人们必须采取的第一步非常相似，以遵循Langlands-Frenkel-Ngo方法对朗兰兹的“超越内窥镜”的提议。我目前的建议有三个相辅相成的方向，它们源于这些想法：1.解决在我的博士导师T.C.Hales于1999年发起的对p-进群进行调和分析的计划中仍然存在的一些悬而未决的问题。2.与J.Achter在Siegel式公式和轨道积分与某些局域密度之间的关系方面的进一步进展；3.试图用伊古萨的思想来理解所谓的基本函数及其轨道积分。这一方向仍处于推测阶段，很大程度上是通过与W·卡塞尔曼的对话了解的。