Uniform estimates and asymptotics for p-adic orbital integrals and characters

这项提案围绕着对与当地朗兰兹通信有关的各种背景下出现的措施进行深入了解这一主题展开。我们的研究对象是非阿基米德局部域上的代数群和李代数，以及它们在调和分析中出现的分布。调和分析通常从给定(局部紧)群G上的Haar测度开始，它直到常数倍数都是唯一的。我的许多工作源于两个非常基本的考虑：第一，Haar测度本身的性质，第二，具有一定算术意义的Haar测度的不同正规化之间的相互作用。本提案建立在两个由这些共同主题联合起来的项目上。 一、动机调和分析。这是由T.Hales于1999年发起的一个长期项目的延续，该项目通过用“动机积分”取代通常的关于Haar度量的积分，对“独立于p”的p-add群进行调和分析。这使得人们可以一致地、独立地处理非阿基米德局部恒等式及其上的函数。最初，这种方法的主要应用是在特征为零且具有足够大的正特征的局部域之间传递积分恒等式，例如基本引理(如文[11])。在最近与R.Cluckers和I.Halupczok的工作中，我们证明了关于分析性质的新的转移原理，例如，可积性和有界性。这种方法也产生一致的轨道积分的p估计，这在数论中被证明是非常有用的。[34]))。拟议的研究旨在将这些新的估计扩展到p-adi群的调和分析中产生的更大类别的函数和分布，包括Harish-Chandra字符，从而导致新的应用。 II.同源类目的大小。2003年，E.-U·Gekeler研究了“素数有限域上给定的椭圆曲线有多少个有理点？”这个问题。他给出了一个基于概率启发式的明确答案，这个启发式太强了，不可能是字面上的真实，这看起来有点神秘。在与J.Achter的这个项目中，我们的目的是通过与Langland-Kottwitz公式建立显式和非常自然的联系来解释Gekeler公式，该公式通过adelic轨道积分来表示主要极化的阿贝尔变种的同源类的大小。然后，我们计划将Gekeler的计算从椭圆曲线推广到阿贝尔变种。类似于Gekeler研究的概率分布与轨道积分的联系允许人们就它们的渐近行为提出许多问题。我们希望能够使用提案第一部分中的轨道积分的统一估计来表述和回答其中的一些渐近问题。

This proposal is centered around the main theme of fine understanding of the measures that arise in the various contexts associated with the local Langlands correspondence. Our objects of study are algebraic groups and Lie algebras over non-Archimedean local fields, and the distributions that arise in their harmonic analysis. Harmonic analysis usually starts with a Haar measure on a given (locally compact) group G, which is unique up to a constant multiple. A lot of my work stems from two very basic considerations: first, the nature of the Haar measure itself, and second, the interplay between different normalizations of Haar measures that have some arithmetic significance. The present proposal is built around two projects united by these common themes. I. MOTIVIC HARMONIC ANALYSIS.  This is a continuation of the long-term project, initiated by T. Hales in 1999, of making harmonic analysis on p-adic groups `independent of p' by means of  replacing the usual integration with respect to Haar measure by `motivic integration'. This allows one to treat non-Archimedean local  fields and functions on them  uniformly and field-independently. Initially, the main application of this approach was transfer of  integral identities, such as the Fundamental Lemma,  between local fields of characteristic zero and of sufficiently large positive characteristic (as in [11]).  In the recent work with R. Cluckers and I. Halupczok, we proved new transfer principles for analytic properties, e.g., integrability and boundedness. This method also yields uniform in p estimates for orbital integrals, which turned out to be very useful in Number theory (cf. [34]). Proposed research aims to expand these new  estimates to a larger class of functions and distributions arising in harmonic analysis on p-adic groups, including Harish-Chandra characters, leading to new applications.  II. SIZES OF ISOGENY CLASSES. In 2003, E.-U. Gekeler studied the question "how likely is a given elliptic curve over a prime finite field to have a given number of rational points?". He gave an explicit answer based on a probabilistic heuristic that was too strong to be literally true, which appeared somewhat mysterious. In this project with J. Achter, we aim to provide an explanation for Gekeler's formula by making an explicit and very natural connection with Langlands-Kottwitz formula that expresses the size of an isogeny class of principally polarized abelian varieties in terms of an adelic orbital integral. Then we plan to extend Gekeler's computations from elliptic curves to abelian varieties. The connection of probability distributions analogous to the ones studied by Gekeler with orbital integrals allows one to ask a number of questions about their asymptotic behaviour. We hope to be able to formulate and answer some of these asymptotic questions using uniform estimates for orbital integrals from Part I of the proposal.