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

该提案围绕着对与当地朗兰兹通信相关的各种上下文中出现的措施的良好理解的主题。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，以及它们调和分析中出现的分布。谐波分析通常从给定（局部紧化）群G上的哈尔测度开始，该群在常数倍数以内是唯一的。我的很多工作都源于两个非常基本的考虑：第一，哈尔测度本身的性质，第二，哈尔测度的不同归一化之间的相互作用，这些归一化具有一定的算术意义。目前的提案是围绕两个项目建立的，这些共同的主题结合在一起。动力谐波分析。这是一个长期项目的延续，由T. Hales于1999年发起，通过用“动机积分”取代关于哈尔测度的通常积分，对“独立于p”的p进群进行谐波分析。这使得人们可以统一地、独立地对待非阿基米德局部场和它们上的函数。最初，这种方法的主要应用是在特征为零的局部域和具有足够大的正特征域（如[11]）之间的积分恒等式的转移，例如基本引理。在最近与R. Cluckers和I. Halupczok的工作中，我们证明了解析性质的新转移原理，例如可积性和有界性。这种方法也为轨道积分提供了一致的p估计，这在数论中是非常有用的（参见[34]）。拟议的研究旨在将这些新的估计扩展到p进群谐波分析中产生的更大类别的函数和分布，包括Harish-Chandra特征，从而导致新的应用。* * *二世。等基因类的大小。2003年，欧盟与美国达成协议。Gekeler研究了这样一个问题：“在素数有限域上，给定的椭圆曲线有多少可能具有给定数量的有理点？”他给出了一个明确的答案，这个答案基于一种概率启发，这种启发太过强烈，不太可能是真的，这看起来有点神秘。在这个与J. Achter合作的项目中，我们的目标是通过与Langlands-Kottwitz公式建立明确而非常自然的联系来解释Gekeler公式，Langlands-Kottwitz公式用adelic轨道积分表示主极化阿贝尔变体的等同系类的大小。然后我们计划将Gekeler的计算从椭圆曲线扩展到阿贝尔曲线。类似于Gekeler所研究的概率分布与轨道积分的联系，允许人们对它们的渐近行为提出一些问题。我们希望能够利用第一部分中轨道积分的一致估计来表述和回答这些渐近问题中的一些。********

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.********