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

这个建议是围绕着一个主题，即在与当地朗兰兹对应关系相关的各种背景下出现的措施的精细理解。我们的研究对象是代数群和李代数在非阿基米德局部领域，以及分布，出现在他们的谐波分析。调和分析通常从给定（局部紧）群G上的Haar测度开始，该测度在常数倍以下是唯一的。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，具有算术意义的哈尔测度的不同归一化之间的相互作用。本提案是围绕由这些共同主题结合在一起的两个项目提出的。 I.动机谐波分析。 这是T. Hales在1999年提出了对p-adic群进行调和分析的方法，该方法通过用motivic积分代替通常的Haar测度积分来实现。这使得人们可以处理非阿基米德的局部领域和功能，他们一致和外地独立。最初，这种方法的主要应用是转移积分恒等式，如基本引理，在特征为零的局部域和足够大的正特征域之间（如[11]）。 在最近的工作与R。克拉克和我。Halupczok，我们证明了新的转移原理的分析性质，例如，可积性和有界性。这种方法也产生了轨道积分的一致p估计，这在数论中非常有用（参见。[34]）。拟议的研究旨在扩大这些新的估计，以更大的一类函数和分布产生的谐波分析的p-adic组，包括哈里什-钱德拉字符，导致新的应用。 二.同源类的大小。2003年，E.联合盖克勒研究的问题“有多大可能是一个给定的椭圆曲线在一个总理有限领域有一个给定数量的合理点？".他给出了一个基于概率启发的明确答案，这个答案太强了，以至于不可能是真的，这看起来有点神秘。在这个项目与J. Achter，我们的目标是提供一个解释盖克勒的公式作出明确的和非常自然的联系与朗兰兹-Kottwitz公式，表示的大小iskal-class的主要极化阿贝尔品种的一个adelic轨道积分。然后，我们计划扩大Gekeler的计算从椭圆曲线阿贝尔品种。连接的概率分布类似的研究Gekeler与轨道积分允许一个问一些问题，他们的渐近行为。我们希望能够制定和回答一些这些渐近问题使用统一估计轨道积分的第一部分的建议。

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.