Equidistribution of Hecke Eigenforms and Related Problems

本文主要研究椭圆Hecke特征形式的定量等分布性质及其在厄米对称空间的高维算术商上的推广，以及与L值的非零性和次凸性有关的自同构形式及其L函数的解析方面的问题，以期从解析的角度对它们有更深的理解。提出的问题是最近解析数论研究和发展的核心。这一建议强调了研究这些问题的不同方法，并指出了与其他学科的有趣联系。自同构L函数是数学中的基本对象。拟研究的均匀分布问题起源于量子力学，并与自同构L函数理论密切相关。我们的方法使用了调和分析和数论的先进工具，结合了自同构表示理论的新进展。这是我们希望在当前的提案中追求的跨不同领域的思想的丰富的相互作用和相互作用。

This proposal is centered on problems concerning quantitative equidistribution properties of elliptic Hecke eigenforms and its generalization to higher dimensional arithmetic quotients of Hermitian symmetric spaces, as well as the analytic aspects of automorphic forms and their L-functions related to nonvanishing and subconvexity of the L-values, with the aim to gain deeper understanding of them from analytic perspective. The problems proposed lie at the heart of recent research and development in analytic number theory. This proposal emphasizes different approaches to study these problems, and points interesting connections with other subjects.Automorphic L-function is fundamental object in mathematics. The equidistribution problems proposed to study have origin from quantum mechanics, and is closely related to the theory of automorphic L-function. They are at the interface of number theory, analysis, and geometry.Our approach employs advanced tools from harmonic analysis and number theory, in combination with the new progress from the theory of automorphic representation.It is the rich interplay and interaction of ideas across different fields that we want to pursue in the current proposal.