Distribution of Hecke eigenvalues for automorphic representations

我的计划是专注于通过研究自守L-函数的自守表示的Hecke特征值的分布。L-函数的研究有着丰富的历史，可以追溯到黎曼zeta函数，其中函数的分析性质被证明对应于算术信息（例如，黎曼zeta函数的欧拉乘积在1处的发散意味着素数的无限性）。我特别感兴趣的是朗兰兹函数性图和Hecke特征值分布结果之间的联系。一个相关的例子是塞尔在伽罗瓦表示的设定中的工作，他在那里证明了佐藤-泰特猜想是由对称幂L函数的某些分析性质所暗示的。我的计划旨在从两个不同的角度发展新的结果Hecke特征值的分布。该程序的一个方面考虑了以下问题：给定一个数域上GL（n）的两个不同的尖点自守表示，关于有限未分歧地方的集合S的大小，它们的相关Hecke特征值不同，可以说些什么？Jacquet-Shalika的一个经典结果通过证明S是无穷大而给出了对这个问题的回答，这被称为强重数1定理。通过Ramakrishnan、Murty-Rajan、Rajan和其他人的工作，在这些问题上取得了进一步的进展。我计划通过演示如何递增地使用函性结果来逐步转换为关于集合S的大小的更强的陈述来填充这幅图。其目的是量化朗兰兹计划内的结构和强多重性的改进之间的关系。另一个方面包括研究与单个自守表示相关联的Hecke特征值序列。我们固定一个常数，并询问尖点自守表示有多少次Hecke特征值等于该常数。类似的问题已提出塞尔和朗特罗特在设置伽罗瓦表示。在早期的工作中，我提供了一个答案，在GL（2）设置的问题塞尔应用结果的自同构的对称权力，并获得上界的出现一个固定的常数作为Hecke特征值。我的目标是改进上述战略，加强边界。结合这一改进的另一个结果是，它应该服从于大的对称功率的应用，因此使我能够建立一个连续的边界的基础上增量假设的对称功率升降机的约束自同构。我还计划获得无条件的结果发生的Hecke特征值在不同地区的复杂的平面，从早期的工作，我和其他技术的基础上。

My program is focused on the distribution of Hecke eigenvalues for automorphic representations via the study of automorphic L-functions. The study of L-functions has a rich history, going back to the Riemann zeta function, where analytic properties of the function were shown to correspond to arithmetic information (for example, the divergence of the Euler product for the Riemann zeta function at 1 implies the infinitude of primes). My particular interest lies in the connection between the Langlands functoriality conjectures and distribution results for Hecke eigenvalues. A related example is Serre's work in the setting of Galois representations, where he showed that the Sato-Tate conjecture is implied by certain analytic properties of symmetric power L-functions. My program aims to develop new results on the distribution of Hecke eigenvalues from two different perspectives. One aspect of the program considers the following question: Given two distinct cuspidal automorphic representations for GL(n) over a number field, what can be said about the size of the set S of finite unramified places at which their associated Hecke eigenvalues differ? A classical result of Jacquet-Shalika gave a response to this question by showing that S was infinite, which is known as the strong multiplicity one theorem. Further progress on such questions took place through the work of Ramakrishnan, Murty-Rajan, Rajan, and others. I plan to fill out this picture by demonstrating how the incremental use of functoriality results translates into progressively stronger statements about the size of the set S. The aim is to quantify the rapport between conjectures within the Langlands program and refinements of strong multiplicity one. Another aspect consists of studying the sequence of Hecke eigenvalues associated to a single automorphic representation. We fix a constant and ask how often a cuspidal automorphic representation has a Hecke eigenvalue equal to that constant. Analogous questions have been raised by Serre and Lang-Trotter in the setting of Galois representations. In earlier work, I provided an answer in the GL(2) setting to the question of Serre by applying results on the automorphy of symmetric powers and obtaining upper bounds on the occurrence of a fixed constant as a Hecke eigenvalue. I aim to improve on the above strategy and strengthen the bounds. Another consequence of incorporating this improvement is that it should be amenable to the application of large symmetric powers and therefore enable me to establish a succession of bounds based on incremental assumptions about the conjectured automorphy of the symmetric power lifts. I also plan to obtain unconditional results about the occurrence of Hecke eigenvalues in different regions of the complex plane, building on techniques from earlier work of mine and others.