Distribution of Hecke eigenvalues for automorphic representations

我的项目主要是通过研究自同构l函数来研究自同构表示的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ζ函数，其中函数的解析性质被证明与算术信息相对应（例如，黎曼ζ函数在1处的欧拉积的散度意味着无穷素数）。我特别感兴趣的是朗兰兹泛函猜想与赫克特征值的分布结果之间的联系。一个相关的例子是Serre在伽罗瓦表示的背景下的工作，他证明了对称幂l函数的某些解析性质隐含了Sato-Tate猜想。我的项目旨在从两个不同的角度发展关于赫克特征值分布的新结果。该方案的一个方面考虑了以下问题：给定GL(n)在一个数域上的两个不同的倒自同构表示，关于它们相关的Hecke特征值不同的有限未分叉位置的集合S的大小可以说些什么？Jacquet-Shalika的一个经典结果通过证明S是无限的给出了对这个问题的回答，这个结果被称为强多重性定理。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。我计划通过演示功能结果的增量使用如何转化为关于集合s大小的逐渐更强的陈述来填补这一图景。目的是量化朗兰兹纲领中的猜想与强多重性1的改进之间的关系。另一方面包括研究与单个自同构表示相关的Hecke特征值序列。我们固定一个常数，然后问一个倒轴自同构表示有多少次赫克特征值等于那个常数。Serre和Lang-Trotter在伽罗瓦表征的背景下也提出了类似的问题。在早期的工作中，我通过应用对称幂自同构的结果并获得固定常数作为Hecke特征值出现的上界，在GL(2)设置中给出了Serre问题的答案。我的目标是改进上述策略并加强边界。结合这一改进的另一个结果是，它应该适用于大对称幂的应用，因此使我能够基于关于对称幂提升的推测自同构的增量假设建立一系列界限。我还计划在我和其他人早期工作的技术基础上，获得关于复平面不同区域Hecke特征值出现的无条件结果。

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.