Distribution of Hecke eigenvalues for automorphic representations

我的程序是通过研究自同构的L函数来研究自同构表示的Hecke本征值的分布。对L函数的研究有很长的历史，可以追溯到黎曼Zeta函数，在那里函数的解析性质被证明对应于算术信息(例如，Riemann Zeta函数的欧拉积在1的发散性意味着素数的无穷大)。我特别感兴趣的是朗兰兹函数猜想和Hecke本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，他证明了佐藤泰特猜想隐含在对称幂L函数的某些解析性质中。我的计划旨在从两个不同的角度发展关于Hecke本征值分布的新结果。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同构表示，关于有限个未分支位置的集合S的大小，它们相关的Hecke本征值不同，可以说什么？Jacquet-Shalika的一个经典结果回答了这个问题，证明了S是无穷的，这就是所谓的强多重数一定理。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。我计划通过演示函数性结果的增量使用如何转化为关于集合S的大小的逐步更强有力的声明来填充这张图。目的是量化朗兰兹计划内的猜想和强重数一的精化之间的融洽关系。另一个方面包括研究与单个自同构表示相关的Hecke本征值序列。我们固定一个常量，并询问一个尖端自同构表示的Hecke本征值等于该常量的频率。在伽罗瓦表示的背景下，Serre和Lang-Trotter也提出了类似的问题。在以前的工作中，我通过应用关于对称幂自同构的结果，得到了固定常数作为Hecke特征值出现的上界，从而在GL(2)集上给出了Serre问题的答案。我的目标是改进上述战略并加强界限。结合这一改进的另一个结果是，它应该服从于大的对称幂的应用，从而使我能够基于关于对称功率提升的猜想的自同构的增量假设来建立一系列的界限。我还计划在我和其他人早期工作的技术的基础上，获得关于Hecke本征值在复平面的不同区域出现的无条件结果。

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.