Analytic problems around automorphic forms and L-functions

理解质数分布的主要工具之一是通过黎曼ζ函数的性质。ζ函数是l函数最基本的例子，l函数是一种数学结构，它同时结合了所有质数的算术信息。更一般的l -函数，如狄利克雷l -函数，对于理解等差数列中的素数很有用。研究l函数的主要方法之一是将它们放入族中，比如所有狄利克雷l函数的族，并通过这个框架来观察它们的性质。本提案中提出的大部分工作涉及新l -函数族性质的发展。其中一个主要目标是更好地理解这些l函数的大小，特别是在使用以前的方法无法访问的某些范围内。PI将继续指导并与本科生合作，特别是通过德州农工大学。这些机会对于学生准备研究生学习非常重要，特别是对于来自非博士学位授予机构的本科生以及在STEM领域代表性不足的人群。PI还将继续建议博士生研究与l函数族及其矩有关的问题。PI将研究新的自同构形式及其相关的l函数族，特别是通过l函数的矩和大筛不等式。PI计划研究l -函数的高矩，以便在导体下降家族中具有挑战性但重要的l -函数方面取得进展。提议者还将通过使用新版本的相对轨迹公式来研究l函数的较窄族，该公式根据其局部行为隔离小族。在一个相关的脉络中，提议者将研究自同构形式族的大筛不等式，有两个主要目标。一个目标是在一些新的、狭窄的科中建立大的筛界。第二个目标是开发启发式方法，用于推测更一般家族的大筛子的大小。PI将指导博士生研究l函数的矩问题，包括窄族和高次l函数的矩问题。申请人将与他的本科生一起学习新形式的Dedekind求和。所采用的方法将是解析数论中的技术，如泛函方程，指数和和积分，以及自同态形式的谱理论，包括Arthur-Selberg迹公式和相对迹公式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the main tools for understanding the distribution of prime numbers is through properties of the Riemann zeta function. The zeta function is the most fundamental example of an L-function, which is a mathematical construction that combines arithmetical information about all the primes at once. More general L-functions, such as Dirichlet L-functions, are useful for understanding primes in arithmetic progressions. One of the main ways that L-functions are studied is by placing them into families, such as the family of all Dirichlet L-functions, and viewing their properties through this framework. Much of the proposed work in this proposal concerns the development of properties of new families of L-functions. One of the main goals is to better-understand the size of these L-functions, especially in certain ranges that have been inaccessible using previous methods. The PI will continue to mentor and collaborate with undergraduate students, particularly through the Texas A&M REU. Such opportunities are important for preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The PI will also continue to advise PhD students to work on problems related to families of L-functions and their moments.The PI will study new families of automorphic forms and their associated L-functions, especially via moments of L-functions and large sieve inequalities. The PI plans to study high moments of L-functions in order to make progress on the challenging but important L-functions in conductor-dropping families. The proposer will also study narrower families of L-functions through the use of new versions of the relative trace formula that isolate small families based on their local behavior. In a related vein, the proposer will study large sieve inequalities for families of automorphic forms, with two main goals. One objective is to establish large sieve bounds in some of the new, narrow families. A second goal is to develop heuristics for conjecturing the size of a large sieve bound for more general families. The PI will mentor PhD students on problems on moments of L-functions for both narrow families and for higher degree L-functions. The proposer will study newform Dedekind sums with his undergraduate students. The methods employed will be techniques from analytic number theory such as functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.