Special Values of L-functions

L 函数的特殊值

• 批准号: RGPIN-2018-06313
• 负责人: Hamieh, Alia
• 金额: $ 1.53万
• 依托单位: University of Northern British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758985
• 关键词: Special; Values; functions

The subject of L-functions has its origin in the study of the Riemann zeta function, and is largely driven by the Riemann hypothesis, a conjecture that continues to elude mathematicians for more than 150 years. Over the last 50 years, a significant part of research in number theory has been centred around the special values of L-functions. This trend will surely continue in view of the bounty of conjectures that this topic encompasses and the major implications they would have on our understanding of number fields, representations, Abelian varieties and automorphic forms. For example, the vanishing or nonvanishing of L-functions and their twists at the center of the critical strip have rich arithmetic implications, as it is vividly portrayed by the Birch and Swinnerton-Dyer Conjecture and its generalizations.This research is primarily motivated by questions regarding the nonvanishing of L-functions in the critical strip. A crucial step in answering such questions is the study of the moments associated with these values; a topic that is interesting in its own right and plays a pivotal role in my research. A conjecture of Chowla (1965) states that the central values of Dirichlet L-functions are nonzero. While this conjecture remains open, there has been remarkable progress towards it (e.g. Soundararajan positive proportion result). Likewise, it is also believed that the central values of modular L-functions are nonvanishing unless there is a trivial (e.g. sign in the functional equation) or an arithmetical reason for these values to vanish. In this research, I explore families of L-functions associated with Hilbert modular forms. I am especially focused on the convolution of two such forms which are varying in the weight or the level aspects. A major component of this research program is to study various moments of certain families of central L-values associated with Hilbert modular forms of varying weight. This will allow us to achieve strong results regarding the size of these values and their non-vanishing. In this research, I also study the value-distribution of certain families of L-functions. On this front, I made some progress pertaining to the particular family associated with cubic Hecke characters. I plan to extend my results to include higher order characters. I also plan to investigate this problem for quadratic twists of automorphic L-functions using the modern probabilistic framework. Another objective of this research program is to establish a mean value theorem for certain long Dirichlet polynomials. This topic is intimately related to moments of L-functions and convolution sums. Finally, I plan to continue my research on half-integral weight modular forms by computing their spaces in more general settings.This research program will enrich our understanding of various aspects of the special values of L-functions while moving this beautiful area of number theory forward.

L函数的主题起源于对黎曼Zeta函数的研究，并在很大程度上受到黎曼假设的推动，这一猜想一直是数学家们150多年来一直未能理解的。在过去的50年里，数论的很大一部分研究都集中在L函数的特殊价值上。鉴于这一主题所包含的大量猜想，以及它们对我们对数域、表示、阿贝尔簇和自同构形的理解的主要影响，这一趋势肯定会继续下去。例如，像Birch和Swinnerton-Dyer猜想及其推广所生动地刻画的那样，L函数及其在临界带中心的扭曲的消失或不消失具有丰富的算术含义。回答这些问题的关键一步是研究与这些价值观相关的时刻；这是一个本身就很有趣的话题，在我的研究中扮演着关键角色。Chowla(1965)的一个猜想指出，Dirichlet L-函数的中心值不为零。虽然这一猜想仍然存在，但已经取得了显著的进展(例如，Soundararajan正比例结果)。同样，人们还认为，模L函数的中心值不为零，除非存在平凡的(例如，函数方程中的符号)或算术原因使这些值消失。在这项研究中，我研究了与Hilbert模形式相关的L函数族。我特别关注这两种形式的卷积，这两种形式在重量或水平方面是不同的。这项研究计划的一个主要组成部分是研究某些L中心族的各种矩--与变权的希尔伯特模形式有关的值。这将使我们能够在这些价值的大小和它们的不消失方面取得强有力的结果。在这项研究中，我还研究了某些L函数家族的值分布。在这方面，我在与立方黑克角色相关的特定家庭方面取得了一些进展。我计划将我的结果扩展到包括更高顺序的字符。我还计划用现代概率框架来研究自同构L函数的二次扭曲问题。本研究计划的另一个目的是建立某些长Dirichlet多项式的中值定理。这一主题与L的矩密切相关--函数与卷积和。最后，我计划通过在更一般的环境下计算半整权模形式的空间来继续我的研究。这个研究程序将丰富我们对L函数的各个方面的特殊价值的理解，同时推动这个美丽的数论领域向前发展。