Special Values of L-functions

L 函数的主题起源于黎曼 zeta 函数的研究，很大程度上是由黎曼假设驱动的，这一猜想在 150 多年来一直困扰着数学家。在过去的 50 年里，数论研究的很大一部分都集中在 L 函数的特殊值上。考虑到该主题包含的大量猜想以及它们对我们对数域、表示、阿贝尔簇和自守形式的理解的重大影响，这种趋势肯定会继续下去。例如，L-函数的消失或不消失以及它们在关键带中心的扭曲具有丰富的算术含义，正如伯奇和斯温纳顿-戴尔猜想及其推广所生动地描述的那样。 这项研究的主要动机是关于关键带中 L 函数不消失的问题。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；这个主题本身很有趣，并且在我的研究中发挥着关键作用。 Chowla (1965) 的猜想指出，Dirichlet L 函数的中心值不为零。虽然这个猜想仍然存在，但已经取得了显着的进展（例如 Soundararajan 正比例结果）。同样，人们还认为，模 L 函数的中心值是非零的，除非存在微不足道的（例如函数方程中的符号）或算术原因导致这些值消失。在这项研究中，我探索了与希尔伯特模形式相关的 L 函数族。我特别关注权重或级别方面不同的两种此类形式的卷积。该研究计划的一个主要组成部分是研究与不同权重的希尔伯特模形式相关的某些中心 L 值族的各种矩。这将使我们能够在这些值的大小及其不消失方面取得强有力的成果。在这项研究中，我还研究了某些 L 函数族的值分布。在这方面，我在与立方赫克字符相关的特定家族方面取得了一些进展。我计划扩展我的结果以包括更高阶的字符。我还计划使用现代概率框架研究自同构 L 函数的二次扭曲问题。该研究计划的另一个目标是建立某些长狄利克雷多项式的中值定理。这个主题与 L 函数矩和卷积和密切相关。最后，我计划通过在更一般的设置中计算半积分权模形式的空间来继续我的研究。 这项研究计划将丰富我们对 L 函数特殊值的各个方面的理解，同时推动数论这个美丽的领域向前发展。

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.