标题： 关于Dawsey和McCarthy的一些超几何模性猜想 显示英文标题

标题： On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy

摘要： 在最近的工作中，作者与Allen、Long和Tu合作，开发了显式超几何模性方法（EHMM），该方法确立了二维和三维情况下一大类超几何伽罗瓦表示的模性。 EHMM的一个重要应用是从超几何背景中构造了一个显式的eta-商族，我们称之为$\mathbb{K}_{2}$函数。 在本文中，我们引入了一个类似的eta-商族，我们称之为$\mathbb{K}_{3}$函数。 这些$\mathbb{K}_{3}$函数是利用由Jonathan和Peter Borwein最初开发的一阶立方theta函数理论构造的。 然后，我们使用$\mathbb{K}_{3}$函数在EHMM中解决了一些Dawsey和McCarthy的超几何模性猜想。 此外，我们提供了对$L$-值的$\mathbb{K}_{3}$函数的应用以及对广义Paley图的研究。 显示英文摘要

摘要： In recent work, the author, in collaboration with Allen, Long, and Tu, developed the Explicit Hypergeometric Modularity Method (EHMM), which establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. One important application of the EHMM is the construction of an explicit family of eta-quotients, which we call the $\mathbb{K}_{2}$ functions, from the hypergeometric background. In this article, we introduce an analogous family of eta-quotients, which we call the $\mathbb{K}_{3}$ functions. These $\mathbb{K}_{3}$ functions are constructed using the theory of weight one cubic theta functions originally developed by Jonathan and Peter Borwein. We then use the $\mathbb{K}_{3}$ functions in the EHMM to resolve several hypergeometric modularity conjectures of Dawsey and McCarthy. Further, we provide applications to special $L$-values of the $\mathbb{K}_{3}$ functions and to the study of generalized Paley graphs.