On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy

Abstract: 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.