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.