On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy

TL;DR

This paper introduces new eta-quotients called $\\mathbb{K}_{3}$ functions, constructed via cubic theta functions, and uses them within the Explicit Hypergeometric Modularity Method to prove hypergeometric modularity conjectures and explore related number theory applications.

Contribution

It develops the $\\mathbb{K}_{3}$ functions and applies the EHMM to resolve conjectures by Dawsey and McCarthy, advancing the understanding of hypergeometric modularity.

Findings

Resolution of several hypergeometric modularity conjectures

Construction of new eta-quotients from cubic theta functions

Applications to special L-values and generalized Paley graphs

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.