The Explicit Hypergeometric-Modularity Method II

TL;DR

This paper applies an explicit hypergeometric modularity method to derive special modular forms and construct Galois representations, linking hypergeometric formulas with automorphic forms and their L-functions.

Contribution

It introduces a novel application of the hypergeometric-modularity method to produce new modular forms and Galois representations with automorphic L-functions.

Findings

Derived a class of weight three modular forms called $\\mathbb{K}_2$-functions.

Constructed degree four Galois representations with automorphic L-functions.

Extended Galois representations to $G_{\mathbb{Q}}$ with matching L-functions.

Abstract

In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this method from a motivic viewpoint through some known hypergeometric well-poised formulae of Whipple and McCarthy. We first use the method to derive a class of special weight three modular forms, labeled as $\mathbb{K}_2$-functions. Then using well-poised hypergeometric formulae we further construct a class of degree four Galois representations of the absolute Galois groups of the corresponding cyclotomic fields. These representations are then shown to be extendable to $G_{\mathbb{Q}}$ and the $L$-function of each extension coincides with the $L$-function of an automorphic form.