The Explicit Hypergeometric-Modularity Method II

2024/11/22 by Allen, Michael, Grove, Brian, Long, Ling +1 · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.15116

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 \mathbbK₂-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_ℚ and the L-function of each extension coincides with the L-function of an automorphic form.

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