Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

TL;DR

This paper constructs explicit eta-quotients related to hypergeometric modularity and expresses Fourier coefficients of certain Hecke eigenforms in terms of finite field period functions, revealing new identities.

Contribution

It introduces two explicit eta-quotient families derived from hypergeometric theory and links Fourier coefficients of specific modular forms to finite field period functions.

Findings

Expressed Fourier coefficients of weight two and four Hecke eigenforms in terms of finite field period functions.

Constructed eta-quotients $\, ext{ extbackslash mathbb{K}_4} ext{ and } ext{ extbackslash mathbb{K}_5}$ from hypergeometric backgrounds.

Derived new identities relating modular form coefficients to special values of finite field Appell series.

Abstract

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we construct two explicit families of eta-quotients, which we call the $\mathbb{K}_4$ and $\mathbb{K}_5$ functions, from the hypergeometric background. These $\mathbb{K}_4$ and $\mathbb{K}_5$ functions are constructed using the theory of weight $1/2$ Jacobi theta functions and their cubic analogues, respectively. Using these constructions, we then express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions. As an application, we obtain new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series $F_1^p$ and $F_2^p$.