Hypergeometric Functions over Finite Fields and Modular Forms: A Survey and New Conjectures

TL;DR

This survey reviews hypergeometric functions over finite fields, their applications in algebraic geometry and modular forms, and proposes 13 new conjectures linking these areas.

Contribution

It provides a comprehensive overview of existing results and introduces 13 new conjectures connecting hypergeometric functions and modular forms.

Findings

Summarizes key results in hypergeometric functions over finite fields

Highlights connections to Fourier coefficients of modular forms

Proposes 13 new conjectures in the field

Abstract

Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely used in, character sum evaluations and counting points on algebraic varieties. More interestingly, perhaps, are their links to Fourier coefficients of modular forms. In this paper, we outline the main results in this area and also conjecture 13 new relations.