Fields of definition of finite hypergeometric functions

TL;DR

This paper explores the properties and definitions of finite hypergeometric functions over finite fields, addressing restrictions on parameters and proposing methods to extend their applicability.

Contribution

It introduces a symmetry-based approach and extends exponential sums to relax the parameter restrictions in finite hypergeometric functions.

Findings

Symmetry in hypergeometric parameters allows broader definitions.

Extension of exponential sums reduces restrictions on parameters.

Enhanced understanding of finite hypergeometric functions' properties.

Abstract

Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide $q-1$. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponential sums to circumvent this problem as much as posssible.