Symmetry of hypergeometric functions over finite fields and geometric interpretation

TL;DR

This paper introduces hypergeometric functions over finite fields, establishes a symmetry property analogous to classical cases, and provides a geometric proof using algebraic varieties.

Contribution

It defines finite field hypergeometric functions, proves a symmetry property, and offers a geometric interpretation through algebraic variety isomorphisms.

Findings

Established a finite field analogue of classical hypergeometric symmetry.

Constructed algebraic varieties whose rational points relate to hypergeometric functions.

Provided a geometric proof of the symmetry using isomorphisms between varieties.

Abstract

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms between certain algebraic varieties. The numbers of rational points on these varieties are hypergeometric functions over finite fields.