Hypergeometric functions over finite fields

TL;DR

This paper defines and explores properties of hypergeometric functions over finite fields, establishing analogies with classical functions and applying them to zeta functions of K3-surfaces.

Contribution

It introduces a new definition of hypergeometric functions over finite fields and proves key formulas, linking them to classical hypergeometric functions.

Findings

Established fundamental properties and formulas for hypergeometric functions over finite fields

Derived summation, transformation, and product formulas

Applied results to zeta functions of K3-surfaces

Abstract

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport-Hasse multiplication formula for Gauss sums.