Dwork hypersurfaces of degree six and Greene's hypergeometric function

TL;DR

This paper derives a formula for counting rational points on degree six Dwork hypersurfaces over finite fields using Greene's hypergeometric function, extending previous formulas for degree four and relating to other known results.

Contribution

It introduces a new formula for degree six Dwork hypersurfaces using Greene's hypergeometric function, generalizing earlier degree four results and connecting with Miyatani's formula.

Findings

Derived a point-counting formula for degree six Dwork hypersurfaces

Extended Goodson's degree four formula to degree six

Established relations with Miyatani's formula

Abstract

In this paper, we give a formula for the number of rational points on the Dwork hypersurfaces of degree six over finite fields by using Greene's finite-field hypergeometric function, which is a generalization of Goodson's formula for the Dwork hypersurfaces of degree four. Our formula is also a higher-dimensional and a finite field analogue of Matsumoto-Terasoma-Yamazaki's formula. Furthermore, we also explain the relation between our formula and Miyatani's formula.