A Complete Hypergeometric Point Count Formula for Dwork Hypersurfaces

TL;DR

This paper derives a comprehensive formula for counting points on Dwork hypersurfaces over finite fields using hypergeometric functions, extending previous work and providing explicit results for specific cases.

Contribution

It proves a new point count formula for Dwork hypersurfaces in terms of finite field hypergeometric functions, including explicit formulas for Dwork threefolds, and extends previous results to even degrees.

Findings

Point counts expressed via hypergeometric functions for odd degrees.

Conjecture for similar formulas when degrees are even.

Explicit formula for Dwork threefolds.

Abstract

We extend our previous work on hypergeometric point count formulas by proving that we can express the number of points on families of Dwork hypersurfaces $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d$$ over finite fields of order $q\equiv 1\pmod d$ in terms of Greene's finite field hypergeometric functions. We prove that when $d$ is odd, the number of points can be expressed as a sum of hypergeometric functions plus $(q^{d-1}-1)/(q-1)$ and conjecture that this is also true when $d$ is even. The proof rests on a result that equates certain Gauss sum expressions with finite field hypergeometric functions. Furthermore, we discuss the types of hypergeometric terms that appear in the point count formula and give an explicit formula for Dwork threefolds.