Hypergeometric Functions and Relations to Dwork Hypersurfaces

TL;DR

This paper derives formulas for counting points on Dwork hypersurfaces over finite fields using hypergeometric functions, connecting these counts to period integrals and extending results to higher dimensions.

Contribution

It introduces hypergeometric point count formulas for Dwork hypersurfaces over finite fields and explores their relation to period integrals, extending to higher dimensions.

Findings

Point counts expressed via Greene's hypergeometric functions.

Hypergeometric formulas developed for all odd primes.

Connection established between period integrals and Frobenius traces.

Abstract

We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's $p$-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.