Counting Points on Dwork Hypersurfaces and $p$-adic Gamma Function

TL;DR

This paper expresses the point count on Dwork hypersurfaces over finite fields using McCarthy's $p$-adic hypergeometric function, linking algebraic geometry with $p$-adic analysis.

Contribution

It provides a new formula connecting point counts on Dwork hypersurfaces to $p$-adic hypergeometric functions for odd primes.

Findings

Point counts are expressed via $p$-adic hypergeometric functions.

The formula applies to finite fields with specific congruence conditions.

Links between algebraic geometry and $p$-adic analysis are established.

Abstract

We express the number of points on the Dwork hypersurface $$X_{\lambda}^d: x_1^d+x_2^d+\cdots +x_d^d=d\lambda x_1x_2\cdots x_d$$ over a finite field of order $q \not \equiv 1 \pmod{d}$ in terms of McCarthy's $p$-adic hypergeometric function for any odd prime $d$.