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

Abstract

We express the number of points on the Dwork hypersurface $$X_{\lambda }^d:x_1^d+x_2^d+\cdots +x_d^d=d\lambda x_1{x}_2\cdots x_d$$ over a finite field of order $q\not\equiv 1(\mathrm{mod}d)$ in terms of McCarthy's $p$-adic hypergeometric function for any odd prime $d$.