Number of 𝔽q-points on diagonal hypersurfaces and hypergeometric function

Abstract

Let [Formula: see text] denote the family of monomial deformations of diagonal hypersurface over a finite field [Formula: see text] given by [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The Dwork hypersurface is the case when [Formula: see text], that is, [Formula: see text]. Formulas for the number of [Formula: see text]-points on the Dwork hypersurfaces in terms of McCarthy’s [Formula: see text]-adic hypergeometric functions are known. In this paper, we provide a formula for the number of [Formula: see text]-points on [Formula: see text] in terms of McCarthy’s [Formula: see text]-adic hypergeometric function which holds for [Formula: see text].