Number of ${\mathbb{F}}_q$-points on Diagonal hypersurfaces and hypergeometric function

Abstract

Let $D_{\lambda }^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field ${\mathbb{F}}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\ge 2$, $h_i\ge 1$, ${\sum }_{i=1}^n{h}_i=d$, and $\gcd (d,h_1,h_2,\dots ,h_n)=1$. The Dwork hypersurface is the case when $d=n$, that is, $h_1=h_2=\cdots =h_n=1$. Formulas for the number of ${\mathbb{F}}_q$-points on the Dwork hypersurfaces in terms of McCarthy's $p$-adic hypergeometric functions are known. In this article we provide a formula for the number of ${\mathbb{F}}_q$-points on $D_{\lambda }^d$ in terms of McCarthy's $p$-adic hypergeometric function which holds for $d\ge n$.

13 pages