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

TL;DR

This paper derives a formula for counting points on a family of diagonal hypersurfaces over finite fields using McCarthy's p-adic hypergeometric functions, extending known results for Dwork hypersurfaces.

Contribution

It generalizes existing formulas for Dwork hypersurfaces to a broader class of monomial deformations with arbitrary degree d ≥ n.

Findings

Provides a new formula for point counts on $D______$ hypersurfaces.

Extends the application of McCarthy's p-adic hypergeometric functions.

Connects geometric properties of hypersurfaces with hypergeometric functions.

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_\lambda^d: X_1^d+X_2^d+\cdots+X_n^d=\lambda d X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$, $h_i\geq1$, $\sum_{i=1}^n h_i=d$, and $\gcd(d,h_1,h_2,\ldots, h_n)=1$. 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 $d\geq n$.