The number of $\mathbb{F}_q$-points on diagonal hypersurfaces with monomial deformation

Abstract

We consider the family of diagonal hypersurfaces with monomial deformation $$D_{d, λ, h}: x_1^d + x_2^d \dots + x_n^d - d λ\, x_1^{h_1} x_2^{h_2} \dots x_n^{h_n}=0$$ where $d = h_1+h_2 +\dots + h_n$ with $\gcd(h_1, h_2, \dots h_n)=1$. We first provide a formula for the number of $\mathbb{F}_{q}$-points on $D_{d, λ, h}$ in terms of Gauss and Jacobi sums. This generalizes a result of Koblitz, which holds in the special case ${d \mid {q-1}}$. We then express the number of $\mathbb{F}_{q}$-points on $D_{d, λ, h}$ in terms of a $p$-adic hypergeometric function previously defined by the author. The parameters in this hypergeometric function mirror exactly those described by Koblitz when drawing an analogy between his result and classical hypergeometric functions. This generalizes a result by Sulakashna and Barman, which holds in the case $\gcd(d,{q-1})=1$. In the special case $h_1 = h_2 = \dots =h_n = 1$ and $d=n$, i.e., the Dwork hypersurface, we also generalize a previous result of the author which holds when $q$ is prime.