Abstract

We consider the family of diagonal hypersurfaces with monomial deformation
$${D}_{d,\lambda ,h}:{x}_{1}^{d}+{x}_{2}^{d}+\cdots +{x}_{n}^{d}d\lambda {x}_{1}^{{h}_{1}}{x}_{2}^{{h}_{2}}\dots {x}_{n}^{{h}_{n}}=0,$$ 
where
$d={h}_{1}+{h}_{2}+\cdots +{h}_{n}$ with
$\mathrm{gcd}({h}_{1},{h}_{2},\dots ,{h}_{n})=1$. We first provide a formula
for the number of
${\mathbb{\mathbb{F}}}_{q}$points
on
${D}_{d,\lambda ,h}$ in terms of
Gauss and Jacobi sums. This generalizes a result of Koblitz, which holds in the special case
$d\mid q1$. We then express the
number of
${\mathbb{\mathbb{F}}}_{q}$points
on
${D}_{d,\lambda ,h}$ 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
$\mathrm{gcd}(d,q1)=1$. In the
special case
${h}_{1}={h}_{2}=\cdots ={h}_{n}=1$
and
$d=n$, i.e.,
the Dwork hypersurface, we also generalize a previous result of the author which holds
when
$q$
is prime.

Keywords
diagonal hypersurface, Gauss sum, Jacobi sum, finite field
hypergeometric function, $p$adic hypergeometric function,
counting points

Mathematical Subject Classification
Primary: 11G25, 33E50
Secondary: 11S80, 11T24, 33C99

Milestones
Received: 9 December 2023
Accepted: 4 April 2024
Published: 30 April 2024

© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY). 
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