Zeta function factorisation, Dwork hypersurfaces, hypergeometric hypersurfaces

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements, $\psi$ a non-zero element of ${\mathbb{F}}_q$, and $n$ an integer $\ge 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over ${\mathbb{F}}_q$ defined by $X_{\psi }\text{\hspace{0.17em}⁣}:x_1^n+\mathrm{.}\mathrm{.}\mathrm{.}+x_n^n-n\psi x_1\mathrm{.}\mathrm{.}\mathrm{.}{x}_n=0$ has, when $n$ is prime and $X_{\psi }$ is non singular (i.e. when $\psi^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\le n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_{\psi }$ and of some varieties of the preceding type and then compare them. This article answers, at least when $n$ is prime, a question asked by D. Wan in his article “Mirror Symmetry for Zeta Functions”.