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 $\geq 3$ 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 $X_\psi \colon x_1^n+...+x_n^n - n \psi x_1... x_n=0$ 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 $\leq n-4$ 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".