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Let $\mathbb{F}_q$ be a finite field with $q$ elements, $\psi$ a non-zeroelement of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim ofthis 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 $\psi^n \neq1$), an explicit decomposition in factors coming from affine varieties of odddimension $\leq n-4$ which are of hypergeometric type. The method we useconsists in counting separately the number of points of $X_\psi$ and of somevarieties of the preceding type and then compare them. This article answers, atleast when $n$ is prime, a question asked by D. Wan in his article "MirrorSymmetry for Zeta Functions".

Zeta function factorisation, Dwork hypersurfaces, hypergeometric hypersurfaces