Isotypic decomposition of the cohomology and factorization of the zeta functions of Dwork hypersurfaces

In this article, we study the representation of a group of automorphisms into the ℓ-adic cohomology of Dwork hypersurfaces (by a method similar to what Brünjes did for Fermat hypersurfaces) and show that it comes from representations defined over Q. This allows us to obtain a factorization of the zeta function of Dwork hypersurfaces. Compared to Kloosterman's factorization (obtained through the use of p-adic Monsky–Washnitzer cohomology), our factorization is slightly finer and we are able to explain the observation made by Candelas, de la Ossa and Rodriguez-Villegas in the quintic threefold case concerning the decomposition of each factor over some finite extension of Q.