On Dwork cohomology for singular hypersurfaces

Let Z be a projective hypersurface over a finite field. With no smoothness assumption, we relate the p-adic cohomology spaces constructed by Dwork in his study of the zeta function of Z to the rigid homology spaces of Z. The key result is a general theorem based on the Fourier transform for arithmetic D-modules which extends to the rigid context results proved in the algebraic one by Adolphson and Sperber, and Dimca, Maaref, Sabbah and Saito. If V, V' are dual vector bundles over a smooth p-adic formal scheme X, u is a section of V' , Z the zero locus of its reduction mod p, this theorem gives an identification between the overconvergent local cohomology of the structure sheaf of X with supports in Z and the relative rigid cohomology of V with coefficients in the Dwork isocrystal associated to u. Thanks to this result, we also give an interpretation of a canonical filtration on the Dwork complexes in terms of the rigid homology spaces of the intersections of Z with intersections of coordinate hyperplanes.