On Dwork cohomology for singular hypersurfaces

Abstract

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 (cf. [29], [30], [31]), to the rigid homology spaces of Z. The key result is a general theorem based on the Fourier transform for D X , Q-modules [40], which extends to the rigid context results proved in the algebraic one by Adolphson and Sperber [3], and Dimca, Maaref, Sabbah and Saito [27]. If V, V ′ are dual vector bundles over a smooth p-adic formal scheme X , u : X → V ′ a section, Z the zero locus of its reduction mod p, this theorem gives an identification between the overconvergent local cohomology of OX , Q 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. 2000 Mathematics Subject Classification: 13N10, 14F30, 14F40, 14G10, 14J70, 16S32, 32C38.