Dwork cohomology, de Rham cohomology, and hypergeometric functions

Abstract

In the 1960s, Dwork developed a p -adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p -analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher codimension has been an open question. In this article, we settle the case of smooth, affine, complete intersections.