A generalization of the Hasse-Witt matrix of a hypersurface

The Hasse-Witt matrix of a hypersurface in Pn over a finite field of characteristic p gives essentially complete mod p information about the zeta function of the hypersurface. But if the degree d of the hypersurface is ≤n, the zeta function is trivial mod p and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all d. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.