# A generalization of the Hasse-Witt matrix of a hypersurface

## TL;DR

This paper extends the classical Hasse-Witt matrix to provide meaningful mod p information about the zeta function of hypersurfaces of any degree, including cases where the traditional matrix is trivial.

## Contribution

The authors generalize the classical Hasse-Witt matrix to yield nontrivial zeta function congruences for all degrees, and analyze its differential equations and invertibility.

## Key findings

- The generalized matrix provides nontrivial mod p zeta function information for all degrees.
- The matrix satisfies specific differential equations.
- It is shown to be generically invertible.

## Abstract

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ 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 $\leq n$, 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.