# Hasse–Witt matrices for polynomials, and applications

### Régis Blache

INSPÉ de la Guadeloupe, Les Abymes, Guadeloupe

## Abstract

In a classical paper, Manin gives a congruence [15, Theorem 1] for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve $C$, defined over the finite field $\mathbf{F}_{q}$, $q=p^m$, in terms of its Hasse–Witt matrix. The aim of this article is to prove a congruence similar to Manin’s one, valid for any $L$-function $L(f,T)$ associated to the exponential sums over affine space attached to an additive character of $\mathbf{F}_q$, and a polynomial $f$. In order to do this, we define a Hasse–Witt matrix $\mathrm{HW}(f)$, which depends on the characteristic $p$, the set $D$ of exponents of $f$, and its coefficients. We also give some applications to the study of the Newton polygons of Artin–Schreier (hyperelliptic when $p=2$) curves, and zeta functions of varieties.

## Cite this article

Régis Blache, Hasse–Witt matrices for polynomials, and applications. Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 117–152

DOI 10.4171/RSMUP/74