Hasse–Witt matrices for polynomials, and applications

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.