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 , defined over the finite field , , 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 -function associated to the exponential sums over affine space attached to an additive character of , and a polynomial . In order to do this, we define a Hasse–Witt matrix , which depends on the characteristic , the set of exponents of , and its coefficients. We also give some applications to the study of the Newton polygons of Artin–Schreier (hyperelliptic when ) 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. 117–152DOI 10.4171/RSMUP/74