Hasse–Witt matrices for polynomials, and applications

  • Régis Blache

    INSPÉ de la Guadeloupe, Les Abymes, Guadeloupe
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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 CC, defined over the finite field Fq\mathbf{F}_{q}, q=pmq=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 LL-function L(f,T)L(f,T) associated to the exponential sums over affine space attached to an additive character of Fq\mathbf{F}_q, and a polynomial ff. In order to do this, we define a Hasse–Witt matrix HW(f)\mathrm{HW}(f), which depends on the characteristic pp, the set DD of exponents of ff, and its coefficients. We also give some applications to the study of the Newton polygons of Artin–Schreier (hyperelliptic when p=2p=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