Hasse-Witt matrices for polynomials, and applications
Summary: In a classical paper, \textit{Yu. I. Manin} gives a congruence [Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802), Theorem 1] for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve \(C\), defined over the finite field \(\mathbb{F}_q\), \(q=p^m\), 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 \(\mathbb{F}_q\), 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.
zeta and \(L\)-functions in characteristic \(p\)
scientific article; zbMATH DE number 7367433
scientific article