Hasse-Witt matrices for polynomials, and applications
type: Item
instance of: scholarly article
title: Hasse-Witt matrices for polynomials, and applications
zbMATH Open document ID: 1479.11157
DOI: 10.4171/RSMUP/74
author: Régis Blache
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2021-07-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11M38
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 7367433
zbMATH Keywords: zeta and \(L\)-functions in characteristic \(p\)
zbMATH Keywords: \(p\)-adic estimates for character sums
zbMATH Keywords: Newton polygons of curves
MaRDI profile type: Publication
cites work: $p$-adic estimates for exponential sums and the theorem of Chevalley-Warning
cites work: Hasse invariants and mod \(p\) solutions of \(A\)-hypergeometric systems
cites work: A generalization of the Hasse-Witt matrix of a hypersurface
cites work: Zeroes of Polynomials Over Finite Fields
cites work: Notes on Crystalline Cohomology. (MN-21)
cites work: Newton stratification for polynomials: the open stratum
cites work: Valuation of exponential sums and the generic first slope for Artin-Schreier curves
cites work: On Exponential Sums in Finite Fields
cites work: Q3222966
cites work: On the zeta function of a hypersurface. I
cites work: On a Theorem of Ax
cites work: Algebraic solutions of differential equations (p-curvature and the Hodge filtration)
cites work: Q5183631
cites work: Q3861527
cites work: Q3285702
cites work: Frobenius and the Hodge filtration
cites work: The Hasse-Witt-matrix of special projective varieties
cites work: Tight bounds for Chevalley–Warning–Ax–Katz type estimates, with improved applications
cites work: Q3338379
cites work: Problems from the Workshop on Automorphisms of Curves
cites work: Q3340959
cites work: Completely continuous endomorphisms of \(p\)-adic Banach spaces
cites work: Families of supersingular curves in characteristic 2.
cites work: Q4534506
cites work: Q4421499
cites work: On the p-adic Theory of Exponential Sums
cites work: Newton polygons of zeta functions and \(L\) functions
cites work: Asymptotic variation of L functions of one-variable exponential sums
full work available at URL: https://doi.org/10.4171/rsmup/74
OpenAlex ID: W3157339146
description: scientific article; zbMATH DE number 7367433