A study of Hasse-Witt matrices by local methods
type: Item
instance of: scholarly article
title: A study of Hasse-Witt matrices by local methods
zbMATH Open document ID: 0637.14017
DOI: 10.1007/BF01215655
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174017
review text: Let K be an algebraic function field in one variable of genus \(g\) with a perfect constant field k of prime characteristic \(p.\) In a classic paper \textit{H. Hasse} and \textit{E. Witt} [Monatsh. Math. Phys. 43, 477-492 (1936; Zbl 0013.34102)] described the abelian unramified extension of K of degree \(p\) in terms of a matrix depending on a choice of g points in general position and local parameters at them. In this paper we modify this procedure to obtain a description in terms of a matrix analogous to the one of Hasse-Witt but depending on a choice of only one rational prime and a local parameter at it. Such a local description not only simplifies practical computations but it is also of theoretical interest: our matrix, like the original one of Hasse-Witt, represents the Cartier operator, and we get bounds for the ranks of this operator and its iterates in terms of Weierstrass gap sequences at primes of K. We obtain also a generalization, for higher genus, of a formula of Atkin and Swinnerton-Dyer for elliptic function fields.
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 4037992
zbMATH Keywords: algebraic function field
zbMATH Keywords: Cartier operator
zbMATH Keywords: Weierstrass gap sequences
zbMATH Keywords: Hasse-Witt matrix
author: Karl-Otto Stöhr
author: Paulo Henrique Viana
MaRDI profile type: Publication
cites work: Q5302599
cites work: Q3931656
cites work: Weierstrass points on certain non-classical curves
cites work: Zyklische unverzweigte Erweiterungskörper vom Primzahlgradep über einem algebraischen Funktionenkörper der Charakteristikp
cites work: Algebraic curves with non-classical types of gap sequences for genus three and four
cites work: Q4075566
cites work: Zur arithmetischen Theorie der algebraischen Funktionen. II: Allgemeine Theorie der Weierstraßpunkte
cites work: Q3277020
cites work: Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. II: Ein spezieller Typ von Funktionenkörpern
cites work: Weierstrass Points and Curves Over Finite Fields
cites work: A formula for the Cartier operator on plane algebraic curves.
cites work: Genus Change in Inseparable Extensions of Function Fields
OpenAlex ID: W2084913135
description: scientific article; zbMATH DE number 4037992