On the Deuring-Shafarevich formula
type: Item
instance of: scholarly article
title: On the Deuring-Shafarevich formula
zbMATH Open document ID: 1237.14034
author: Daisuke Shiomi
published in: Tokyo Journal of Mathematics
publication date: 2012-04-03T00:00:00Z
review text: The author gives a new proof of the Deuring-Shafarevich formula. Let \(K\) be a global function field over a finite field of characteristic \(p\). Let \(L/K\) be a geometric cyclic extension of degree \(p\). Let \(\lambda_L\) (resp. \(\lambda_K\)) denote the Hasse-Witt invariant of \(L\) (resp. \(K\)). The formula mentioned above has the following form: \(\lambda_L-1=p(\lambda_K-1)+\sum_{P\in S_K}(e_P-1)\deg_KP\), where \(S_K\) is the set of primes of \(K\), \(e_P\) is the ramification index of \(P\) in \(L/K\), and \(\deg_KP\) is the degree of \(P\). He proves this formula by using the fact that \(\lambda_K=\deg\bar{Z}_K(X)\), where \(\bar{Z}_K(X)\in\mathbb{F}_p[X]\) denotes the reduction of the zeta polynomial \(Z_K(X)\), and by focusing on the behavior of the polynomial under the field extension.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 11M38
zbMATH DE Number: 6020922
zbMATH Keywords: Jacobians
zbMATH Keywords: Zeta functions
zbMATH Keywords: Hasse-Witt invariant
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1327931387
OpenAlex ID: W1992255817
cites work: Q3222966
cites work: Automorphismen und Divisorenklassen der Ordnung \(l\) in algebraischen Funktionenkörpern
cites work: Explicit Class Field Theory for Rational Function Fields
cites work: Class numbers of cyclotomic function fields
cites work: Some consequences of the Riemann hypothesis for varieties over finite fields
cites work: On a theorem of M. Deuring and I. R. Šafarevič
cites work: Q2776773
cites work: The p-rank of Artin-Schreier curves
cites work: Sur l'image de l'application d'Abel-Jacobi de Bloch
DOI: 10.3836/TJM/1327931387
description: scientific article; zbMATH DE number 6020922