Unramified Galois coverings of algebraic curves
type: Item
instance of: scholarly article
title: Unramified Galois coverings of algebraic curves
zbMATH Open document ID: 0846.14014
published in: Journal of Number Theory
publication date: 1996-09-26T00:00:00Z
review text: Let \(X\) be a complete nonsingular irreducible algebraic curve of genus \(g \) defined over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(\pi_1 (X)\) be its algebraic fundamental group, and \(\Gamma_g\) the topological fundamental group of (any) compact Riemann surface of genus \(g\). It is well known since Grothendieck, that a finite group \(G\) of order prime to \(p\) can be realized as the Galois group of an unramified cover of \(X\) if and only if it is a quotient of \(\Gamma_g\). Other information on the structure of \(\pi_1 (X)\) comes from the Hasse-Witt invariant of \(X\), i.e. the \(\mathbb{F}_p\)-dimension of the \(p\)-torsion subgroup of the Jacobian variety of \(X\). In this paper the author counts the number of unramified Galois coverings of \(X\) whose Galois group is isomorphic to an extension of a group \(G\) of order prime to \(p\) by a finite group \(H\), which is an irreducible \(\mathbb{F}_p [G]\)-module. This counting depends on some invariants attached to a Galois cover \(Y \to X\) of group \(G\), called generalized Hasse-Witt invariants, arising in the canonical decomposition of the \(p\)-torsion space of the Jacobian variety \(J_Y\) of \(Y\). -- Some particular cases had already been treated by \textit{S. Nakajima} [in: Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 69-88 (1983; Zbl 0529.14016)] and \textit{H. Katsurada} [J. Math. Soc. Japan 31, 101-125 (1979; Zbl 0401.14004)], whose results are generalized in the present paper.
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 798698
zbMATH Keywords: algebraic curve
zbMATH Keywords: algebraic fundamental group
zbMATH Keywords: topological fundamental group
zbMATH Keywords: Hasse-Witt invariant
zbMATH Keywords: Jacobian variety
zbMATH Keywords: number of unramified Galois coverings
author: Amílcar Pacheco
reviewed by: Pietro Corvaja
MaRDI profile type: Publication
full work available at URL: jnth.1995.1087
OpenAlex ID: W2037777024
DOI: 10.1006/JNTH.1995.1087
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description: scientific article; zbMATH DE number 798698