The group of automorphisms of cyclic extensions of rational function fields
type: Item
instance of: scholarly article
title: The group of automorphisms of cyclic extensions of rational function fields
zbMATH Open document ID: 0938.11056
published in: Journal of Algebra
publication date: 2000-06-12T00:00:00Z
review text: Let \(K = k(t)\) be a rational function field over an algebraically closed field of characteristic \(p \geq 0\). We have that the group of \(k\)-automorphisms of \(K\), \(\Aut_k K\) is isomorphic to \(PGL(2,k)\) the projective linear group. In his Ph.D. dissertation, \textit{R. Brandt} [Essen: Univ. (Gesamthochschule) Essen, Fachbereich Mathematik, Diss. (1988; Zbl 0716.11058)] studied the group of automorphisms of function fields \(F\) which are cyclic extensions of \(K\) with \(\text{Gal}(F/K) \cong C _q\) the cyclic group of prime order \(q \neq p\). In the paper under review the author studies function fields \(F\) with \(k\)-automorphism groups \(\Aut_k F \cong G\) which are cyclic extensions of order \(n\) of \(K\). It is assumed that \(p\) does not divide \(n\), that all the ramified places in \(F/K\) are fully ramified and that \(\text{Gal}(F/K)\) is normal in \(G\). This last condition is met when the genus of \(F\), \(g_F\), is larger than \((n-1) ^2\) or equivalently, when the number of ramified places, \(s\), in \(F/K\) satisfies \(s > 2 n\). The finite subgroups of \(PGL(2,k)\) are described by \textit{R. Valentini} and \textit{M. Madan} in [J. Reine Angew. Math. 318, 156-177 (1980; Zbl 0426.12016)]. In that paper, Valentini and Madan found the ramification type in each possible case. Using this description and the ramification type, the author determines the structure of \(G\) in term of generators and relations when \(G/C _n\) is isomorphic to \(C _n, D _n, A _4, A _5, S _4\) or a semidirect product of an elementary abelian group by a cyclic one, and in terms of cohomology class of the group extension in all other cases. It turns out that in most cases the ramification type in \(F/K\) and in \(K/F ^G\) determines the group structure of the extension \(G\). However there are cases of function fields with the same ramification type but different automorphism groups. It is also proved that for every finite subgroup of \(PGL(2, k)\) there is a cyclic Kumm…
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 1340596
zbMATH Keywords: Group of automorphisms
zbMATH Keywords: algebraic function fields
zbMATH Keywords: Galois extensions
author: Aristides Kontogeorgis
reviewed by: Gabriel D. Villa Salvador
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W1983478549
cites work: Strongly Branched Coverings of Closed Riemann Surfaces
cites work: Footnotes to a paper of Beniamino Segre. (The number of \(g^1_d\)'s on a general \(d\)-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves)
cites work: Q4040773
cites work: Die Automorphismengruppen hyperelliptischer Kurven. (The groups of automorphisms of hyperelliptic curves)
cites work: Endliche Gruppen I
cites work: The group of automorphisms of the function fields of the curve \(x^n+y^m+1=0\)
cites work: Equivalence problem and automorphism groups of certain compact Riemann surfaces
cites work: Algebraic function fields and codes
cites work: Q3856855
cites work: Q3861516
cites work: Q5587110
DOI: 10.1006/JABR.1998.7804
description: scientific article; zbMATH DE number 1340596