Automorphisms of ramified coverings of Riemann surfaces
type: Item
instance of: scholarly article
title: Automorphisms of ramified coverings of Riemann surfaces
zbMATH Open document ID: 0845.30027
author: Masaharu Tanabe
published in: Kodai Mathematical Journal
publication date: 1996-09-18T00:00:00Z
review text: Let \(M\), \(N\) be Riemann surfaces, and \(\pi: M\to N\) be an unlimited, ramified, finite-sheeted covering map. We say that \((M, \pi, N)\) is of excluded type if \(N= \mathbb{C}\) and the image set of ramification points of \(\pi\) consists of one point on \(N\). The main result of the paper is the following. Suppose that the above-mentioned covering map \(\pi: M\to N\) is of non-excluded type. Then there exist a Riemann surface \(S\) and unlimited covering maps \(\sigma: M\to S\) and \(q: S\to N\) such that \(\pi= q\circ \sigma\), where \(q\) is unramified, and for any \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\) there exists a \(T_*\in \Aut(S)\) which satisfies \(\sigma\circ T= T_*\circ \sigma\). The following corollaries are of special interest. Corollary 1. Let \(\pi: M\to N\) be an unlimited, ramified, \(l\)-sheeted covering, and of non-excluded type. Suppose that \(l\) is a prime number. Then for any arbitrary \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\), there exists a \(T_*\in \Aut(N)\) which satisfies \(\pi\circ T= T_*\circ \pi\). Corollary 2. Let \(\pi: M\to N\) be an unlimited, ramified, finite-sheeted covering, and of non-excluded type. Suppose that \(N\) is simply connected. Then for an arbitrary \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\), there exists a \(T_*\in \Aut(N)\) which satisfies \(\pi\circ T= T_*\circ \pi\).
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 840319
zbMATH Keywords: Riemann surface
reviewed by: Alexander Mednykh
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043425
OpenAlex ID: W1975655896
cites work: Q3931656
cites work: Q5666814
cites work: Q5568135
cites work: Observations on Morphisms of Closed Riemann Surfaces
DOI: 10.2996/KMJ/1138043425
description: scientific article; zbMATH DE number 840319