The University of Southampton
University of Southampton Institutional Repository

Uniform dessins of low genus

Uniform dessins of low genus
Uniform dessins of low genus
It is known that every dessin (map or hypermap) corresponds to a finite index subgroup of a triangle group and can be embedded naturally into some Riemann surface [JSl, JS3]. A dessin is uniform if its (hyper)vertices all have the same valency, its (hyper)edges all have the same valency, and its (hyper)faces all have the same valency; uniform dessins correspond to torsion-free subgroups of triangle groups. By the theorems of Belyi [Bel] and Wolfart [Wol], a compact Riemann surface X is defined over the field of algebraic numbers Q if and only if X carries a dessin (also see [Gro]). In this thesis we study the uniform dessins of genus g < 3 and investigate their connections with algebraic curves, Belyi's Theorem, and the absolute Galois group Gal(Q/Q). An elliptic curve of modulus r can be uniformized by a finite index subgroup of a Euclidean triangle group if and only if r G Q(i) or r £ Q(p); these elliptic curves are said to have Euclidean Belyi uniformizations and naturally carry the uniform dessins of genus 1. Using results from number theory, it is proved that there are only five rational elliptic curves with Euclidean Belyi uniformizations. A classification of the genus 1 uniform maps is given which extends the notation for genus 1 regular maps found in [CMo]. Formulae are derived for the number of genus 1 uniform maps with a given number of vertices, and the refiexible maps are described. Belyi functions are computed in a number of cases, and arbitrarily large Galois orbits of genus 1 uniform dessins are constructed. The existence of two uniform maps of genus g > 1 lying on conformally equivalent Riemann surfaces is considered. This leads naturally to the study of arithmetic Fuchsian groups [Vi] and motivates the definitions of arithmetic and non-arithmetic maps. General results are proved for non-arithmetic maps, and specific examples are given in the arithmetic case.
University of Southampton
Syddall, Robert I.
a123b76c-e1f2-4a02-8e0f-9410cac9e218
Syddall, Robert I.
a123b76c-e1f2-4a02-8e0f-9410cac9e218
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b

Syddall, Robert I. (1997) Uniform dessins of low genus. University of Southampton, Doctoral Thesis, 155pp.

Record type: Thesis (Doctoral)

Abstract

It is known that every dessin (map or hypermap) corresponds to a finite index subgroup of a triangle group and can be embedded naturally into some Riemann surface [JSl, JS3]. A dessin is uniform if its (hyper)vertices all have the same valency, its (hyper)edges all have the same valency, and its (hyper)faces all have the same valency; uniform dessins correspond to torsion-free subgroups of triangle groups. By the theorems of Belyi [Bel] and Wolfart [Wol], a compact Riemann surface X is defined over the field of algebraic numbers Q if and only if X carries a dessin (also see [Gro]). In this thesis we study the uniform dessins of genus g < 3 and investigate their connections with algebraic curves, Belyi's Theorem, and the absolute Galois group Gal(Q/Q). An elliptic curve of modulus r can be uniformized by a finite index subgroup of a Euclidean triangle group if and only if r G Q(i) or r £ Q(p); these elliptic curves are said to have Euclidean Belyi uniformizations and naturally carry the uniform dessins of genus 1. Using results from number theory, it is proved that there are only five rational elliptic curves with Euclidean Belyi uniformizations. A classification of the genus 1 uniform maps is given which extends the notation for genus 1 regular maps found in [CMo]. Formulae are derived for the number of genus 1 uniform maps with a given number of vertices, and the refiexible maps are described. Belyi functions are computed in a number of cases, and arbitrarily large Galois orbits of genus 1 uniform dessins are constructed. The existence of two uniform maps of genus g > 1 lying on conformally equivalent Riemann surfaces is considered. This leads naturally to the study of arithmetic Fuchsian groups [Vi] and motivates the definitions of arithmetic and non-arithmetic maps. General results are proved for non-arithmetic maps, and specific examples are given in the arithmetic case.

Text
Syddall - Version of Record
Available under License University of Southampton Thesis Licence.
Download (7MB)

More information

Published date: 1 July 1997

Identifiers

Local EPrints ID: 426659
URI: http://eprints.soton.ac.uk/id/eprint/426659
PURE UUID: fb190e34-9b22-480f-970f-0fadff2aac83

Catalogue record

Date deposited: 10 Dec 2018 17:30
Last modified: 13 Aug 2019 16:30

Export record

Contributors

Author: Robert I. Syddall
Thesis advisor: David Singerman

University divisions

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×