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Modular subgroups, dessins d’enfants and elliptic K3 surfaces

Part of: Arithmetic algebraic geometry Surfaces and higher-dimensional varieties Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 August 2013

Yang-Hui He ,
John McKay  and
James Read
Yang-Hui He
Affiliation:
Department of Mathematics,City University,Northampton Square,London EC1V 0HB,United Kingdom email yang-hui.he@merton.ox.ac.uk School of Physics,NanKai University,Tianjin, 300071,PR China email yang-hui.he@merton.ox.ac.uk Merton College,University of Oxford,Oxford OX1 4JD,United Kingdom email yang-hui.he@merton.ox.ac.uk
John McKay
Affiliation:
Department of Mathematics and Statistics,Concordia University,1455 de Maisonneuve Boulevard,West Montreal, Quebec, H3G 1M8,Canada email mckay@encs.concordia.ca
James Read
Affiliation:
Oriel College,University of Oxford,Oxford OX1 4EW,United Kingdom email james.read@oriel.ox.ac.uk

Abstract

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We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

MSC classification

Secondary: 11G32: Dessins d'enfants, Belyĭ theory 11F06: Structure of modular groups and generalizations; arithmetic groups 14J32: Calabi-Yau manifolds 14J28: $K3$ surfaces and Enriques surfaces
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 271 - 318
Copyright
© The Author(s) 2013 

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