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Modular subgroups, dessins d’enfants and elliptic K3 surfaces

Published online by Cambridge University Press:  01 August 2013

Yang-Hui He
Affiliation:
Department of Mathematics,City University,Northampton Square,London EC1V 0HB,United Kingdom email [email protected] School of Physics,NanKai University,Tianjin, 300071,PR China email [email protected] Merton College,University of Oxford,Oxford OX1 4JD,United Kingdom email [email protected]
John McKay
Affiliation:
Department of Mathematics and Statistics,Concordia University,1455 de Maisonneuve Boulevard,West Montreal, Quebec, H3G 1M8,Canada email [email protected]
James Read
Affiliation:
Oriel College,University of Oxford,Oxford OX1 4EW,United Kingdom email [email protected]

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.

Type
Research Article
Copyright
© The Author(s) 2013 

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