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Modular subgroups, dessins d’enfants and elliptic K3 surfaces. (English) Zbl 1298.11058

In this work, the authors study the conjugacy classes of genus zero, torsion free modular subgroups. The ramification data and dessins d’enfants for these are computed. In a special case of index 36 subgroups the corresponding Calabi-Yau threefolds are identified. Moreover, using the classification of external, semi-elliptic K3 surfaces with six singular fibres and associated dessins (by R. Miranda and U. Persson [Math. Z. 201, No. 3, 339–361 (1989; Zbl 0694.14019)] and by F. Beukers and H. Montanus [in: Number theory and polynomials. Proceedings of the workshop, Bristol, UK, 2006. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 352, 33–51 (2008; Zbl 1266.11078)]) the authors study whether they can be assigned to subgroups of modular group, not necessarily congruence. Using elliptic \(j\)-invariants as Belyi maps and “cartographic groups”, the authors find the generating set for a representative of the class of subgroup for each case.

MSC:

11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
11F06 Structure of modular groups and generalizations; arithmetic groups
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
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