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Bourjaily, Jacob L. He, Yang-Hui McLeod, Andrew J. von Hippel, Matt and Wilhelm, Matthias 2018. Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms. Physical Review Letters, Vol. 121, Issue. 7,

Shabat, G. 2017. Calculating and Drawing Belyi Pairs. Journal of Mathematical Sciences, Vol. 226, Issue. 5, p. 667.

He, Yang-Hui and Read, James 2015. Dessins d’enfants in N = 2 $$\mathcal{N}=2$$ generalised quiver theories. Journal of High Energy Physics, Vol. 2015, Issue. 8,

He, Yang-Hui McKay, John and Read, James 2013. Modular subgroups, dessins d’enfants and elliptic K3 surfaces. LMS Journal of Computation and Mathematics, Vol. 16, Issue. , p. 271.

He, Yang-Hui and McKay, John 2013. N=2 gauge theories: Congruence subgroups, coset graphs, and modular surfaces. Journal of Mathematical Physics, Vol. 54, Issue. 1, p. 012301.

Schütt, Matthias 2010. K3 surfaces with Picard rank 20. Algebra & Number Theory, Vol. 4, Issue. 3, p. 335.

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• Print publication year: 2008
• Online publication date: May 2010

### Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps

Summary

Abstract. In this paper we give an introduction to Belyi-maps and Grothendieck's dessins d'enfant. In addition we provide an explicit method to compute the Belyi-maps corresponding to all semi-stable families of elliptic curves with six singular fibers.

Introduction

In a paper by Miranda and Persson, the authors study semi-stable elliptic fibrations over ℙ1 of K3-surfaces with 6 singular fibres. In their paper the authors give a list of possible fiber types for such fibrations. It turns out that there are 112 cases. The corresponding J-invariant is a so-called Belyifunction. More particularly, J is a rational function of degree 24, it ramifies of order 3 in every point above 0, it ramifies of order 2 in every point above 1, and the only other ramification occurs above infinity. To every such map we can associate a so-called ‘dessin d'enfant’ (a name coined by Grothendieck) which in its turn uniquely determines the Belyi map. If f: C → ℙ1 is a Belyi map, the dessin is the inverse image under f of the real segment.

Several papers, e.g., have been devoted to the calculation of some of the rational J-invariants for the Miranda-Persson list. It turns out that explicit calculations quickly become too cumbersome (even for a computer) if one is not careful enough. The goal of this paper is to compute all J-invariants corresponding to the Miranda-Persson list.

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