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.