On Elliptic K3 Surfaces and Dessins d'Enfants

We classify subgroups of $\text{SL}(2,\mathbb{Z})$ up to conjugacy, which occur as monodromy groups of elliptically fibered K3 surfaces following a general strategy proposed by Bogomolov and Tschinkel. The essential step is the factorisation of the functional invariant $j$ with second factor a Belyi function of maximal possible degree and the classification of the corresponding subgroups $\bar{\mathrm{\Gamma }}$ of $\text{PSL}(2,\mathbb{Z})$ using dessins d’enfants.