Modular Subgroups, Dessins d'Enfants and Elliptic K3 Surfaces

TL;DR

This paper explores the classification of modular subgroups, dessins d'enfants, and elliptic K3 surfaces, revealing new connections between subgroup data and geometric structures like Calabi-Yau threefolds and extremal K3 surfaces.

Contribution

It provides a detailed analysis of genus zero, torsion-free modular subgroups, their ramification data, and links to Calabi-Yau and K3 surfaces, including explicit subgroup representatives.

Findings

Classification of 33 modular subgroups and their dessins d'enfants.

Identification of Calabi-Yau threefolds associated with index 36 subgroups.

Analysis of 112 semi-stable elliptic fibrations as extremal K3 surfaces.

Abstract

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 with K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over 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.