| Résume | Holomorphicity of the Schwarzian of a Hauptmodul leads to the classification of torsion-free genus zero congruence subgroups of the modular group. The subgroups are of index $m = 6,12,24, 36,48, 60$, yielding Beauville's ($m = 12$) six extremal elliptic surfaces. There are $2 + m/6$ singular fibres. A trivalent graph encodes the cusp widths of each Hauptmodul. (This is joint work with Abdellah Sebbar.) Reference: CRM Proceedings on Moonshine \& Related Topics (2002). |
