Arithmetic semistable elliptic surfaces. (English) Zbl 1191.11010

McKay, John (ed.) et al., Proceedings on Moonshine and related topics. Proceedings of the workshop, Montréal, Canada, May 1999. Dedicated to the memory of Chih-Han Sah. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2879-7/pbk). CRM Proc. Lect. Notes 30, 119-130 (2001).
The authors list all torsion-free genus zero congruence subgroups of \(\text{PSL}_2(\mathbb Z)\) together with their indices and their cusp widths. They give the \(J\)-invariants of all the modular elliptic surfaces over \({\mathbb P}^1\) arising arising from such a subgroup, called arithmetic elliptic surfaces, which are semistable. To understand the situation at the level of cosets, they describe the action of the generators \(x\) and \(y\) as permutation images of the generating transformations \(S\) and \(ST\) of the modular group, where \[ S:\tau\mapsto \frac{-1}{\tau},\quad T:\tau\mapsto\tau+1,\quad (\text{Im}\,\tau>0). \] If the index of a subgroup is \(\mu\), then \(x\) and \(y\) generate a transitive permutation subgroup on the \(\mu\) cosets. If \(x\) and \(y\) act fixed-point freely, the subgroup is torsion-free. Moreover, the disjoint cycle decomposition of the permutation \(xy\) provides the genus as well as the cusp widths for each subgroup. There is a bijection between these data and Schreier coset graphs with \(\mu\) nodes which have no loops and whose \(xy\)-circuit lengths correspond both to the disjoint cycle lengths of \(xy\) and also to the cusp widths. The authors give the graphs of all the subgroups in question, from which the generating permutations on the cosets may be derived.
For the entire collection see [Zbl 0980.00029].


11F06 Structure of modular groups and generalizations; arithmetic groups
11F23 Relations with algebraic geometry and topology
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations