Arithmetic semistable elliptic surfaces. (Q2782401)

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 NEWLINE\[NEWLINES:\tau\mapsto \frac{-1}{\tau},\quad T:\tau\mapsto\tau+1,\quad (\text{Im}\,\tau>0).NEWLINE\]NEWLINE 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.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].