On Rademacher's conjecture: congruence subgroups of genus zero of the modular group

Abstract

We list all genus zero congruence subgroups of PSL 2( Z) . There are altogether 132 of them (up to conjugation in PSL 2( Z) ). Geometrical invariants (genus, v 2, v 3, number of cusps, index in PSL 2( Z) ), fundamental polygons, Farey symbol, and independent generators of all such groups are determined. Generators of the function fields associated to such groups are also determined ( http://www.math.nus.edu.sg/~matlml/).