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/ ).