Noncongruence Subgroups, Covers and Drawings

Summary

The theory of congruence subgroups, more precisely the theory of the action of congruence subgroups of the modular group on the upper half plane, is an area of mathematics in which the mathematical structure is well understood and wonderfully intricate, and beautiful numbers appear as if by magic (see for instance [GZ]). When one turns to noncongruence subgroups, the mathematical structure must be built with fewer bricks, since in particular the Hecke theory is missing, but though the structure is more mysterious it seems to be almost as rich, and the ‘ballet of numbers’ continues to be just as beautiful. There is an enormous literature on the action of general discontinuous groups on the upper half plane; and there is an enormous literature concerned with the arithmetic theory of the action of congruence subgroups of the modular group; in contrast, the arithmetic theory of subgroups which are not congruence subgroups is surprisingly little developed. In a neighbouring area, there is a beautiful corpus of recent work concerned with the arithmetic of Galois coverings of the projective line, ramified in a prescribed way above a finite set of places; a motivation for this has been its application to the inverse Galois problem.