Modular functions arising from some finite groups

Abstract:

In [2] Conway and Norton have assigned a "Thompson series" of the form \[ {q^{ - 1}} + {H_1}q + {H_2}{q^2} + \ldots \] to each element m of the Fischer-Griess "Monster" group M and conjectured that these functions are Hauptmoduls for certain genus-zero modular groups. We have found, for a large number of values of N, all the genus-zero groups between ${\Gamma _0}(N)$ and $PSL(2,R)$ that have Hauptmoduls of the above form, and this provides the necessary verification that the series assigned in [2] to particular elements of M really are such Hauptmoduls. (Atkin and Fong [1] have recently verified that ${H_n}(m)$ really is a character of M for all n.) We compute Thompson series for various finite groups and discuss the differences between these groups and M. We find that the resulting Thompson series are all Hauptmoduls for suitable genus-zero subgroups of $PSL(2,R)$.