N = 2 super Yang-Mills and subgroups of SL(2, Z)

We discuss SL(2, Z) subgroups appropriate for the study of N = 2 super Yang-Mills with N_f = 2 n flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular, uniqueness arguments are sufficient to construct the SU(3) curve, up to two numerical constants, which can be fixed by making some assumptions about strong coupling behavior. We also discuss the situation for higher groups. We also include a derivation of the closed form β-function for the SU(2) and SU(3) theories without matter, and the massless theories with N_f = n.