Abstract. We characterize the 171 discrete subgroups of PSL 2 (R) occurring in Monstrous Moonshine in terms of their group-theoretic properties alone. The main resultLet M denote the largest sporadic simple group known as the Monster. It was predicted to exist by Fischer and Griess in the early 1970s and was constructed by Griess a few years later [14]. In their Monstrous Moonshine paper [7], Conway and Norton expand observations of McKay and Thompson to formulate the Moonshine conjecture:(1) To each cyclic group m , m ∈ M, is associated a function:such that each q-coefficient is the value of a character (known as the head character) of M at m. (2) Each f m is a principal modulus for a certain genus zero congruence group commensurable with the modular group, Γ = PSL 2 (Z). In fact, for each m , Conway and Norton proposed two discrete subgroups of PSL 2 (R) attached to f m defined as follows.The first group, F m , is the invariance group of f m . Let N be the level of F m , that is, the least N for which Γ(N ) ⊂ F (m). Write h = N/n where n is the order of m. It happens that h is always an integer such that h | 24 and h 2 | N . The second group, E m , is the subgroup of PSL 2 (R) whose elements multiply f m by hth roots of unity. Hence F m is a subgroup of E m . There are 171 distinct f m of which only 48 occur with h > 1. In [7], it is proposed that the E m coincide with congruence subgroups Γ 0 (n|h) + · · · , and F m is a subgroup of index h in E m with a similar parametrization of the form Γ 0 (n||h) + · · · (see Section 2 for the definitions). When h = 1, we have E m = F m and it is easy to exhibit a principal modulus for the corresponding parametrization (referred to as fundamental); these principal moduli are given by the η-products and theta functions of Table 3 in [7]. The fundamental elements can be modified to provide principal moduli for the parametrizations of F m even when h = 1; see Section 6 of [7]. To verify that

In this paper, a close connection is established between the geometry of certain genus zero Fuchsian groups and the analytic properties of the automorphic forms obtained by applying a certain differential operator to the Hauptmoduls of the groups.

We study and classify all the conjugacy classes of the genus zero congruence subgroups of PSL 2 (R) with no elliptic elements. We show that it suffices to classify those inside the modular group and determine them completely. We also discuss an application to modular curves.

We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.

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