On the discrete groups of Moonshine

Abstract

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

Rademacher sums, moonshine and gravity

“…Proposition 6.2) that the function R (−1) Γ,p|q be invariant for this subgroup Γ 0 (nh). Our moduli interpretation of the n h-type groups, together with the invariance and symmetry conditions on Rademacher sums just described, facilitate a reformulation of the conditions of [7]. We thus arrive at a new characterization (cf.…”

“…However, in spite of all the subsequent spectacular developments related to the Monster, the consensus of the experts (cf. e.g., [7]), is that "the real nature of moonshine is still remote".…”

“…A group theoretic characterization of the functions arising as McKay-Thompson series was given in [7]. There the authors showed that a holomorphic function on H coincides with T g (z) for some g ∈ M if and only if it is the normalized hauptmodul for a group Γ satisfying each of the following properties.…”

“…Monstrous moonshine associates a group Γ g commensurable with G(Z) and having width one at infinity to each (conjugacy class of) element(s) g in the Monster group M. In the article [7] the set of groups {Γ g | g ∈ M} is characterized in purely group theoretic terms. In this section we describe a reformulation of this characterization in terms of normalized Rademacher sums, and a certain family of moduli problems for solid tori with conformal structure on the boundary.…”

“…In light of Lemma 6.4 we may reformulate the fourth condition of the main theorem of [7] as follows.…”