The automorphism group of Leech lattice and elliptic modular functions (Q1062089)

Certain finite groups G with a representation \(\rho\) of degree d \(| 24\) over \({\mathbb{Q}}\) are shown to have moonshine properties in the sense of \textit{J. H. Conway} and \textit{S. P. Norton} [Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)]. This means that generalized characters \(a_ n\) of G, \(n\in {\mathbb{N}}\), are found such that for each \(\sigma\in G\) the series \(q^{-1}+\sum a_ n(\sigma) q^ n\) defines a modular function \(j_{\sigma}\) for some discrete subgroup of \(SL_ 2({\mathbb{R}})\) which contains a group \(\Gamma_ 0(N)\) and has a function field of genus zero generated by \(j_{\sigma}\). The functions \(j_{\sigma}\) are obtained here by various manipulations of Dedekind's eta-function with the exponents \(r_ t\in {\mathbb{Z}}\) in the characteristic polynomial \(\prod (x^ t-1)^{r_ t}\) of \(\rho\) (\(\sigma)\).