Replication formulae for \(n| h\)-type Hauptmoduls
In Monstrous Moonshine, to each of the 172 conjugacy classes of cyclic subgroups of the Monster, \(\mathbb M\), corresponds a Hauptmodul and its \(q\)-expansion at its cusp at {infinity}. There are necessarily relations among the infinitely many \(q\)-coefficients of the series, these are called replication formulae. There is also the important replication power map which maps a Hauptmodul \(f_{\langle g\rangle}\) to \(f_{\langle g^k\rangle}\). The groups of the title are described and Hauptmoduln given, and shown to satisfy the replication formulae. A consequence is that the \(q\)-coefficients are generalized characters of \(\mathbb M\). This work should be compared with the unpublished [\textit{M. Koike}, On the replication formulae and Hecke operators (Nagoya University Preprint) (1987)].
scientific article; zbMATH DE number 868022