On the discrete groups of Mathieu moonshine (Q2875940)

Moonshine is a mysterious relationship between modular forms and sporadic finite simple groups. Such a phenomenon for the Mathieu group was first noticed by \textit{T. Eguchi} et al. [Exp. Math. 20, No. 1, 91--96 (2011; Zbl 1266.58008)] in connection with a \(K3\) surface. In the paper under review, the authors show that the space of cusp forms for the Hecke congruence group \(\Gamma_0(n)\) of weight \(3/2\) with a certain multiplier is one dimensional if and only if \(n\) is the order of an element of the Mathieu group \(M_{23}\). The result includes the assertion that for a prime \(p\), \(\Gamma_0(p)\) has a unique cusp form of weight \(3/2\) with the multiplier if and only if \(p\) divides the order of the Mathieu group \(M_{24}\). Perspectives on future developments are also discussed.NEWLINENEWLINEFor the entire collection see [Zbl 1285.00037].