Xavier Gonzalez

Moonshine for all finite groups

``Monstrous Moonshine'' began with the distinguished observation by McKay in 1978 that $1 + 196883 = 196884$. This observation was one of many that related the Fourier coefficients of Klein's $j$-invariant to the dimensions of the irreducible representations of the Monster group $\mathbb(M)$. Ultimately, in 1992 Borcherds proved the conjecture made by Conway and Norton in 1979 relating representation of $\mathbb(M)$ to Hauptmoduln of certain genus $0$ modular curves. In part for this achievement, Borcherds won the Fields Medal in 1998. More recently, moonshine---the connection between representations of finite groups and distinguished modular forms---has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups.

This talk presents the history of moonshine and the results of DeHority, Vafa, Van Peski, and the speaker in \emph{Moonshine for all finite groups} (Res. Math. Sci. \mathbf{5}, 2018). For every finite group $G$, we give constructions of infinitely many graded infinite-dimensional $\C[G]$-modules where the McKay-Thompson series for a conjugacy class $[g]$ is a weakly holomorphic modular function properly on $\Gamma_0(\ord(g))$. As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level $\ord(g)$ as the graded trace functions for any conjugacy class $[g]$, and completely classify the finite abelian groups with this property. In particular, these include $(\Z / 5 \Z)^5$ and $(\Z / 7 \Z)^4$, which are not subgroups of the Monster.