Monstrous Moonshine - History

History

Specifically, Conway and Norton, following an initial observation by John McKay, found that the Fourier expansion of j(τ) (sequence A000521 in OEIS), with τ denoting the half-period ratio, could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (sequence A001379 in OEIS):

where, and

Conway and Norton formulated conjectures concerning the functions obtained by replacing the traces on the identity by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus zero. In other words, if G_g is the subgroup of SL₂(R) which fixes, then the quotient of the upper half of the complex plane by G_g is a sphere with a finite number of points removed, corresponding to the cusps of G_g.

It turns out that lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by Richard Ewen Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac-Moody superalgebras. Borcherds won the Fields medal for his work, and more connections between M and the j-function were subsequently discovered.