The Jack Daniels problem

Abstract

In 1975 Ogg offered a bottle of Jack Daniels for an explanation of the fact that the prime divisors of the order of the monster M are the primes p for which the characteristic p supersingular j-invariants are all defined over Fp. This coincidence is often suggested as the first hint of monstrous moonshine, the deep unexpected interplay between the monster and modular functions. We revisit Ogg's problem, and we point out (using existing tools) that the moonshine functions for order p elements give the set of characteristic p supersingular j-invariants (apart from 0 and 1728). Furthermore, we discuss this coincidence of the two seemingly unrelated sets of primes using the first principles of moonshine.