The Jack Daniels Problem

TL;DR

This paper explores the intriguing connection between prime divisors related to the monster group and supersingular j-invariants, providing insights into the deep interplay known as monstrous moonshine.

Contribution

It demonstrates that moonshine functions for order p elements correspond to supersingular j-invariants in characteristic p, clarifying a longstanding mathematical coincidence.

Findings

Moonshine functions match supersingular j-invariants for order p elements.

The coincidence involves primes where these invariants are defined.

The paper offers a first-principles discussion of this prime coincidence.

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 are the primes p for which the characteristic p supersingular j-invariants are all defined over the field with p elements. This coincidence is often suggested to be 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.