Modular Curve - Relation With The Monster Group

Relation With The Monster Group

Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the monstrous moonshine conjectures. First several coefficients of q-expansions of their Hauptmoduln were computed already in 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster.

Another connection is that the modular curve corresponding to the normalizer Γ₀(p)+ of Γ₀(p) in SL(2,R) has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are precisely the prime factors of the order of the monster group. The result about Γ₀(p)+ is due to Jean-Pierre Serre, Andrew Ogg and John G. Thompson in the 1970s, and the subsequent observation relating it to the monster group is due to Ogg, who wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.

The relation runs very deep and as demonstrated by Richard Borcherds, it also involves generalized Kac–Moody algebras. Work in this area underlined the importance of modular functions that are meromorphic and can have poles at the cusps, as opposed to modular forms, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.