Editing Modular curve (section)

== 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.  The first several coefficients of the ''q''-expansions of their Hauptmoduln were computed already in the 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]] Γ<sub>0</sub>(''p'')<sup>+</sup> of [[modular group Gamma0|Γ<sub>0</sub>]](''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 [[supersingular prime (moonshine theory)|supersingular primes in moonshine theory]], i.e. the prime factors of the order of the [[monster group]]. The result about Γ<sub>0</sub>(''p'')<sup>+</sup> 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.<ref>{{harvtxt|Ogg|1974}}</ref>

The relation runs very deep and, as demonstrated by [[Richard Borcherds]], it also involves [[generalized Kac–Moody algebra]]s. Work in this area underlined the importance of [[modular function|modular ''functions'']] that are meromorphic and can have poles at the cusps, as opposed to [[modular form|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.