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.
Read more about this topic: Modular Curve
Famous quotes containing the words relation with, relation, monster and/or group:
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“Nothing is poetical if plain daylight is not poetical; and no monster should amaze us if the normal man does not amaze.”
—Gilbert Keith Chesterton (1874–1936)
“My routines come out of total unhappiness. My audiences are my group therapy.”
—Joan Rivers (b. 1935)