Monstrous Moonshine

Conway, John H.; Norton, Simon

doi:10.1112/blms/11.3.308

Cited by 525 publications

(938 citation statements)

References 7 publications

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“…Then the 174 genus zero groups are subgroups of P SL(2, R) generated by Γ 0 (N ) together with a set of Atkin-Lehner involutions (see [19] For a long and glory history of this group, see the paper of Conway-Norton [17]. Based on some remarkable observations of MacKay and Thompson, Conway-Norton conjectured that there exist a natural Z-graded representation V of M with the following property:…”

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

“…To prove our assertion above, we will use Schwarz' theorem on the Riemann mapping and the fact that the Thompson series above are hauptmoduls for the following genus zero groups Γ 0 (1), Γ 0 (2)+, Γ 0 (3)+, Γ 0 (4)+ (see [17] on notations). If G is a genus zero group and h(t) its normalized hauptmodul, then it is easy to check that the expression {h(t), t}/(2h ′ (t) 2 ) is a meromorphic function which is invariant under G. Since h(t) is a generator of the function field for G, it follows that {h(t), t}/(2h ′ (t) 2 ) is a rational expression Q in h(t).…”

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

“…At the time the conjecture was made, neither M nor V had been known to exist, though the evidence for them was overwhelming. On some hypotheses, the first few coefficients of the Thompson series were computed in [17] and were seen to coincide with the coefficients of the appropriate hauptmoduls. (see table 4 (see table 4 of [17]).…”

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

“…Let's us recall a few facts about genus zero functions and the Thompson series (see [17].) Let H be the upper half plane, G be a discrete subgroup of P SL(2, R) which acts on H by linear fractional transformations.…”

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

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Arithmetic properties of mirror map and quantum coupling

Lian

Yau²

1996

Commun.Math. Phys.

109

170

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We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the J-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field Q(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced mod p. Under the mirror hypothesis and an integrality assumption, we derive mod p congruences for the Fourier coefficients.For the quintics, we deduce (at least for 5 |d) that the degree d instanton numbers n d are divisible by 5 3 -a fact first conjectured by Clemens.Revised 12/94 ⋄

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Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

Section: Relations With Genus Zero Functions and The Thompson Seriesmentioning

confidence: 99%

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Arithmetic properties of mirror map and quantum coupling

Lian

Yau²

1996

Commun.Math. Phys.

109

170

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show abstract

“…This normalizer has acquired its importance in several areas of mathematics. For instance, the genus zero subgroups of N 0 .N / have a mysterious correspondence to the conjugacy classes of the monster simple group [6,7]. Moreover, the normalizer N 0 .N / played an important role in the work on Weierstrass points on the modular curve X 0 .N / associated to 0 .N / [14] and on ternary quadratic forms [15].…”

Section: Introductionmentioning

confidence: 99%