Monstrous Moonshine

Abstract

A quick summary of the recent amazing discoveries about the Fischer-Griess "MONSTER" simple group.

Arithmetic properties of mirror map and quantum coupling

“…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:…”

“…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).…”

“…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]).…”

“…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.…”

Normalizers of intermediate congruence subgroups of the Hecke subgroups

“…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].…”