Math 2.0

Number theory Automorphic forms Theorem

Monstrous moonshine

Builds on Monster groupGroup representationj-invariantCongruence subgroup

Statement

Let 𝕄\mathbb{M} be the Monster group. Let J(τ)=j(τ)−744=q−1+∑n≥1c(n)qnJ(\tau) = j(\tau) - 744 = q^{-1} + \sum_{n \ge 1} c(n) q^n with q=e2πiτq = e^{2\pi i \tau}, where jj is the j-invariant. There is a graded complex representation V♮=⨁n≥−1VnV^\natural = \bigoplus_{n \ge -1} V_n of 𝕄\mathbb{M}, with each VnV_n finite-dimensional, such that:

  1. dim⁡V−1=1\dim V_{-1} = 1, V0=0V_0 = 0, and dim⁡Vn=c(n)\dim V_n = c(n) for all n≥1n \ge 1.
  2. For each g∈𝕄g \in \mathbb{M}, the McKay–Thompson series Tg(τ)=∑n≥−1tr⁡(g∣Vn) qnT_g(\tau) = \sum_{n \ge -1} \tr(g \mid V_n)\, q^n is the normalised Hauptmodul of a genus zero subgroup Γg\Gamma_g of SL2(ℝ)\mathrm{SL}_2(\R). That is, TgT_g is invariant under Γg\Gamma_g and Tg=q−1+O(q)T_g = q^{-1} + O(q). Also, TgT_g induces an isomorphism from the compactification of Γg\ℍ\Gamma_g \backslash \mathbb{H} onto the Riemann sphere.
  3. Each Γg\Gamma_g contains the congruence subgroup Γ0(N)\Gamma_0(N) for some N≥1N \ge 1.

Remarks

The representation V♮V^\natural is the moonshine module, constructed by Frenkel, Lepowsky and Meurman as a vertex operator algebra with an action of 𝕄\mathbb{M}. The statement was conjectured by Conway and Norton, after McKay observed that 196884=1+196883196884 = 1 + 196883, where 196883196883 is the smallest dimension of a nontrivial irreducible representation of 𝕄\mathbb{M}. Borcherds proved it in 1992. He constructed the monster Lie algebra from V♮V^\natural. Its denominator identity shows that the series TgT_g satisfy the replication formulas, which determine them from finitely many coefficients. These coefficients were then checked against the known Hauptmoduln. For g=1g = 1, the series T1T_1 is JJ itself, the Hauptmodul of SL2(ℤ)\mathrm{SL}_2(\Z).

Going deeper

For a student

The coefficient identity 196884=1+196883196884 = 1 + 196883 is the first visible trace of the theorem, at n=1n = 1. The proof goes through vertex operator algebras and generalized Kac–Moody algebras, which link finite groups, modular functions and string theory. Analogous phenomena, such as umbral moonshine, were found later for other groups.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Congruence subgroup, concept, no evidence levelCongruence subgroupconceptGriess algebra, object, no evidence levelGriess algebraobjectGroup representation, concept, no evidence levelGroup representationconceptIrreducible representation, concept, no evidence levelIrreducible represent…conceptOrder of a group, concept, no evidence levelOrder of a groupconceptSimple group, concept, no evidence levelSimple groupconceptSporadic group, concept, no evidence levelSporadic groupconceptj-invariant, concept, no evidence levelj-invariantconceptMonster group, object, no evidence levelMonster groupobjectMonstrous moonshine, statement, no evidence levelMonstrous moonshinestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Congruence subgroup concept
  2. Griess algebra object
  3. Group representation concept
  4. Irreducible representation concept
  5. Order of a group concept
  6. Simple group concept
  7. Sporadic group concept
  8. j-invariant concept
  9. Monster group object
  10. Monstrous moonshine statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Monstrous_moonshine
About this page
TagE0Y7Y75T6C
URIm2://E0Y7Y75T6C
Kindstatement, theorem
WritingW0 stub
MSC11F22, 20D08, 17B69
StatusThis copy of the site has no build data. Run m2 status E0Y7Y75T6C.
AboutMonster group, j-invariant
UsesMonster group, Group representation, j-invariant, Congruence subgroup
Same as
Changed2 November 2026

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