---
tag: E0Y7Y75T6C
kind: statement
role: theorem
name: Monstrous moonshine
msc: [11F22, 20D08, 17B69]
uses: [C9VY20TAPT, WYGRXCG2RN, FEC2XH822R, 1H6R856P3C]
about: [C9VY20TAPT, FEC2XH822R]
sources:
  - url: https://en.wikipedia.org/wiki/Monstrous_moonshine
same_as:
  wikidata: Q392440
  wikipedia: Monstrous moonshine
---

## Statement

Let $\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].
Let $J(\tau) = j(\tau) - 744 = q^{-1} + \sum_{n \ge 1} c(n) q^n$ with $q = e^{2\pi i \tau}$, where $j$ is the [[j-invariant|FEC2XH822R]].
There is a graded complex [[representation|WYGRXCG2RN]] $V^\natural = \bigoplus_{n \ge -1} V_n$ of $\mathbb{M}$, with each $V_n$ finite-dimensional, such that:

1. $\dim V_{-1} = 1$, $V_0 = 0$, and $\dim V_n = c(n)$ for all $n \ge 1$.
2. For each $g \in \mathbb{M}$, the McKay–Thompson series $T_g(\tau) = \sum_{n \ge -1} \tr(g \mid V_n)\, q^n$ is the normalised Hauptmodul of a genus zero subgroup $\Gamma_g$ of $\mathrm{SL}_2(\R)$.
That is, $T_g$ is invariant under $\Gamma_g$ and $T_g = q^{-1} + O(q)$.
Also, $T_g$ induces an isomorphism from the compactification of $\Gamma_g \backslash \mathbb{H}$ onto the Riemann sphere.
3. Each $\Gamma_g$ contains the [[congruence subgroup|1H6R856P3C]] $\Gamma_0(N)$ for some $N \ge 1$.

## Remarks

The representation $V^\natural$ is the moonshine module, constructed by Frenkel, Lepowsky and Meurman as a vertex operator algebra with an action of $\mathbb{M}$.
The statement was conjectured by Conway and Norton, after McKay observed that $196884 = 1 + 196883$, where $196883$ is the smallest dimension of a nontrivial irreducible representation of $\mathbb{M}$.
Borcherds proved it in 1992.
He constructed the monster Lie algebra from $V^\natural$.
Its denominator identity shows that the series $T_g$ satisfy the replication formulas, which determine them from finitely many coefficients.
These coefficients were then checked against the known Hauptmoduln.
For $g = 1$, the series $T_1$ is $J$ itself, the Hauptmodul of $\mathrm{SL}_2(\Z)$.

## Explanations

### L1 human

The Monster is the largest sporadic finite simple group, an object from pure algebra.
The $j$-function is a classical function from complex analysis and number theory.
Moonshine says that the coefficients of $j$ are dimensions of natural pieces of one representation of the Monster.
Even more, twisting by each element of the Monster gives another function of the same special kind.

### L2 human

The coefficient identity $196884 = 1 + 196883$ is the first visible trace of the theorem, at $n = 1$.
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.