{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].\nLet $J(\\tau) = j(\\tau) - 744 = q^{-1} + \\sum_{n \\ge 1} c(n) q^n$ with $q = e^{2\\pi i \\tau}$, where $j$ is the [[j-invariant|FEC2XH822R]].\nThere 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:\n\n1. $\\dim V_{-1} = 1$, $V_0 = 0$, and $\\dim V_n = c(n)$ for all $n \\ge 1$.\n2. 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 $\\mathrm{SL}_2(\\R)$.\nThat is, $T_g$ is invariant under $\\Gamma_g$ and $T_g = q^{-1} + O(q)$.\nAlso, $T_g$ induces an isomorphism from the compactification of $\\Gamma_g \\backslash \\mathbb{H}$ onto the Riemann sphere.\n3. Each $\\Gamma_g$ contains the [[congruence subgroup|1H6R856P3C]] $\\Gamma_0(N)$ for some $N \\ge 1$.\n\n## Remarks\n\nThe representation $V^\\natural$ is the moonshine module, constructed by Frenkel, Lepowsky and Meurman as a vertex operator algebra with an action of $\\mathbb{M}$.\nThe 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 $\\mathbb{M}$.\nBorcherds proved it in 1992.\nHe constructed the monster Lie algebra from $V^\\natural$.\nIts denominator identity shows that the series $T_g$ satisfy the replication formulas, which determine them from finitely many coefficients.\nThese coefficients were then checked against the known Hauptmoduln.\nFor $g = 1$, the series $T_1$ is $J$ itself, the Hauptmodul of $\\mathrm{SL}_2(\\Z)$.\n\n## Explanations\n\n### L1 human\n\nThe Monster is the largest sporadic finite simple group, an object from pure algebra.\nThe $j$-function is a classical function from complex analysis and number theory.\nMoonshine says that the coefficients of $j$ are dimensions of natural pieces of one representation of the Monster.\nEven more, twisting by each element of the Monster gives another function of the same special kind.\n\n### L2 human\n\nThe coefficient identity $196884 = 1 + 196883$ is the first visible trace of the theorem, at $n = 1$.\nThe proof goes through vertex operator algebras and generalized Kac–Moody algebras, which link finite groups, modular functions and string theory.\nAnalogous phenomena, such as umbral moonshine, were found later for other groups.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "C9VY20TAPT",
      "FEC2XH822R"
    ],
    "kind": "statement",
    "msc": [
      "11F22",
      "20D08",
      "17B69"
    ],
    "name": "Monstrous moonshine",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q392440",
      "wikipedia": "Monstrous moonshine"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Monstrous_moonshine"
      }
    ],
    "tag": "E0Y7Y75T6C",
    "uses": [
      "C9VY20TAPT",
      "WYGRXCG2RN",
      "FEC2XH822R",
      "1H6R856P3C"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/E0Y7Y75T6C/",
  "links": [
    {
      "field": "about",
      "target": "C9VY20TAPT"
    },
    {
      "field": "about",
      "target": "FEC2XH822R"
    },
    {
      "field": "uses",
      "target": "C9VY20TAPT"
    },
    {
      "field": "uses",
      "target": "WYGRXCG2RN"
    },
    {
      "field": "uses",
      "target": "FEC2XH822R"
    },
    {
      "field": "uses",
      "target": "1H6R856P3C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/E0Y7Y75T6C.md",
  "msc": [
    "11F22",
    "20D08",
    "17B69"
  ],
  "name": "Monstrous moonshine",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E0Y/E0Y7Y75T6C.md",
  "status": null,
  "tag": "E0Y7Y75T6C",
  "uri": "m2://E0Y7Y75T6C",
  "url": "https://math2.tamnd.com/wiki/monstrous-moonshine/",
  "writing": {
    "tag": "E0Y7Y75T6C",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}