{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{S} = \\operatorname{Res}_{\\C/\\R} \\mathbb{G}_m$ be the Deligne torus, so that $\\mathbb{S}(\\R) = \\C^\\times$.\nLet $G$ be a connected [[reductive group|2XS84PWE3S]] over $\\Q$ and let $X$ be a $G(\\R)$-conjugacy class of homomorphisms $h \\colon \\mathbb{S} \\to G_\\R$.\nThe pair $(G, X)$ is a *Shimura datum* if for every $h \\in X$:\n\n1. The [[Hodge structure|ZSM083M97C]] on the Lie algebra of $G_\\R$ defined by $\\operatorname{Ad} \\circ h$ has only the types $(-1, 1)$, $(0, 0)$ and $(1, -1)$.\n2. Conjugation by $h(i)$ is a Cartan involution of the adjoint group $G^{\\mathrm{ad}}_\\R$.\n3. $G^{\\mathrm{ad}}$ has no factor defined over $\\Q$ on which the projection of $h$ is trivial.\n\nLet $\\mathbb{A}_f$ be the finite [[adeles|SEH68BA1QQ]] of $\\Q$ and let $K \\subseteq G(\\mathbb{A}_f)$ be a compact open subgroup.\nThe *Shimura variety* of level $K$ is\n$$\\operatorname{Sh}_K(G, X) = G(\\Q) \\backslash \\big( X \\times G(\\mathbb{A}_f) / K \\big),$$\nwith $G(\\Q)$ acting diagonally and $K$ acting on the right of $G(\\mathbb{A}_f)$.\n\n## Remarks\n\nThe conditions make $X$ a finite union of Hermitian [[symmetric spaces|PAA93R4WYD]], and $\\operatorname{Sh}_K(G, X)$ is a finite disjoint union of quotients $\\Gamma \\backslash X^+$ by arithmetic subgroups $\\Gamma$.\nBy the theorem of Baily and Borel, it is a [[quasi-projective variety|EQSHJPE7H8]] over $\\C$, and it is smooth when $K$ is small enough.\nBy work of Shimura, Deligne and others it has a canonical model over a number field, the reflex field.\n\nExample: let $G = \\mathrm{GL}_2$ and let $X$ be the conjugacy class of $h(a + bi) = \\begin{pmatrix} a \u0026 -b \\\\ b \u0026 a \\end{pmatrix}$.\nThen $X$ is identified with $\\C \\setminus \\R$, the union of the upper and lower half-planes, and the Shimura varieties $\\operatorname{Sh}_K(\\mathrm{GL}_2, X)$ are finite unions of [[modular curves|T5W0CMS0DM]].\nFor $G = \\mathrm{GSp}_{2g}$ one obtains the Siegel modular varieties, which parametrize principally polarized abelian varieties of dimension $g$ with level structure.\n\nNon-example: $G = \\mathrm{GL}_3$ admits no Shimura datum, because the symmetric space of $\\mathrm{SL}_3(\\R)$ has no invariant complex structure, so clauses 1 and 2 cannot both hold.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QT4CF7ZKK5",
    "kind": "definition",
    "msc": [
      "14G35",
      "11G18"
    ],
    "name": "Shimura variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Shimura_variety"
      }
    ],
    "tag": "53HPW59SJC",
    "uses": [
      "2XS84PWE3S",
      "SEH68BA1QQ",
      "ZSM083M97C",
      "PAA93R4WYD",
      "EQSHJPE7H8"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/53HPW59SJC/",
  "links": [
    {
      "field": "defines",
      "target": "QT4CF7ZKK5"
    },
    {
      "field": "uses",
      "target": "2XS84PWE3S"
    },
    {
      "field": "uses",
      "target": "SEH68BA1QQ"
    },
    {
      "field": "uses",
      "target": "ZSM083M97C"
    },
    {
      "field": "uses",
      "target": "PAA93R4WYD"
    },
    {
      "field": "uses",
      "target": "EQSHJPE7H8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/53HPW59SJC.md",
  "msc": [
    "14G35",
    "11G18"
  ],
  "name": "Shimura variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/53H/53HPW59SJC.md",
  "status": null,
  "tag": "53HPW59SJC",
  "uri": "m2://53HPW59SJC",
  "url": "https://tamnd.github.io/math2/wiki/shimura-variety-definition/",
  "writing": {
    "tag": "53HPW59SJC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}