{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a number field with [[adele ring|SEH68BA1QQ]] $\\mathbb A$, and let $G$ be a connected reductive group over $F$.\nLet $G(\\mathbb A)^1$ be the common kernel of the maps $x \\mapsto \\abs{\\chi(x)}$ for the rational characters $\\chi$ of $G$, and assume that $G(F) \\backslash G(\\mathbb A)^1$ is compact.\nLet $R$ be the right regular representation of $G(\\mathbb A)^1$ on $L^2(G(F) \\backslash G(\\mathbb A)^1)$.\nFor a smooth compactly supported function $f$ on $G(\\mathbb A)^1$, let $R(f) = \\int f(y) R(y) \\, dy$, with a fixed [[Haar measure|VY0HXD3GC4]].\nThen $R(f)$ is a [[trace class operator|Z2CP530WDE]].\nIts trace is given by\n$$\\sum_{\\pi} m(\\pi) \\operatorname{tr} \\pi(f) = \\sum_{\\gamma} \\operatorname{vol}\\big(G_\\gamma(F) \\backslash G_\\gamma(\\mathbb A)^1\\big) \\int_{G_\\gamma(\\mathbb A) \\backslash G(\\mathbb A)^1} f(x^{-1} \\gamma x) \\, dx.$$\nHere $\\pi$ runs over the irreducible unitary representations of $G(\\mathbb A)^1$, and $m(\\pi)$ is the multiplicity of $\\pi$ in $R$.\nThe index $\\gamma$ runs over representatives of the [[conjugacy classes|0DEBXAJ5SA]] of $G(F)$, and $G_\\gamma$ is the centralizer of $\\gamma$.\nBoth sides equal $\\operatorname{tr} R(f)$.\n\n## Remarks\n\nThis compact-quotient case is the direct generalisation of the [[Selberg trace formula|BDAWKPBNQP]].\nThe proof writes $R(f)$ as an integral operator with kernel $K(x, y) = \\sum_{\\gamma \\in G(F)} f(x^{-1} \\gamma y)$ and integrates the kernel over the diagonal.\nGrouping the terms by conjugacy class gives the geometric side; the discrete decomposition of $L^2$ gives the spectral side.\nFor general $G$, such as $\\mathrm{GL}_n$ with $n \\ge 2$, the quotient is not compact and $R(f)$ is not trace class.\nArthur's trace formula then replaces both sides by truncated distributions $J_{\\mathrm{geom}}(f) = J_{\\mathrm{spec}}(f)$, which contain extra terms from the continuous spectrum and from non-semisimple conjugacy classes.\nThe spectral side encodes [[automorphic forms|3C73HND6XA]], the geometric side encodes orbital integrals, and comparing the formulas for two groups is the main tool for proving cases of Langlands functoriality.\n\n## Explanations\n\n### L1 human\n\nThe formula computes one number, the trace of an operator on functions on an adelic quotient, in two ways.\nOne way lists the automorphic representations with their multiplicities.\nThe other way lists the conjugacy classes of rational elements of the group, weighted by volumes and integrals.\nEquating them turns questions about automorphic forms into questions about conjugacy classes, which are often easier.\n\n### L2 human\n\nThe Selberg trace formula is the case of a cocompact Fuchsian group, seen through the real place only.\nJacquet and Langlands used a trace formula comparison to relate automorphic forms on $\\mathrm{GL}_2$ and on a quaternion algebra.\nThe stabilization of the trace formula, completed with the fundamental lemma, allowed Arthur to classify the automorphic representations of classical groups.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "3C73HND6XA"
    ],
    "kind": "statement",
    "msc": [
      "11F72",
      "11F70",
      "22E55"
    ],
    "name": "Arthur–Selberg trace formula",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4800826",
      "wikipedia": "Arthur–Selberg trace formula"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Arthur%E2%80%93Selberg_trace_formula"
      }
    ],
    "tag": "BCETMS8K3X",
    "uses": [
      "BDAWKPBNQP",
      "SEH68BA1QQ",
      "VY0HXD3GC4",
      "Z2CP530WDE",
      "0DEBXAJ5SA",
      "3C73HND6XA"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/BCETMS8K3X/",
  "links": [
    {
      "field": "about",
      "target": "3C73HND6XA"
    },
    {
      "field": "uses",
      "target": "BDAWKPBNQP"
    },
    {
      "field": "uses",
      "target": "SEH68BA1QQ"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "Z2CP530WDE"
    },
    {
      "field": "uses",
      "target": "0DEBXAJ5SA"
    },
    {
      "field": "uses",
      "target": "3C73HND6XA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BCETMS8K3X.md",
  "msc": [
    "11F72",
    "11F70",
    "22E55"
  ],
  "name": "Arthur–Selberg trace formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BCE/BCETMS8K3X.md",
  "status": null,
  "tag": "BCETMS8K3X",
  "uri": "m2://BCETMS8K3X",
  "url": "https://tamnd.github.io/math2/wiki/arthur-selberg-trace-formula/",
  "writing": {
    "tag": "BCETMS8K3X",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}