{
  "id": "1208.1945",
  "version": "v3",
  "published": "2012-08-09T15:19:07.000Z",
  "updated": "2014-11-15T05:01:03.000Z",
  "title": "Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions",
  "authors": [
    "Sug Woo Shin",
    "Nicolas Templier"
  ],
  "comment": "Appendix A by Robert Kottwitz; Appendix B by Raf Cluckers, Julia Gordon and Immanuel Halupczok",
  "categories": [
    "math.NT",
    "math.LO",
    "math.RT"
  ],
  "abstract": "We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at infinity. Let $^{L}G=\\hat G \\rtimes \\mathrm{Gal}(\\bar F/F)$ be the associated $L$-group and $r:{}^L G\\to \\mathrm{GL}(d,\\mathbb{C})$ a continuous homomorphism which is irreducible and does not factor through $\\mathrm{Gal}(\\bar F/F)$. The families under consideration consist of discrete automorphic representations of $G(\\mathbb{A}_F)$ of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak [Progr. Math. 70 (1987), 321--331.] and Serre [J. Amer. Math. Soc. 10 (1997), no. 1, 75--102.]. As an application we study the distribution of the low-lying zeros of the associated family of $L$-functions $L(s,\\pi,r)$, assuming from the principle of functoriality that these $L$-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the Unitary, Symplectic and Orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius--Schur indicator to determine this Symmetry type. If $r$ is not isomorphic to its dual $r^\\vee$ then the Symmetry type is Unitary. Otherwise there is a bilinear form on $\\mathbb{C}^d$ which realizes the isomorphism between $r$ and $r^\\vee$. If the bilinear form is symmetric (resp. alternating) then $r$ is real (resp. quaternionic) and the Symmetry type is Symplectic (resp. Orthogonal).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-13T13:42:28.000Z",
      "journal": null,
      "doi": null,
      "authors": [
        "Sug-Woo Shin",
        "Nicolas Templier"
      ]
    },
    {
      "version": "v3",
      "updated": "2014-11-15T05:01:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "low-lying zeros",
      "sato-tate theorem",
      "symmetry type",
      "admits discrete series representations",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1945S"
    }
  }
}