{
  "id": "1301.5702",
  "version": "v2",
  "published": "2013-01-24T05:24:53.000Z",
  "updated": "2014-01-20T19:57:49.000Z",
  "title": "Low-lying zeroes of Maass form $L$-functions",
  "authors": [
    "Levent Alpoge",
    "Steven J. Miller"
  ],
  "comment": "version 2.1, 19 pages, fixed some typos. To appear in IMRN",
  "doi": "10.1093/imrn/rnu012",
  "categories": [
    "math.NT"
  ],
  "abstract": "The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the limit as $kN \\to\\infty$ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in $(-2,2)$. Exceeding $(-1,1)$ is important as the three orthogonal groups are indistinguishable for support up to $(-1,1)$ but are distinguishable for any larger support. We study the other family of ${\\rm GL}_2$ automorphic forms over $\\mathbb{Q}$: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order $M$ at the origin. For test functions with Fourier transform supported inside $\\left(-2 + \\frac{2}{2M+1}, 2 - \\frac{2}{2M+1}\\right)$, we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-20T19:57:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11M41",
      "15A52"
    ],
    "keywords": [
      "maass form",
      "one-level density",
      "low-lying zeroes",
      "test function",
      "scaling limit"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5702A"
    }
  }
}