{
  "id": "1911.08310",
  "version": "v1",
  "published": "2019-11-19T14:44:11.000Z",
  "updated": "2019-11-19T14:44:11.000Z",
  "title": "Low-lying zeros in families of holomorphic cusp forms: the weight aspect",
  "authors": [
    "Lucile Devin",
    "Daniel Fiorilli",
    "Anders Södergren"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $\\phi$ satisfying the condition supp$(\\widehat \\phi) \\subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\\widehat \\phi$ reaches the point $1$. In particular the first of these terms involves the quantity $\\widehat \\phi(1)$ which appeared in previous work of Fouvry--Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of GRH for $\\text{GL}(2)$ automorphic $L$-functions. Finally, when supp$(\\widehat \\phi)\\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the $L$-functions Ratios Conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-19T14:44:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "holomorphic cusp forms",
      "low-lying zeros",
      "weight aspect",
      "orthogonal symmetry type",
      "functions ratios conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}