{
  "id": "2308.06359",
  "version": "v1",
  "published": "2023-08-11T19:33:07.000Z",
  "updated": "2023-08-11T19:33:07.000Z",
  "title": "Low-Lying Zeros of a Thin Family of Automorphic $L$-Functions in the Level Aspect",
  "authors": [
    "Matthew Kroesche"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We calculate the one-level density of thin subfamilies of a family of Hecke cuspforms formed by twisting the forms in a smaller family by a character. The result gives support up to 1, conditional on GRH, and we also find several of the lower-order main terms. In addition, we find an unconditional result that has only slightly lower support. A crucial step in doing so is the establishment of an on-average version of the Weil bound that applies to twisted Kloosterman sums. Moreover, we average over these thin subfamilies by running over the characters in a coset, and observe that any amount of averaging at all is enough to allow us to get support greater than 1 and thus distinguish between the SO(even) and SO(odd) symmetry types. Finally, we also apply our results to nonvanishing problems for the families studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-11T19:33:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "level aspect",
      "low-lying zeros",
      "thin family",
      "automorphic",
      "thin subfamilies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}