{
  "id": "2404.05561",
  "version": "v1",
  "published": "2024-04-08T14:35:09.000Z",
  "updated": "2024-04-08T14:35:09.000Z",
  "title": "Local analysis of the Kuznetsov formula and the density conjecture",
  "authors": [
    "Edgar Assing",
    "Valentin Blomer",
    "Paul D. Nelson"
  ],
  "comment": "48 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove Sarnak's spherical density conjecture for the principal congruence subgroup of SL(n, Z) of arbitrary level. Applications include a complete version of Sarnak's optimal lifting conjecture for principal congruence subgroups of SL(n, Z), as well as a transfer of the density theorem to certain co-compact situations. The main ingredients are new lower bounds for Whittaker functions and strong estimates for the cardinality of ramified Kloosterman sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-08T14:35:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kuznetsov formula",
      "local analysis",
      "principal congruence subgroup",
      "sarnaks spherical density conjecture",
      "sarnaks optimal lifting conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}