{
  "id": "2407.13464",
  "version": "v1",
  "published": "2024-07-18T12:37:35.000Z",
  "updated": "2024-07-18T12:37:35.000Z",
  "title": "Critical values of $L$-functions of residual representations of $\\mathrm{GL}_4$",
  "authors": [
    "Johannes Droschl"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\\mathrm{GL}_4(\\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\\mathrm{GL}_2'(\\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-18T12:37:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F70",
      "11F75"
    ],
    "keywords": [
      "critical values",
      "residual representations",
      "cuspidal automorphic representations",
      "rational multiple",
      "jacquet-langlands correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}