{
  "id": "2409.14933",
  "version": "v1",
  "published": "2024-09-23T11:41:10.000Z",
  "updated": "2024-09-23T11:41:10.000Z",
  "title": "Adjoint $L$-functions, congruence ideals, and Selmer groups over $\\mathrm{GL}_n$",
  "authors": [
    "Ho Leung Fong"
  ],
  "comment": "33 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we relate $L(1,\\pi,\\mathrm{Ad}^\\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $\\pi$ of $\\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,\\pi,\\mathrm{Ad}^\\circ)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-23T11:41:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F75",
      "11F33"
    ],
    "keywords": [
      "congruence ideals",
      "selmer groups",
      "cohomological cuspidal automorphic representations",
      "automorphic forms",
      "deduce relationships"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}