{
  "id": "2210.16291",
  "version": "v1",
  "published": "2022-10-28T17:37:24.000Z",
  "updated": "2022-10-28T17:37:24.000Z",
  "title": "On the local $L^2$-Bound of the Eisenstein Series",
  "authors": [
    "Subhajit Jana",
    "Amitay Kamber"
  ],
  "comment": "47 pages",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. In this work, we show an upper bound of the squared $L^2$-norms of the unitary Eisenstein series restricted to a fixed compact domain, of \\emph{poly-logarithmic} strength on an average, for a large class of reductive groups. Moreover, as an application of our method, we prove the optimal lifting property for $\\mathrm{SL}_n(\\mathbb{Z}/q\\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\\mathrm{SL}_n(\\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-28T17:37:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "11F72",
      "22E55"
    ],
    "keywords": [
      "unitary eisenstein series",
      "reductive groups",
      "principal congruence subgroup",
      "number fields",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}