{
  "id": "2003.04611",
  "version": "v1",
  "published": "2020-03-10T09:52:36.000Z",
  "updated": "2020-03-10T09:52:36.000Z",
  "title": "Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\\mathbb{Q}$",
  "authors": [
    "Joachim Schwermer"
  ],
  "comment": "27 pages, no figures",
  "categories": [
    "math.NT",
    "math.KT"
  ],
  "abstract": "The cohomology $H^*(\\Gamma, E) $ of a torsion-free arithmetic subgroup $\\Gamma$ of the special linear $\\mathbb{Q}$-group $\\sG = SL_n$ may be interpreted in terms of the automorphic spectrum of $\\Gamma$. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes $\\{\\sP\\}$ of associate proper parabolic $\\mathbb{Q}$-subgroups of $\\sG$. Each summand $H^*_{\\mathrm{\\{P\\}}}(\\Gamma, E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $\\{\\sP\\}$. The cohomology $H^*(\\Gamma, E) $ vanishes above the degree given by the cohomological dimension $\\mathrm{cd}(\\Gamma) = \\frac{n(n-1)}{2}$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $\\{\\sP\\}$ for which the corresponding summand $H^{\\mathrm{cd}(\\Gamma)}_{\\mathrm{\\{\\sP\\}}}(\\Gamma, E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H^{\\mathrm{cd}(\\Gamma)}_{\\mathrm{\\{\\sQ\\}}}(\\Gamma, \\C)$. Finally, in the case of a principal congruence subgroup $\\Gamma(q)$, $q = p^{\\nu} > 5$, $p\\geq 3$ a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes $\\{\\sQ\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-10T09:52:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F75",
      "11F67",
      "22E40"
    ],
    "keywords": [
      "eisenstein series",
      "degree cohomology",
      "associate classes",
      "torsion-free arithmetic subgroup",
      "associate proper parabolic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}