{
  "id": "1711.07281",
  "version": "v1",
  "published": "2017-11-20T12:19:43.000Z",
  "updated": "2017-11-20T12:19:43.000Z",
  "title": "On Poincaré series of half-integral weight",
  "authors": [
    "Sonja Žunar"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "We use Poincar\\'e series of $ K $-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $ \\mathrm{SL}_2(\\mathbb R) $ to construct a spanning set for the space of cusp forms $ S_m(\\Gamma,\\chi) $, where $ \\Gamma $ is a discrete subgroup of finite covolume in the metaplectic cover of $ \\mathrm{SL}_2(\\mathbb R) $, $ \\chi $ is a character of $ \\Gamma $ of finite order, and $ m\\in\\frac52+\\mathbb Z_{\\geq0} $. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $ f\\in S_m(\\Gamma,\\chi) $. Using this last result, we construct a Poincar\\'e series $ \\Delta_{\\Gamma,k,m,\\xi,\\chi}\\in S_m(\\Gamma,\\chi) $ that corresponds, in the sense of the Riesz representation theorem, to the linear functional $ f\\mapsto f^{(k)}(\\xi) $ on $ S_m(\\Gamma,\\chi) $, where $ \\xi\\in\\mathbb C_{\\Im(z)>0} $ and $ k\\in\\mathbb Z_{\\geq0} $. Under some additional conditions on $ \\Gamma $ and $ \\chi $, we provide the Fourier expansion of cusp forms $ \\Delta_{\\Gamma,k,m,\\xi,\\chi} $ and their expansion in a series of classical Poincar\\'e series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-20T12:19:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F12",
      "11F37"
    ],
    "keywords": [
      "half-integral weight",
      "cusp forms",
      "metaplectic cover",
      "finite matrix coefficients",
      "riesz representation theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}