{
  "id": "math/0412213",
  "version": "v2",
  "published": "2004-12-10T13:55:08.000Z",
  "updated": "2005-12-09T09:10:36.000Z",
  "title": "On the SL(2) period integral",
  "authors": [
    "U. K. Anandavardhanan",
    "Dipendra Prasad"
  ],
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let E/F be a quadratic extension of number fields. For a cuspidal representation $\\pi$ of SL(2,A_E), we study the non-vanishing of the period integral on SL(2,F)\\SL(2,A_F). We characterise the non-vanishing of the period integral of $\\pi$ in terms of $\\pi$ being generic with respect to characters of E\\A_E which are trivial on A_F. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL(2,A_E) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation $\\pi$ on SL(2,A_E) which is abstractly SL(2,A_F) distinguished but none of the elements in the global L-packet determined by $\\pi$ is distinguished by SL(2,A_F).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-12-09T09:10:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "22E55"
    ],
    "keywords": [
      "cuspidal representation",
      "local invariant functionals",
      "local constituent admits",
      "number fields",
      "global l-packet"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12213A"
    }
  }
}