{
  "id": "0707.1708",
  "version": "v1",
  "published": "2007-07-11T21:08:53.000Z",
  "updated": "2007-07-11T21:08:53.000Z",
  "title": "On certain period relations for cusp forms on GL_n",
  "authors": [
    "A. Raghuram",
    "Freydoon Shahidi"
  ],
  "comment": "40 pages. This preprint is also up on preprint server of the Erwin Schrodinger Institute as preprint number 1928. The URL is http://www.esi.ac.at/Preprint-shadows/esi1928.html",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\pi$ be a regular algebraic cuspidal automorphic representation of ${\\rm GL}_n({\\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $\\pi$. These periods were originally defined by Harder when $n=2$, and later by Mahnkopf when $F = {\\mathbb Q}$. In the first part of the paper we analyze the behaviour of these periods upon twisting $\\pi$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $\\phi_{\\chi}$ is the cusp form associated to a character $\\chi$ of a quadratic extension, then we relate the periods of $\\phi_{\\chi^n}$ to those of $\\phi_{\\chi}$, and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power $L$-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power $L$-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-07-11T21:08:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F70",
      "22E55"
    ],
    "keywords": [
      "cusp form",
      "period relations",
      "regular algebraic cuspidal automorphic representation",
      "algebraic hecke characters",
      "symmetric fourth power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1708R"
    }
  }
}