{
  "id": "1211.1241",
  "version": "v5",
  "published": "2012-11-06T14:34:27.000Z",
  "updated": "2015-02-18T21:08:08.000Z",
  "title": "A specialisation of the Bump-Friedberg $L$-function",
  "authors": [
    "Nadir Matringe"
  ],
  "comment": "The paper is under the form that will appear in Canad. Math. Bull",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the restriction of the Bump-Friedberg integrals to affine lines $\\{(s+\\alpha,2s),s\\in\\C\\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\\alpha,\\pi)L(2s,\\Lambda^2,\\pi)$ which we denote by $L^{lin}(s,\\pi,\\alpha)$ for this abstract, when $\\pi$ is a cuspidal automorphic representation of $GL(k,A)$ for $A$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\\pi$, the partial $L$-function $L^{lin,S}(s,\\pi,\\alpha)$ has a pole at 1/2, if and only if $\\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\\pi$ has a twisted global period if and only if $L(\\alpha+1/2,\\pi)\\neq 0$ and $L(1,\\Lambda^2,\\pi)=\\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\\geq 1/2$ when $Re(\\alpha)$ is $\\geq 0$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-07-31T15:17:07.000Z",
      "comment": "The paper should be more readable. We added an appendix",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-02-18T21:08:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67"
    ],
    "keywords": [
      "cuspidal automorphic representation",
      "specialisation",
      "simple theory",
      "number field",
      "twisted global period"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.1241M"
    }
  }
}