{
  "id": "1501.01565",
  "version": "v1",
  "published": "2015-01-07T17:21:18.000Z",
  "updated": "2015-01-07T17:21:18.000Z",
  "title": "The Theta Correspondence, Periods of Automorphic Forms and Special Values of Standard Automorphic L-Functions",
  "authors": [
    "Patrick Walls"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $\\sigma$ of a symplectic group $G$ lifts to a cuspidal representation $\\pi = \\theta_{\\psi}(\\sigma)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $\\sigma$ are linked to periods of automorphic forms in $\\pi$. Consequently, when our results are combined with the Rallis inner product formula in the convergent range or the second term range, we prove a special value formula for the standard automorphic $L$-function $L(s,\\sigma,\\chi_V)$ attached to $\\sigma$ (and twisted by the character $\\chi_V$ of $V$) at the point $s_{m,2n}=m/2-(2n+1)/2$ in terms of the Fourier coefficients of automorphic forms in $\\sigma$ and periods of forms in $\\pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-07T17:21:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "automorphic forms",
      "standard automorphic l-functions",
      "theta correspondence",
      "cuspidal representation",
      "fourier coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101565W"
    }
  }
}