{
  "id": "1702.05004",
  "version": "v1",
  "published": "2017-02-16T15:04:17.000Z",
  "updated": "2017-02-16T15:04:17.000Z",
  "title": "Integral representation and critical $L$-values for holomorphic forms on $GSp_{2n} \\times GL_1$",
  "authors": [
    "Ameya Pitale",
    "Abhishek Saha",
    "Ralf Schmidt"
  ],
  "comment": "43 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$, we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. The proof of this application uses our recent structure theorem [arXiv:1501.00524] for the space of nearly holomorphic Siegel modular forms of degree 2 and arbitrary level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-16T15:04:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral representation",
      "holomorphic forms",
      "arbitrary level",
      "holomorphic siegel modular forms",
      "holomorphic vector-valued siegel cusp form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}