{
  "id": "1512.07204",
  "version": "v1",
  "published": "2015-12-22T19:01:35.000Z",
  "updated": "2015-12-22T19:01:35.000Z",
  "title": "Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level",
  "authors": [
    "Martin Dickson",
    "Ameya Pitale",
    "Abhishek Saha",
    "Ralf Schmidt"
  ],
  "comment": "38 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We formulate an explicit refinement of B\\\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms. We also compute and write down the relevant $\\varepsilon$-factors for all dihedral twists of non-supercuspidal representations of $GSp_4$ over a non-archimedean local field. By comparing this with the conditions for a local Bessel model to exist, we verify the local Gan-Gross-Prasad conjecture in these cases for the generic L-packets and also demonstrate its severe failure for the non-generic L-packets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-22T19:01:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "siegel modular forms",
      "explicit refinement",
      "squarefree level",
      "böcherers conjecture",
      "fourier coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151207204D"
    }
  }
}