{
  "id": "1310.6991",
  "version": "v1",
  "published": "2013-10-25T17:55:41.000Z",
  "updated": "2013-10-25T17:55:41.000Z",
  "title": "Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields",
  "authors": [
    "Jose Ignacio Burgos Gil",
    "Ariel Pacetti"
  ],
  "comment": "26 pages, 4 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\\Om_K$ its ring of integers. Let $\\Gamma$ be a congruence subgroup of $\\SL_2(\\Om_K)$ and $M_{(k_1,k_2)}(\\Gamma)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $\\Gamma$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $\\Gamma$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \\in M_{(k_1,k_2)}(\\Gamma)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to the same statement in the Sturm case, i.e. suppose that all the Fourier coefficients of the form $G$ lie in a finite extension of $\\Q$, and let $\\id{p}$ be a prime ideal in such extension, whose norm is unramified in $K$; suppose furthermore that the Fourier expansion coefficients of $G$ lie in the ideal $\\id{p}$ for all the elements in $S$, then they all lie in the ideal $\\id{p}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-25T17:55:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F41"
    ],
    "keywords": [
      "hilbert modular forms",
      "real quadratic field",
      "sturm bounds",
      "fourier expansion coefficients",
      "second result corresponds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.6991B"
    }
  }
}