{
  "id": "math/9806011",
  "version": "v1",
  "published": "1998-06-03T12:27:49.000Z",
  "updated": "1998-06-03T12:27:49.000Z",
  "title": "The Modular Form of the Barth-Nieto Quintic",
  "authors": [
    "V. Gritsenko",
    "K. Hulek"
  ],
  "comment": "20 pages, Latex2e RIMS Preprint 1203",
  "categories": [
    "math.AG"
  ],
  "abstract": "Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \\Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \\Delta_1^3. The form \\Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \\Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \\Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \\Delta_1 is a weight 3 cusp form with respect to the group \\Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\\cal A}_3(2).",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-06-03T12:27:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular form",
      "barth-nieto quintic",
      "parametrizes heisenberg invariant kummer surfaces",
      "abelian surfaces",
      "moduli space"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......6011G"
    }
  }
}