{
  "id": "1306.1183",
  "version": "v3",
  "published": "2013-06-05T17:39:16.000Z",
  "updated": "2016-04-21T14:37:54.000Z",
  "title": "Hyperelliptic Schottky Problem and Stable Modular Forms",
  "authors": [
    "Giulio Codogni"
  ],
  "comment": "Final version, title changed, to appear in Documenta Mathematica",
  "categories": [
    "math.AG"
  ],
  "abstract": "It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-11T16:28:19.000Z",
      "title": "Non-perturbative Schottky problem and stable equations for the hyperelliptic locus",
      "abstract": "Given an integer g and an even unimodular positive definite lattice $\\Lambda$, one can construct the classical Theta series $\\Theta_{\\Lambda,g}$, which is a degree $g$ modular form. If we fix the lattice and we package all these modular forms together, we may interpret this as a character $\\Theta_{\\Lambda}$ on the monoid $\\A_{\\infty}=\\bigcup\\A_g$. We then consider differences of Theta series. It is known that none of these vanish on $\\M_{\\infty}=\\bigcup\\M_g$, on the other hand, we are able to exhibit many non-trivial differences of Theta series vanishing on $\\h_{\\infty}=\\bigcup \\h_g$. Furthermore, we describe the behaviour of differences of Theta series associated to rank 24 lattices. One of the main ingredients is a precise description of the tangent space to the boundary of the Satake compactification.",
      "comment": "Comments are welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2016-04-21T14:37:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H42",
      "32G20"
    ],
    "keywords": [
      "non-perturbative schottky problem",
      "hyperelliptic locus",
      "stable equations",
      "modular form",
      "unimodular positive definite lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1237169,
      "adsabs": "2013arXiv1306.1183C"
    }
  }
}