{
  "id": "math/0211120",
  "version": "v1",
  "published": "2002-11-06T19:25:53.000Z",
  "updated": "2002-11-06T19:25:53.000Z",
  "title": "Quaternions, polarizations and class numbers",
  "authors": [
    "Victor Rotger"
  ],
  "comment": "To appear in Crelle",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study abelian varieties $A$ with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number $\\pi_0(A)$ of isomorphism classes of principal polarizations on $A$ in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that $\\pi_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-06T19:25:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class numbers",
      "simple abelian varieties",
      "non-isomorphic principal polarizations",
      "totally real number field",
      "totally indefinite quaternion algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11120R"
    }
  }
}