{
  "id": "math/0002232",
  "version": "v3",
  "published": "2000-02-28T07:04:38.000Z",
  "updated": "2022-12-10T21:14:24.000Z",
  "title": "Isogeny classes of abelian varieties with no principal polarizations",
  "authors": [
    "Everett W. Howe"
  ],
  "comment": "This version highlights an error in the paper, and includes marginal notes pointing out the incorrect result and the error in the proof",
  "journal": "pp. 203--216 in: Moduli of Abelian Varieties (Carel Faber, Gerard van der Geer, and Frans Oort, eds.), Progr. Math 195, Birkh\\\"auser, Basel, 2001",
  "doi": "10.1007/978-3-0348-8303-0_7",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\\ell$ of $k$ of odd prime degree $p$, and an elliptic curve $E$ over $k$ that has no complex multiplication over $k$ and that has no $k$-defined $p$-isogenies to another elliptic curve, we construct a simple $(p-1)$-dimensional abelian variety $X$ over $k$ such that every polarization of every abelian variety isogenous to $X$ has degree divisible by $p^2$. We note that for every odd prime $p$ and every number field $k$, there exist $\\ell$ and $E$ as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field $k$ have positive characteristic and that there be a Galois extension of $k$ with a certain non-abelian Galois group. Note: Theorem 3.2 in this paper is incorrect. The current version of the paper includes comments explaining the mistake.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2000-03-06T18:07:49.000Z",
      "abstract": "We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a field k, a Galois extension l of k of odd prime degree p, and an elliptic curve E over k that has no complex multiplication over k and that has no k-defined p-isogenies to another elliptic curve, we construct a simple (p-1)-dimensional abelian variety X over k such that every polarization of every abelian variety isogenous to X has degree divisible by p^2. We note that for every odd prime p and every number field k, there exist l and E as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field k have positive characteristic and that there be a Galois extension of k with a certain non-abelian Galois group.",
      "comment": "13 pages, AMS-TeX, with updated references. To appear in the volume \"Moduli of Abelian Varieties (Texel Island 1999)\"",
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2022-12-10T21:14:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K02",
      "11G10",
      "14K15"
    ],
    "keywords": [
      "abelian variety",
      "principal polarization",
      "isogeny classes",
      "galois extension",
      "elliptic curve"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......2232H"
    }
  }
}