{
  "id": "1405.3151",
  "version": "v2",
  "published": "2014-05-13T13:58:13.000Z",
  "updated": "2014-12-24T15:30:02.000Z",
  "title": "Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties",
  "authors": [
    "L. Alexander Betts",
    "Vladimir Dokchitser"
  ],
  "comment": "A new appendix by A. Morgan and V. Dokchitser deals with hyperelliptic curves of genus 2 given by C: y^2 = f(x) over p-adic fields, assuming that p is odd and f(x) has no triple roots modulo p. It explains how to read off both the Tamagawa number of the Jacobian of C from the defining equation, and the action of Frobenius on the dual character group of the toric part of its reduction",
  "categories": [
    "math.NT"
  ],
  "abstract": "We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $\\Lambda/e\\Lambda'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the $p$-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties, that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form $y^2=f(x)$, under some simplifying hypotheses.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-13T13:58:13.000Z",
      "abstract": "We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In view of the Raynaud parametrisation, this translates into a purely algebraic problem concerning the number of $H$-invariant points on a quotient of $C_n$-lattices $\\Lambda/e\\Lambda'$ for varying subgroups $H$ of $C_n$ and integers $e$. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions (corresponding to varying $e$) and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the $p$-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties, that is similar to the well-known one for elliptic curves.",
      "comment": "35 pages, 2 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-24T15:30:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11G25",
      "11G40",
      "14K15",
      "20C10"
    ],
    "keywords": [
      "tamagawa numbers",
      "semistable abelian varieties",
      "finite quotients",
      "general local behaviour",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3151B"
    }
  }
}