{
  "id": "1302.1477",
  "version": "v1",
  "published": "2013-02-06T19:02:11.000Z",
  "updated": "2013-02-06T19:02:11.000Z",
  "title": "Arithmetic of abelian varieties with constrained torsion",
  "authors": [
    "Christopher Rasmussen",
    "Akio Tamagawa"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\\ell$-power torsion fields are arithmetically constrained for some rational prime $\\ell$. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$\\ell$ fundamental group of $P^1 - \\{0,1,\\infty\\}$. Under GRH, we demonstrate the set of classes is finite for any fixed $K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of $K/\\Q$ and the dimension of abelian varieties are not too large, through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of $\\ell$) are uniform in the degree of the extension $K/\\Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-06T19:02:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11F80",
      "14K15"
    ],
    "keywords": [
      "abelian varieties",
      "constrained torsion",
      "canonical outer galois representation",
      "power torsion fields",
      "special fiber"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.1477R"
    }
  }
}