{
  "id": "1805.03850",
  "version": "v1",
  "published": "2018-05-10T06:54:12.000Z",
  "updated": "2018-05-10T06:54:12.000Z",
  "title": "Asymptotic lower bound of class numbers along a Galois representation",
  "authors": [
    "Tatsuya Ohshita"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K_n, which is the Galois extension field of K fixed by the stabilizer of T/p^n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p^\\infty])/K for a given abelian variety A with certain conditions in terms of the Mordell-Weil group. We also prove another asymptotic inequality for the cases when A is a Hilbert--Blumenthal or CM abelian variety.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-10T06:54:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11G05",
      "11G10",
      "11R23"
    ],
    "keywords": [
      "class number",
      "asymptotic lower bound",
      "galois representation",
      "asymptotic inequality",
      "explicit lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}