{
  "id": "1511.09026",
  "version": "v1",
  "published": "2015-11-29T15:36:22.000Z",
  "updated": "2015-11-29T15:36:22.000Z",
  "title": "On the invariant factors of class groups in towers of number fields",
  "authors": [
    "Farshid Hajir",
    "Christian Maire"
  ],
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "For a finite abelian p-group A of rank d, we define its (logarithmic) mean exponent to be the base-p logarithm of the d-th root of its cardinality. We study the behavior of the mean exponent of p-class groups in towers of number fields. By combining techniques from group theory with the Tsfasman-Valdut generalization of the Brauer-Siegel Theorem, we construct infinite tamely ramified towers in which the mean exponent of class groups remains bounded. Several explicit examples with p=2 are given. We introduce an invariant M(G) attached to a finitely generated FAb pro-p group G which measures the asymptotic growth of the mean exponent of abelianizations of subgroups of index n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic behavior of the mean exponent of class groups in L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-29T15:36:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R37"
    ],
    "keywords": [
      "mean exponent",
      "number fields",
      "invariant factors",
      "group theory",
      "finitely generated fab pro-p group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151109026H"
    }
  }
}