{
  "id": "math/0112319",
  "version": "v1",
  "published": "2001-12-04T00:00:00.000Z",
  "updated": "2001-12-04T00:00:00.000Z",
  "title": "Extensions of number fields with wild ramification of bounded depth",
  "authors": [
    "Farshid Hajir",
    "Christian Maire"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the length of the ramification filtration becomes large. We show that all p-adic representations of these Galois groups are potentially semistable; thus, a conjecture of Fontaine and Mazur on the structure of tamely ramified Galois p-extensions extends to our case. Further evidence in support of this conjecture is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-12-04T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "number fields",
      "wild ramification",
      "bounded depth",
      "galois group",
      "extend well-known properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....12319H"
    }
  }
}