{
  "id": "0712.2812",
  "version": "v4",
  "published": "2007-12-17T20:48:30.000Z",
  "updated": "2008-10-11T15:59:19.000Z",
  "title": "Prescribing valuations of the order of a point in the reductions of abelian varieties and tori",
  "authors": [
    "Antonella Perucca"
  ],
  "comment": "Final version. To appear on Journal of Number Theory",
  "doi": "10.1016/j.jnt.2008.07.004",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) equals v_l(m).",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-10-11T15:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K15",
      "11G10",
      "14G25",
      "14L15",
      "11R45"
    ],
    "keywords": [
      "abelian variety",
      "prescribing valuations",
      "reductions",
      "smallest algebraic k-subgroup",
      "infinite order"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.2812P"
    }
  }
}