{
  "id": "math/0504215",
  "version": "v1",
  "published": "2005-04-11T12:16:02.000Z",
  "updated": "2005-04-11T12:16:02.000Z",
  "title": "On reduction maps and support problem in K-theory and abelian varieties",
  "authors": [
    "Stefan Baranczuk"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In this paper we consider reduction maps $r_{v} : K_{2n+1}(F)/C_{F} \\to K_{2n+1}(\\kappa_{v})_{l}$ where $F$ is a number field and $C_{F}$ denotes the subgroup of $K_{2n+1}(F)$ generated by $l$-parts (for all primes $l$) of kernels of the Dwyer-Friedlander map and maps $r_{v} : A(F)\\to A_{v}(\\kappa _{v})_{l}$ where $A(F)$ is an abelian variety over a number field. We prove a generalization of the support problem of Schinzel for $K$-groups of number fields: Let $P_{1}, ..., P_{s}, Q_{1}, ..., Q_{s}\\in K_{2n+1}(F)/C_{F}$ be the points of infinite order. Assume that for almost every prime $l$ the following condition holds: for every set of positive integers $m_{1}, ..., m_{s}$ and for almost every prime $v$ $$m_{1} r_{v}(P_{1})+... + m_{s} r_{v}(P_{s})=0 \\mathrm{implies} m_{1} r_{v}(Q_{1})+... + m_{s}r_{v}(Q_{s})= 0. $$ Then there exist $\\alpha_{i}$, $\\beta_{i}\\in \\mathbb{Z} \\setminus \\{0 \\}$ such that $\\alpha_{i} P_{i}+\\beta_{i} Q_{i}=0$ in $B(F)$ for every $i \\in \\{1, ... s\\}$. We also get an analogues result for abelian varieties over number fields. The main technical result of the paper says that if $P_{1}, ..., P_{s}$ are nontorsion elements of $K_{2n+1}(F)/C_{F}$, which are linearly independent over $\\mathbb{Z}$, then for any prime $l$, and for any set $\\{k_{1},... ,k_{s}\\}\\subset \\mathbb{N} \\cup \\{0\\}$, there are infinitely many primes $v$, such that the image of the point $P_{t}$ via the map $r_{v}$ has order equal $l^{k_{t}}$ for every $t \\in \\{1, ..., s \\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-04-11T12:16:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "19F99"
    ],
    "keywords": [
      "abelian variety",
      "support problem",
      "reduction maps",
      "number field",
      "paper says"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4215B"
    }
  }
}