{
  "id": "1811.05631",
  "version": "v1",
  "published": "2018-11-14T04:11:17.000Z",
  "updated": "2018-11-14T04:11:17.000Z",
  "title": "On a reduction map for Drinfeld modules",
  "authors": [
    "Wojciech Bondarewicz",
    "Piotr Krasoń"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\\widehat\\varphi}={\\phi}_{1}^{e_1}\\times \\dots \\times {\\phi}_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\\mathbb F}_{q}[t].$ We assume that the ${\\mathrm{rank}}(\\phi)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\\mathrm{End}}({\\phi}_{i})=A$ for $ i=1,\\dots , t.$ Our main result is the following numeric criterion. Let ${N}={N}_{1}^{e_1}\\times\\dots\\times {N}_{t}^{e_t}$ be a finitely generated $A$ submodule of the Mordell-Weil group ${\\widehat\\varphi}({\\cal O}_{K})={\\phi}_{1}({\\cal O}_{K})^{e_{1}}\\times\\dots\\times {\\phi}_{t}({\\cal O}_{K})^{{e}_{t}},$ and let ${\\Lambda}\\subset N$ be an $A$ - submodule. If we assume $d_{i}\\geq e_{i}$ and $P\\in N$ such that $r_{\\cal W}(P)\\in r_{\\cal W}({\\Lambda}) $ for almost all primes ${\\cal W}$ of ${\\cal O}_{K},$ then $P\\in {\\Lambda}+N_{tor}.$ We also build on the recent results of S.Bara{\\'n}czuk \\cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-14T04:11:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G09",
      "14G05",
      "14G25",
      "11J93"
    ],
    "keywords": [
      "drinfeld modules",
      "reduction map",
      "mordell-weil group",
      "global principle",
      "mordell-weil type groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}