{
  "id": "1012.1825",
  "version": "v1",
  "published": "2010-12-08T18:51:18.000Z",
  "updated": "2010-12-08T18:51:18.000Z",
  "title": "Algebraic equations on the adelic closure of a Drinfeld module",
  "authors": [
    "Dragos Ghioca",
    "Thomas Scanlon"
  ],
  "categories": [
    "math.NT",
    "math.AG",
    "math.LO"
  ],
  "abstract": "Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\\mathbf A}_K$ be a ring of ad\\'{e}les of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $\\Phi:{\\mathbb F}[t] \\to \\operatorname{End}_K({\\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\\mathbf A}_K^g$ as $\\Phi({\\mathbb F}_p[t])$-modules under the diagonal action induced by $\\Phi$. For $\\Gamma \\subseteq K^g$ a finitely generated $\\Phi(\\F_p[t])$-submodule and an affine subvariety $X \\subseteq \\bG_a^g$ defined over $K$, we study the intersection of $X({\\mathbf A}_K)$, the ad\\`{e}lic points of $X$, with $bar{\\Gamma}$, the closure of $\\Gamma$ with respect to the ad\\`{e}lic topology, showing under various hypotheses that this intersection is no more than $X(K) \\cap \\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-08T18:51:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G09",
      "14G17",
      "03C98"
    ],
    "keywords": [
      "drinfeld module",
      "adelic closure",
      "algebraic equations",
      "function field",
      "cofinite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.1825G"
    }
  }
}