{
  "id": "2002.10854",
  "version": "v1",
  "published": "2020-02-25T13:46:02.000Z",
  "updated": "2020-02-25T13:46:02.000Z",
  "title": "Arithmetic complexity revisited",
  "authors": [
    "Igor Nikolaev"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.OA"
  ],
  "abstract": "The arithmetic complexity $c(\\mathscr{A}_{\\theta})$ is a non-commutative measure of the ranks of elliptic curves $\\mathscr{E}(K)=\\mathbf{Z}^r \\oplus \\mathscr{E}_{tors}$. The $c(\\mathscr{A}_{\\theta})$ is equal to the dimension of a connected component $V_{N,k}^0$ of the Brock-Elkies-Jordan variety associated to a periodic continued fraction $\\theta=[b_1,\\dots, b_N, \\overline{a_1,\\dots,a_k}]$. We prove that the $V_{N,k}^0$ is a fiber bundle over the Fermat-Pell conic with the structure group $\\mathscr{E}_{tors}$ and the fiber an $r$-dimensional affine space. The result is used to evaluate the Tate-Shafarevich group of $\\mathscr{E}(K)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-25T13:46:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "46L85"
    ],
    "keywords": [
      "arithmetic complexity",
      "dimensional affine space",
      "tate-shafarevich group",
      "elliptic curves",
      "structure group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}