{
  "id": "2301.12816",
  "version": "v1",
  "published": "2023-01-30T12:13:00.000Z",
  "updated": "2023-01-30T12:13:00.000Z",
  "title": "Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves",
  "authors": [
    "Timo Keller"
  ],
  "comment": "accepted for publication in International Journal of Number Theory",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for N\\'eron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \\to U \\subseteq \\mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme over $S$ stays the same when restricted to $S_x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-30T12:13:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11G05",
      "11G35"
    ],
    "keywords": [
      "mordell-weil rank",
      "abelian scheme",
      "silvermans specialization theorem",
      "neron-severi ranks",
      "base surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}