{
  "id": "2112.02280",
  "version": "v2",
  "published": "2021-12-04T08:37:08.000Z",
  "updated": "2022-01-27T08:05:09.000Z",
  "title": "Birational classification for algebraic tori",
  "authors": [
    "Akinari Hoshi",
    "Aiichi Yamasaki"
  ],
  "comment": "175 pages, corrected typos",
  "categories": [
    "math.AG",
    "math.NT",
    "math.RA"
  ],
  "abstract": "We give a stably birational classification for algebraic tori of dimensions $3$ and $4$ over a field $k$. First, we define the weak stably equivalence of algebraic tori and show that there exist $13$ (resp. $128$) weak stably equivalent classes of algebraic tori $T$ of dimension $3$ (resp. $4$) which are not stably rational by computing some cohomological stably birational invariants, e.g. the Brauer-Grothendieck group of $X$ where $X$ is a smooth compactification of $T$, provided by Kunyavskii, Skorobogatov and Tsfasman. We make a procedure to compute such stably birational invariants effectively and the computations are done by using the computer algebra system GAP. Second, we define the $p$-part of the flabby class $[\\widehat{T}]^{fl}$ as a $Z_p[Sy_p(G)]$-lattice and prove that they are faithful and indecomposable $Z_p[Sy_p(G)]$-lattices unless it vanishes for $p=2$ (resp. $p=2,3$) in dimension $3$ (resp. $4$). The $Z_p$-ranks of them are also given. Third, we give a necessary and sufficient condition for which two not stably rational algebraic tori $T$ and $T^\\prime$ of dimensions $3$ (resp. $4$) are stably birationally equivalent in terms of the splitting fields and the weak stably equivalent classes of $T$ and $T^\\prime$. In particular, the splitting fields of them should coincide if $\\widehat{T}$ and $\\widehat{T}^\\prime$ are indecomposable. Forth, for each $7$ cases of not stably but retract rational algebraic tori of dimension $4$, we find an algebraic torus $T^\\prime$ of dimension $4$ which satisfies that $T\\times_k T^\\prime$ is stably rational. Finally, we give a criteria to determine whether two algebraic tori $T$ and $T^\\prime$ of general dimensions are stably birationally equivalent when $T$ (resp. $T^\\prime$) is stably birationally equivalent to some algebraic torus of dimension up to $4$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-27T08:05:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "12F20",
      "13A50",
      "14E08",
      "20C10",
      "20G15"
    ],
    "keywords": [
      "algebraic torus",
      "birational classification",
      "stably birationally equivalent",
      "weak stably equivalent classes",
      "stably rational"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 175,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}