{
  "id": "2006.00316",
  "version": "v1",
  "published": "2020-05-30T16:56:44.000Z",
  "updated": "2020-05-30T16:56:44.000Z",
  "title": "Calculation of $h^1$ of some Anderson t-motives",
  "authors": [
    "Stefan Ehbauer",
    "Aleksandr Grishkov",
    "Dmitry Logachev"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider Anderson t-motives $M$ of dimension 2 and rank 4 defined by some simple explicit equations parameterized by $2\\times2$ matrices. We use methods of explicit calculation of $h^1(M)$ -- the dimension of their cohomology group $H^1(M)$ ( = the dimension of the lattice of their dual t-motive $M'$) developed in our earlier paper. We calculate $h^1(M)$ for $M$ defined by all matrices having 0 on the diagonal, and by some other matrices. These methods permit to make analogous calculations for most (probably all) t-motives. $h^1$ of all Anderson t-motives $M$ under consideration satisfy the inequality $h^1(M)\\le4$, while in all known examples we have $h^1(M)=0,1,4$. Do exist $M$ of this type having $h^1=2,3$? We do not know, this is a subject of further research.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-30T16:56:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G09"
    ],
    "keywords": [
      "anderson t-motives",
      "simple explicit equations",
      "consideration satisfy",
      "explicit calculation",
      "cohomology group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}