{
  "id": "2506.18776",
  "version": "v1",
  "published": "2025-06-23T15:45:12.000Z",
  "updated": "2025-06-23T15:45:12.000Z",
  "title": "New evidence for Rémond's generalisation of Lehmer's conjecture",
  "authors": [
    "Sara Checcoli",
    "Gabriel Andreas Dill"
  ],
  "comment": "33 pages. Comments are welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of R\\'emond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup $\\Gamma$ of an almost split semiabelian variety $G$, we consider the group of rational points of $G$ over a finite extension of the field generated by the saturated closure of $\\Gamma$, i.e. the division closure of the subgroup generated by $\\Gamma$ and all its images under geometric endomorphisms of $G$. We show that this becomes a free group after one quotients out the saturated closure of $\\Gamma$. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by R\\'emond.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-23T15:45:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20K15",
      "11J95",
      "11R32",
      "11G50"
    ],
    "keywords": [
      "lehmers conjecture",
      "rémonds generalisation",
      "split semiabelian variety",
      "finite rank subgroup",
      "saturated closure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}