{
  "id": "2411.07857",
  "version": "v1",
  "published": "2024-11-12T15:15:11.000Z",
  "updated": "2024-11-12T15:15:11.000Z",
  "title": "17T7 is a Galois group over the rationals",
  "authors": [
    "Raymond van Bommel",
    "Edgar Costa",
    "Noam D. Elkies",
    "Timo Keller",
    "Sam Schiavone",
    "John Voight"
  ],
  "comment": "23 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that the transitive permutation group 17T7, isomorphic to a split extension of $C_2$ by $\\mathrm{PSL}_2(\\mathbb{F}_{16})$, is a Galois group over the rationals. The group arises from the field of definition of the 2-torsion on an abelian fourfold with real multiplication defined over a real quadratic field. We find such fourfolds using Hilbert modular forms. Finally, building upon work of Demb\\'el\\'e, we show how to conjecturally reconstruct a period matrix for an abelian variety attached to a Hilbert modular form; we then use this to exhibit an explicit degree 17 polynomial with Galois group 17T7.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-12T15:15:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12F12",
      "11F80",
      "11F41",
      "14G10"
    ],
    "keywords": [
      "hilbert modular form",
      "real quadratic field",
      "transitive permutation group 17t7",
      "galois group 17t7",
      "explicit degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}