{
  "id": "2109.13718",
  "version": "v2",
  "published": "2021-09-28T13:36:06.000Z",
  "updated": "2022-05-27T16:54:06.000Z",
  "title": "Height functions on Hecke orbits and the generalised André-Pink-Zannier conjecture",
  "authors": [
    "Rodolphe Richard",
    "Andrei Yafaev"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain a lower bounds for the size of Galois orbits of points in a generalised Hecke orbit in terms of these height, assuming a version of the Mumford-Tate conjecture. We then use it to prove the generalised Andr\\'e-Pink-Zannier conjecture under this assumption by implementing the Pila-Zannier strategy.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-05-27T16:54:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64",
      "11G18",
      "11G50",
      "11F80",
      "14L30",
      "20G35",
      "15A16",
      "14G35"
    ],
    "keywords": [
      "generalised andré-pink-zannier conjecture",
      "height function",
      "generalised hecke orbit",
      "pila-zannier strategy",
      "shimura variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}