{
  "id": "2205.10344",
  "version": "v1",
  "published": "2022-05-20T17:56:27.000Z",
  "updated": "2022-05-20T17:56:27.000Z",
  "title": "Hecke orbits on Shimura varieties of Hodge type",
  "authors": [
    "Marco D'Addezio",
    "Pol van Hoften"
  ],
  "comment": "61 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at primes of good reduction, under a mild assumption on the size of the prime. Our proof uses a new generalisation of Serre--Tate coordinates for deformation spaces of central leaves in such Shimura varieties, constructed using work of Caraiani--Scholze and Kim. We use these coordinates to give a new interpretation of Chai--Oort's notion of strongly Tate-linear subspaces of these deformation spaces. This lets us prove upper bounds on the local monodromy of these subspaces using the Cartier--Witt stacks of Bhatt--Lurie. We also prove a rigidity result in the style of Chai--Oort for strongly Tate-linear subspaces. Another main ingredient of our proof is a new result on the local monodromy groups of $F$-isocrystals \"coming from geometry\", which should be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-20T17:56:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18",
      "14G35"
    ],
    "keywords": [
      "shimura varieties",
      "hodge type",
      "strongly tate-linear subspaces",
      "deformation spaces",
      "local monodromy groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}