{
  "id": "2310.07006",
  "version": "v1",
  "published": "2023-10-10T20:49:10.000Z",
  "updated": "2023-10-10T20:49:10.000Z",
  "title": "Mirror symmetry and the Breuil-Mézard Conjecture",
  "authors": [
    "Tony Feng",
    "Bao Le Hung"
  ],
  "categories": [
    "math.NT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "The Breuil-M\\'{e}zard Conjecture predicts the existence of hypothetical \"Breuil-Mezard cycles\" in the moduli space of mod $p$ Galois representations of $\\mathrm{Gal}(\\overline{\\mathbb{Q}}_q/\\mathbb{Q}_q)$ that should govern congruences between mod $p$ automorphic forms. For generic parameters, we propose a construction of Breuil-M\\'{e}zard cycles in arbitrary rank, and verify that they satisfy the Breuil-M\\'{e}zard Conjecture for all sufficiently generic tame types and small Hodge-Tate weights. Our method is purely local and group-theoretic, and completely distinct from previous approaches to the Breuil-M\\'ezard Conjecture. In particular, we leverage new connections between the Breuil-M\\'ezard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T20:49:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mirror symmetry",
      "breuil-mézard conjecture",
      "breuil-mezard conjecture",
      "sufficiently generic tame types",
      "small hodge-tate weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}