{
  "annotation_id": "0712f4b6-eccf-46f4-bbe6-7c18109ba723",
  "date_created": "2026-02-17T05:53:19.948000Z",
  "date_modified": "2026-02-17T05:53:19.948000Z",
  "file_hash": "6dea0652dc33e110a5d35d3d8cf5af8ae72d36c31dd6733d366c650635611394",
  "private": false,
  "record": {
    "abstract": "Let $\\rho$ be an $n$-dimensional representation of $\\mathcal{G}_{\\mathbb{Q}_p}$ over $\\overline{\\mathbb{F}_p}$. When $\\rho$ is generic and a good conjugate, the article \"Conjectures and results on modular representations of $\\mathrm{GL}_n(K)$ for a $p$-adic field $K$\", by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of compatibility with $\\rho$ for an admissible representation of $\\mathrm{GL}_n(\\mathbb{Q}_p)$. In loc. cit., the five authors also question whether one could recover a representation of $\\mathcal{G}_{\\mathbb{Q}_p}^{n-1}$, called $\\overline{L}^{\\boxtimes}(\\rho)$ and constructed from $\\rho$, from some $\\Pi$ compatible with $\\rho$ by using Z\\\u0027abr\\\u0027adi\u0027s functor $\\mathbf{V}_{\\Delta}$. We give a range of results, for an arbitrary $\\Pi$ verifying some \"weak\" compatibilities with $\\rho$, about how badly $\\mathbf{V}_{\\Delta}(\\Pi)$ behaves. In particular, when $\\rho$ is reducible and $n\\geq 3$, no $\\Pi$ compatible with $P_{\\rho}$ can verify $\\mathbf{V}_{\\Delta}(\\Pi)\\simeq \\overline{L}^{\\boxtimes}(\\rho)$.",
    "arxiv_id": "2601.09539",
    "authors": [
      "Nataniel Marquis"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Implications of Breuil-Herzig-Hu-Morra-Schraen\u0027s conjectures on Z\\\u0027abr\\\u0027adi\u0027s functor",
    "url": "https://arxiv.org/abs/2601.09539",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "861e7f44-1aa5-47ee-b366-43fc51ce8883",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}