{
  "abstract": "We prove that the two-dimensional Jacobian conjecture $\\mathrm{JC}_2$ implies the generalized Dixmier conjecture $\\mathrm{DC}'_1$ for the first Weyl algebra $A_1$, which states, in one of several equivalent formulations, that every endomorphism of the two-by-two matrix ring $\\operatorname{Mat}_2(A_1)$ is invertible. For the converse, an endomorphism of $\\mathbb A^2_{\\mathbb C}$ with Jacobian one gives rise to a family of holonomic $A_2$-modules over finite fields. One needs to lift this family to an $A_2$-module in characteristic zero with the prescribed arithmetic support. The existence of such a lift is a special case of Kontsevich's conjecture on holonomic $\\mathcal D$-modules with prescribed arithmetic support. We prove that in this special situation such a lift is unique whenever it exists. If these lifts exists for every endomorphism with Jacobian one, then $\\mathrm{DC}'_1$ implies $\\mathrm{JC}_2$. For every $n\\ge 3$, we construct a Lagrangian subvariety of $T^*\\mathbb{A}^n$ that provides a counterexample to Kontsevich's conjecture. The counterexamples arise from the graph of a noninvertible symplectic endomorphism of $\\mathbb A^4$ constructed by Long. We also present a different strategy for lifting $\\mathcal D$-modules associated with a Jacobian-one endomorphism of $\\mathbb A^2_{\\mathbb C}$ to characteristic zero via ultraproducts and show that the existence of a characteristic-zero lift in this setting is equivalent to the uniform boundedness of the Berest--Wilson invariants of these $A_2$-modules, regarded as left $A_1$-modules. Finally, we study two other conjectures of Kontsevich on arithmetic support, giving a counterexample to one and partial results on the other.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33919",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexey Kanel-Belov"
    },
    {
      "name": "Timofey Fedorov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cwrwkxnvm58jp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33919",
  "fulltext_url": null,
  "id": "cwrwkxnvm58jp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33919",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223864692",
  "pdf_url": "https://arxiv.org/pdf/2610.33919",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cwrwkxnvm58jp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33919",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33919"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture",
  "title_text": "The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/cwrwkxnvm58jp",
  "venue": "arXiv",
  "versions": []
}