{
  "abstract": "Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \\cite{DhillonClassifying}, Kottwitz's local--global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \\[ Br(BG)/Br(k)\\simeq Pic(G), \\] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer--Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \\[ Br^{un}(BG)/ Br(k)\\simeq Sha^1_{\\mathrm{cyc}}(k,\\widehat G), \\qquad \\widehat G=X^*(G_{\\bar{k}}), \\] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02819",
  "author_total": 1,
  "authors": [
    {
      "name": "Ajneet Dhillon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9d7n6q0b61m6n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02819",
  "fulltext_url": null,
  "id": "9d7n6q0b61m6n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02819",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.02819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9d7n6q0b61m6n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Adelic points and unmramified Brauer approximation for classifying stacks",
  "title_text": "Adelic points and unmramified Brauer approximation for classifying stacks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9d7n6q0b61m6n",
  "venue": "arXiv",
  "versions": []
}