{
  "abstract": "We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $μ_p\\not\\subset K$. For $p\\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $μ_p\\not\\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\\bullet(G_K,\\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11390",
  "author_total": 2,
  "authors": [
    {
      "name": "Ambrus Pál"
    },
    {
      "name": "Gereon Quick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fx2jdc0mjb7br.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11390",
  "fulltext_url": null,
  "id": "fx2jdc0mjb7br",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11390",
  "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.AT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221180790",
  "pdf_url": "https://arxiv.org/pdf/2610.11390",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fx2jdc0mjb7br.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.11390",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11390"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Weak and genuine A_n-formality for number fields",
  "title_text": "Weak and genuine A_n-formality for number fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/fx2jdc0mjb7br",
  "venue": "arXiv",
  "versions": []
}