Weak and genuine A_n-formality for number fields

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_pnot⊂ K. For p≥ 5, and for p=3 under a natural local cyclotomic condition, we prove that every associated finite A₃-embedding problem is solvable. In contrast, for every odd p with μ_pnot⊂ K, we show that no compatible family of finite-stage solutions exists, and hence C^•(G_K,𝔽_p) is not genuinely A₃-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 A₃-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.

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BibTeX RIS

BibTeX

@misc{pal2026weak,
  title = {{Weak and genuine A_n-formality for number fields}},
  author = {Ambrus Pál and Gereon Quick},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11390},
  eprint = {2610.11390},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11390},
}

RIS

TY  - UNPB
TI  - Weak and genuine A_n-formality for number fields
AU  - Ambrus Pál
AU  - Gereon Quick
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11390
UR  - https://arxiv.org/abs/2610.11390
AB  - 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_pnot⊂ K. For p≥ 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_pnot⊂ K, we show that no compatible family of finite-stage solutions exists, and hence C^•(G_K,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.
ER  -

Abstract

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