Adelic points and unmramified Brauer approximation for classifying stacks

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)≃ 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)≃ Sha¹_cyc(k,Ĝ), Ĝ=X^*(G_k̅), 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.

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BibTeX RIS

BibTeX

@misc{dhillon2026adelic,
  title = {{Adelic points and unmramified Brauer approximation for classifying stacks}},
  author = {Ajneet Dhillon},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02819},
  eprint = {2609.02819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02819},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - Adelic points and unmramified Brauer approximation for classifying stacks
AU  - Ajneet Dhillon
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02819
UR  - https://arxiv.org/abs/2609.02819
AB  - 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)≃ 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)≃ Sha^1_cyc(k,Ĝ), Ĝ=X^*(G_k̅), 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.
ER  -

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