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PreprintCC BY 4.0arXiv

Unramified cohomology and Brauer--Manin pairing

  • Amalendu Krishna
  • Jitendra Rathore
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Abstract

We show that the Tate module of the ℓ-adic unramified cohomology H3nr(X, Qℓ/Zℓ) of a smooth projective surface X over a local or strictly local field k of residue characteristic prime to ℓ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer--Manin pairing for open subvarieties of X. To prove the vanishing theorem, we establish Bloch's formula for the Chow group of codimension-two cycles on a semistable model of X, and derive applications to the restriction map for such cycles to the closed fiber of the model.

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Cite

BibTeXRIS
BibTeX
@misc{krishna2026unramified,
  title = {{Unramified cohomology and Brauer--Manin pairing}},
  author = {Amalendu Krishna and Jitendra Rathore},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09127},
  eprint = {2609.09127},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09127},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unramified cohomology and Brauer--Manin pairing
AU  - Amalendu Krishna
AU  - Jitendra Rathore
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09127
UR  - https://arxiv.org/abs/2609.09127
AB  - We show that the Tate module of the ℓ-adic unramified cohomology H^3_nr(X, Q_ℓ/Z_ℓ) of a smooth projective surface X over a local or strictly local field k of residue characteristic prime to ℓ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer--Manin pairing for open subvarieties of X. To prove the vanishing theorem, we establish Bloch's formula for the Chow group of codimension-two cycles on a semistable model of X, and derive applications to the restriction map for such cycles to the closed fiber of the model.
ER  -