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Brauer-Manin obstruction for symmetric powers

  • Yang Cao
  • Lin Zhou
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Abstract

For a quasi-projective smooth geometrically integral variety X over a number field, we prove that the Brauer-Manin obstruction for its symmetric powers coincides with the obstruction coming from X. As a consequence, Brauer-Manin obstruction for the symmetric powers cannot provide new cohomological obstruction to weak approximation of zero-cycles for X. The key intermediate ingredients are a Dold-Thom type result for Brauer groups and a precise descent formula for adelic points.

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Cite

BibTeXRIS
BibTeX
@misc{cao2026brauermanin,
  title = {{Brauer-Manin obstruction for symmetric powers}},
  author = {Yang Cao and Lin Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23430},
  eprint = {2610.23430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23430},
}
RIS
TY  - UNPB
TI  - Brauer-Manin obstruction for symmetric powers
AU  - Yang Cao
AU  - Lin Zhou
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23430
UR  - https://arxiv.org/abs/2610.23430
AB  - For a quasi-projective smooth geometrically integral variety X over a number field, we prove that the Brauer-Manin obstruction for its symmetric powers coincides with the obstruction coming from X. As a consequence, Brauer-Manin obstruction for the symmetric powers cannot provide new cohomological obstruction to weak approximation of zero-cycles for X. The key intermediate ingredients are a Dold-Thom type result for Brauer groups and a precise descent formula for adelic points.
ER  -