Brauer groups of conic bundles over elliptic curves

Abdulmuhsin Alfaraj

doi:10.1007/s13366-026-00846-w

Brauer groups of conic bundles over elliptic curves

Abdulmuhsin Alfaraj

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field k. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above k-points that are 2-torsion on the elliptic curve, and the corresponding splitting fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Châtelet surfaces. We investigate Brauer–Manin obstructions to weak approximation coming from the real places on such surfaces.

Original language	English
Journal	Beitrage zur Algebra und Geometrie
Early online date	15 Apr 2026
DOIs	https://doi.org/10.1007/s13366-026-00846-w
Publication status	E-pub ahead of print - 15 Apr 2026

Data Availability Statement

The author declares that the manuscript has no associated data.

Acknowledgements

I would like to thank my supervisor Daniel Loughran for his endless support. I would also like to thank Alexei Skorobogatov for the proof of Proposition 2.13 and Jean-Louis Colliot-Thélène for useful discussions.

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Cite this

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abstract = "We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field k. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above k-points that are 2-torsion on the elliptic curve, and the corresponding splitting fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Ch{\^a}telet surfaces. We investigate Brauer–Manin obstructions to weak approximation coming from the real places on such surfaces.",

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AB - We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field k. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above k-points that are 2-torsion on the elliptic curve, and the corresponding splitting fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Châtelet surfaces. We investigate Brauer–Manin obstructions to weak approximation coming from the real places on such surfaces.

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Brauer groups of conic bundles over elliptic curves

Abstract

Data Availability Statement

Acknowledgements

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