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Bad reduction of the Brauer-Manin obstruction (2015)

Bright, M.

Oxford University Press

In: Journal of the London Mathematical Society

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Publication Date: 2015-05-20

Description: We relate the Brauer group of a smooth variety over a $p$ -adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how this surjectivity can be used to prove vanishing of the Brauer–Manin obstruction on varieties over number fields.

Print ISSN: 0024-6107

Electronic ISSN: 1469-7750

Topics: Mathematics

Published by Oxford University Press

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