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Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 405–421 | Cite as

Evaluating Azumaya algebras on cubic surfaces

  • Martin BrightEmail author
Article

Abstract

Let X be a cubic surface over a p-adic field k. Given an Azumaya algebra on X, we describe the local evaluation map \({X(k) \to \mathbb{Q}/\mathbb{Z}}\) in two cases, showing a sharp dependence on the geometry of the reduction of X. When X has good reduction, then the evaluation map is constant. When the reduction of X is a cone over a smooth cubic curve, then generically the evaluation map takes as many values as possible. We show that such a cubic surface defined over a number field has no Brauer–Manin obstruction. This extends results of Colliot-Thélène, Kanevsky and Sansuc.

Mathematics Subject Classification (2000)

Primary: 11G25 Secondary: 11G35 11D25 14G25 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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