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.
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Bright, M. Evaluating Azumaya algebras on cubic surfaces. manuscripta math. 134, 405–421 (2011). https://doi.org/10.1007/s00229-010-0400-2
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DOI: https://doi.org/10.1007/s00229-010-0400-2