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Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

We consider the reduction of Brauer classes on surfaces over number fields, with a view toward applications to rationality and derived equivalence. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial upon reduction for a set of places of positive natural density. As a consequence, there are cubic fourfolds which become rational upon reduction for a positive proportion of places, and there are twisted derived equivalent K3 surfaces which become derived equivalent upon reduction for a positive proportion of places.