Brauer–Manin obstruction and families of generalised Châtelet surfaces over number fields

Abstract

Over infinitely many number fields [Formula: see text] (including all finite Galois extensions [Formula: see text] of odd degree unramified at 2), we give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] associated to the normic equation [Formula: see text], where [Formula: see text] has a specific form and [Formula: see text] is even and arbitrarily large, to have the property that [Formula: see text] but [Formula: see text]. We also give general sufficient conditions in order for the generalised Châtelet surfaces [Formula: see text] over [Formula: see text] of the same form as above to have the property that [Formula: see text] and [Formula: see text]. As an application, we prove that, for a certain family of generalised Châtelet surfaces over [Formula: see text], a positive proportion (but not 100[Formula: see text]) of its members exhibits a violation of the Hasse principle explained by the Brauer–Manin obstruction.