Arithmetic of Châtelet surface bundles revisited

We study the arithmetic of algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate–Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer–Manin obstruction is not invariant under an arbitrary finite extension of the ground field.