  We study arithmetic of the 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 Ch\^atelet 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.
