Beyond the Manin obstruction

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic curves. By combining the Manin obstruction with descent we define a refinement of the Manin obstruction which for surfaces of the type considered suffices to explain the absence of a rational point.