Weak approximation versus the Hasse principle for subvarieties of abelian varieties

For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational points should be dense in the set of (modified) adelic points surviving finite descent. We show that this follows from the a priori weaker assumption that finite descent is the only obstruction to the existence of rational points. We also prove a similar statement for the obstruction coming from the Mordell-Weil sieve.