The Brauer-Manin obstruction for subvarieties of abelian varieties over   function fields
  We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety. 
Bjorn Poonen, Jose Felipe Voloch