Brauer-Manin obstruction to weak approximation on abelian varieties

Abstract

LetA be an abelian variety defined over a number fieldK. Assume that the Tate-Shafarevich group is finite. We prove that the condition that the topological closure ofA (K) in $$\prod {_{v \in M_K^\infty } A(K_v )} $$ is open is equivalent to the condition that the Brauer-Manin obstruction is the only obstruction to weak approximation.