  Let A be an abelian variety over a number field k. We show that weak
approximation holds in the Weil-Ch\^atelet group of A/k but that it may fail
when one restricts to the n-torsion subgroup. This failure is however
relatively mild; we show that weak approximation holds outside a finite set of
primes which is generically empty. This proves a conjecture of Lang and Tate
that can be seen as an analog of the Grunwald-Wang theorem in class field
theory. The methods apply, for the most part, to arbitrary finite Galois
modules and so may be of interest in their own right.
