Theorem 6 of [this paper][1] gives a generalization of Shafarevich's Theorem:  

>**Theorem**: Let $K$ be a Hilbertian field.  Let $A_{/K}$ be a nontrivial abelian variety.  If $n > 1$ is indivisible by the characteristic of $K$ and $A(K)/nA(K)$ is finite, then $H^1(K,A)$ has infinitely many elements of order $n$.  

As explained in the paper, the hypothesis that $A(K)/nA(K)$ be finite cannot be omitted, but the hypothesis that $n$ is indivisible by the characteristic of $K$ can be weakened to: $A(K^{\operatorname{sep}})$ contains a point of order $n$.  Via the Kummer sequence, the proof quickly reduces to showing that $H^1(K,A[n])$ contains infinitely many elements of order $n$, which is related to [your other question][2] and (yet more closely) to Lior Bary-Soroker's answer to it.





[1]: http://alpha.math.uga.edu/~pete/IFG_Revised.pdf
[2]: http://mathoverflow.net/questions/239493