TITLE: Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
QUESTION [15 upvotes]: Let $E$ be an elliptic curve over $k=\mathbb{Q}$.  Consider $H^1(k,E)$. 
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I would be happy to have an explanation of this fact and/or a reference.
(I usually work with linear algebraic groups. For an abelian linear algebraic 
group $A$ there exists a natural number $d=d(A_{\bar k})$ such that the order of any element of $H^1(k,A)$ is $\le d$.)
EDIT: Cassels, § 10 and § 27, writes that this result was first proved by Shafarevich in 1957.

REPLY [3 votes]: Theorem 6 of this paper 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 and (yet more closely) to Lior Bary-Soroker's answer to it.