Consider some elliptic curve $E$ over a number field $k$. Then for any prime $p$ there is a short exact sequence $$ 0 \to E(k)/pE(k) \to H^1(k,E[p]) \to H^1(k,E)[p] \to 0. $$ Now, $H^1$ has an interpretation in terms of ($k$-equivalence classes of) torsors. My question is:
How should I interpret the first injective arrow in terms of torsors? In other words, given some point $P \in E(k)/pE(k)$, how can I associate to it a torsor whose class is in $H^1(k,E[p])$?
For $P \in E(k)/pE(k)$, consider the set $\{Q \in E(\overline{k}) :pQ = P\}$. This is principal homogeneous under $E[p](\overline{k})$, so it is a good candidate for the $\overline{k}$-points of the torsor we are looking for.
In fact, getting the actual torsor is just a matter of reformulating the above. Identify the point $P$ with $\mathrm{Spec}(k)$. Consider the map $p: E \to E$; consider the fibre $F$ over the point $P = \mathrm{Spec }(k) \hookrightarrow E$. If $p$ is invertible in $k$, then $F/k$ is finite étale of rank $p^2$ over $\mathrm{Spec }(k)$, and you can check that it is equipped with a natural scheme-theoretic action of $E[p]$ making it into a $\text{Gal}(\overline{k}/k)$-torsor of $E[p]$. The image of $P$ in $H^1(k, E[p])$ is precisely the class of this torsor.