Let $G$ be a finite abelian etale group scheme over a number field $k$. Let $E$ be an elliptic curve over $k$ and $C := E\backslash \{O\}$ its affine model of the same equation.


Recall that for a variety $X$, the pointed (etale cohomology) set $H^1(X,G)$ classifies all $X$-torsors under $G$.


Given a torsor $Y \rightarrow E$ under $G$, the pull-back by the canonical inclusion $C \rightarrow E$ gives us an element whose class is in $H^1(C,G)$. Thus we have a map $$H^1(E,G) \rightarrow H^1(C,G)$$ defined by pull-backs.


I want to know if this map has any chance of being an isomorphism. There isn't much in the literature about the cohomology groups $H^i(E,-)$, usually $E$ is on the right, and therefore I hope to get references understanding $E$-torsors under some group rather than some $X$-torsors under $E$.



Torsors over elliptic curves
Indeed this map is an isomorphism. There is a diagram of five-term exact sequences, arising from the Leray spectral sequence for the maps $E\to \text{Spec}(k), C\to \text{Spec}(k)$, from


$0\to H^1(k, H^0(E\_{\overline{k}}, G))\to H^1(E, G)\to H^0(k, H^1(E\_{\overline{k}}, G))\to H^2(k, H^0(E\_{\overline{k}}, G))\to H^2(E, G)$


to the analogous sequence with $C$


$0\to H^1(k, H^0(C\_{\overline{k}}, G))\to H^1(C, G)\to H^0(k, H^1(C\_{\overline{k}}, G))\to H^2(k, H^0(C\_{\overline{k}}, G))\to H^2(C, G)$.


By comparison with the complex-analytic setting, the map $H^i(E\_{\overline{k}}, G)\to H^i(C\_{\overline{k}},G)$ is an isomorphism for $i=0,1$. Hence this map of five-term exact sequences is an isomorphism on the first, third, and fourth terms.


We would like to argue that it is an isomorphism on the second term. But this is immediate from the five lemma.


