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I'm studying elliptic curves on the book of Silverman The Arithmetic of Elliptic Curves. In the appendix the author describes the cohomology groups for finite and profinite groups . In the first case we use any cocycles while in the second ONLY continuous cocycles . However in the bijection between Twists and $ H^1$ or between the Weil-Chatelet group and $ H^1$ ( for example to prove surjectivity ), it seems to me that he DOESN'T use only continuous cocycles but every cocycles. Why? Further, where are the points in which we use the continuity of the cocycles in the book ?

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  • $\begingroup$ Note that if the group $G$ is finite, then every map from $G$ to a $G$-module $M$ is continuous (simply because both $G$ and $M$ are equipped with their discrete topology). So finite group cohomology is a special case of profinite group cohomology with continuous cocycles. $\endgroup$ – paf Jul 4 '16 at 16:03

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