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I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I even do not know the definition of higher Galois cohomologies.

  1. Is there any interesting application of higher cohomologies ($H^2, H^3, H^4$ etc) in the study of elliptic curves?

    1. If so, what are these? Can you describe the idea?

Thank you.

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  • $\begingroup$ This is an issue in many instances of cohomology. Namely, there are very geometric interpretations of $H^0$ and $H^1$ and some for $H^2$ (with a marked increase in sophistication) but no known interpretations for the higher $H^3$s. For example, $H^0$ is just the $K$-points of $E$, $H^1(K,E)$ is just the twists of $E$, but $H^2$ is something more complicated as far as I know ($E$-gerbes?). $\endgroup$ – Alex Youcis Oct 13 '15 at 2:44
  • $\begingroup$ I mean, Galois cohomology comes into play in the study of elliptic curves (mostly) because the observation that if $E$ is an elliptic curve over a number field $K$ then $E(K)$ embeds into $H^1(G_K,T_\ell E)$ for any prime $\ell$ (this is just Kummer theory). As an example whatever $H^3$ may mean it must be a purely global phenomenon since, for example $H^3(K,E[N])=0$ for any ($p$-adic) $K$ local and $E/K$ an elliptic curve. I would be very interested to hear any interpretation of $H^2$ in elementary terms. $\endgroup$ – Alex Youcis Oct 13 '15 at 2:45
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    $\begingroup$ Snow, could you give us some context as to your studies? Depending on the specific situation we might be able to say more. This also might be nice to cross-post to MO. $\endgroup$ – Alex Youcis Oct 20 '15 at 14:22

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