TITLE: Galois cohomologies of an elliptic curve
QUESTION [12 upvotes]: I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
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.

Is there any interesting application of higher cohomologies ($H^2, H^3, H^4$ etc) in the study of elliptic curves?
If so, what are these? Can you describe the idea?

Thank you.

REPLY [14 votes]: $H^2$ appears quite prominently in the duality theory of elliptic curves. For a local field such as a finite extension of $\mathbb Q_p$, one has $H^2(\text{Gal}(\bar K/K),\bar K^*)=\mathbb Q/\mathbb Z$, and for a global field such as a number field, one similarly has $H^2(\text{Gal}(\bar K/K),\mathsf{A}_{\bar K}^*)=\mathbb Q/\mathbb Z$, where $\mathsf{A}_{\bar K}^*$ is the group of ideles. This makes these groups ideal targets for pairings via cup product from, e.g., two $H^1$ groups. Tate's local pairing on $E(K)\times H^1(\text{Gal}(\bar K/K),E(\bar K))$ and the Cassels-Tate pairing on the Shafarevich-Tate group can be formulated in these terms; see Tate's address to the International Congress. An interesting side note is that Tate's definition of the global pairing relies at one point on the fact that $H^3(\text{Gal}(\bar K/K),\bar K^*)=0$ in order to modify a 2-cochain and turn it into a cocycle. So there's a situation where an $H^3$ appears.
You ask "But two introductory textbooks I read used only H0 and H1. I am curious why higher cohomologies did not appear." I'm guessing one of those textbooks is mine. The reason is that the material in the text only needed $H^0$ and $H^1$, and I wanted to keep the exposition as accessible as possible, so I did not see the need to develop the general theory. In principle, the whole book could have been written without any cohomology. But cohomology is very important in modern mathematics, so my idea was to introduce the $H^0$ and $H^1$ very concretely and show the reader how useful they are, which I hoped would help motivate the more general (but more complicated) theory of higher $H^i$'s when the reader eventually ran across them. I will note that in the first paragraph of Appendix B of The Arithmetic of Elliptic Curves, I did provide 4 references for people who wanted to see more of the theory of group cohomology, including the definition and construction of the higher $H^i$'s.