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.