Selmer and Shafarevich-Tate Groups

Question

I'm currently trying to under the Selmer and Shafarevich-Tate Groups from Silverman's Arithmetic of Elliptic Curves (2nd edition), pg. 331 onwards. I have a couple of questions I think is derived from my lack of understanding in Galois cohomology. Having said that, I have worked through Appendix B and are already familiar with definitions and theorems in the appendix.

My questions are as follows:

1. On page 331, how is the fundamental short exact sequence obtained from the long exact sequence in the middle of the page? I have looked through several articles regarding this and most of them will refer to it as "extracting the short exact sequence", but I don't see a theorem or anything that indicates this should be the case.

2. At the top of page 332, there are two rows of short exact sequences. My understanding of the maps between the two rows are that from the left to right, the first map is an inclusion map (which is obvious), but the next two are restriction maps. Is the reason we consider the definition of the Selmer group as it is because these restriction maps are necessarily injective? I don't see anywhere stating that they are necessarily injective but I thought that it might be the case since we want to bound the kernel of the surjective map at the top row by a larger kernel.

3. Finally, why do the definitions of the $\phi$-Selmer group and the Shafarevich-Tate group on page 332 leave out $[\phi]$, which I think should follow from the diagram at the top of page 332? Does it not make a difference?

Answer 1

1. This is just the definition of exact sequence, namely the kernel of a map equals the image of the previous one. So the kernel of the map $E'(K)\to H^1(G_K,E[\phi])$ equals the image of the map $\phi\colon E(K)\to E'(K)$, so by the first isomorphism theorem, you have an injection $E'(K)/E(K)\to H^1(G_K,E[\phi])$. Now apply the same principle: the image of the map $\phi\colon H^1(G_K,E[\phi])\to H^1(G_K,E)$ coincides with the kernel of the next one, and such a kernel is denoted by $H^1(G_K,E)[\phi]$.

2. The other vertical maps are not restriction, but they are a product of restriction maps. I don't know the answer to your question, but I don't think that the restriction map $H^1(G_K,E[\phi])\to H^1(G_{K_v},E[\phi])$ needs to be injective.

3. Yes, for the $\phi$-Selmer group it doesn't make a difference since the image of $\prod_vH^1(G_{K_v},E[\phi])$ is contained in $\prod_vWC(E(K_v))[\phi]$. Regarding the Tate-Shafarevic group, I don't understand your question...