Confusion about homogeneous spaces associated to a cocycle $\xi$ and its localization $\xi_v$.

Question

I have a slight confusion about Selmer group and homogeneous spaces $WC(E/K)$. I am using Silverman's book The Arithmetic of Elliptic Curves. First, there is this commutative diagram

$$\require{AMScd} \begin{CD} 0 @>>> E'(K) / \phi(E(K)) @>\delta_0>> H^1(G_{\bar K / K}, E[\phi]) @>wc_0>> WC(E/K)[\phi] @>>> 0 \\ & @VVV @V \ell VV @V \ell VV \\ 0 @>>> E'(K_v) / \phi(E(K_v)) @>\delta>> H^1(G_v, E[\phi]) @>wc>> WC(E / K_v)[\phi] @>>> 0 \end{CD} $$

where I have named some arrows.

For $\xi \in H^1(G_{\bar K / K}, E[\phi])$, and any $v \in M_K$, we can localize (via $\ell$) to get $\xi_v \in H^1(G_v, E[\phi])$. Then we can associate $wc(\xi_v)$ to a homogeneous space, denoted $C_{\xi_v}$. Question: Is $C_{\xi_v}$ the same curve for all places $v$?

Context for this question: I was trying to understand how computing the Selmer group reduces to checking whether a certain curve has a rational point. From reading the book, my understanding is that we do:

1. Take $\xi \in H^1(G_{\bar K / K}, E[\phi])$.
2. Localize (via $\ell$) to get $\xi_v \in H^1(G_v, E[\phi])$. Then find the homogeneous space $wc(\xi_v)$, which I will denote $C_{\xi_v}$, and check whether it is trivial, which happens if and only if $C_{\xi_v}(K_v) \neq \emptyset$.
3. If $C_{\xi_v}$ is trivial for all $v$, then $\xi$ is an element in the Selmer group.

However, from the examples in Silverman's book, to each $\xi$, he finds one homogeneous space $C_\xi$, and checks whether it is trivial for all $v$, i.e. if $C_\xi(K_v) \neq \emptyset$ for all $v$, then $\xi$ is an element of the Selmer group.

Why do this two ways give the same result? If it is too long to explain, a reference to a book will suffice too. P.S. I have searched and found several similar question but they are not asking the same question.

Thank you for your help!

Answer 1

There is a natural map $WC(E/K) \to WC(E/K_v)$ just by viewing a homogeneous space $C$ for $E$ over $K$ as being defined over $K_v$. The trick is to check that this map agrees with the restriction map on cohomology after we identify $WC(E/k)$ with $H^1(k, E)$.

Following Silverman X3.6 let $p_0 \in C(\bar{K})$ and let $c : \sigma \mapsto p_0^\sigma - p_0$ be the corresponding cocycle representing a class in $H^1(K, E)$. Then $c_v$ is obtained by restricting the domain to $G_{\bar{K}_v/K_v} \subset G_{\bar{K}/K}$. Noting this, and that $p_0 \in C(\bar{K_v})$ we see that (the class of) $c_v$ is precisely the image of $C$ in $H^1(K_v, E)$ when $C$ is viewed as having coefficients in $K_v$.

Recalling that the $\phi$-Selmer group is defined to be the kernel of the map $H^1(K, E[\phi]) \to \prod_v WC(E/K_v)$ then we see that some $\xi \in H^1(K, E[\phi])$ is contained in the $\phi$-Selmer group if (using the notation of OP) $C_\xi$ has a $K_v$-rational point for every place $v$ of $K$.