My question is the following: Let $\phi: E \rightarrow E \prime$ be a non-zero isogeny between elliptic curves, defined over some numberfield K. Then we can define the Selmer-group $Sel^{\phi}(E/K)$. Moreover let $v \in M_K$ be a place of K and $G_v \subset Gal(\bar{K}/K)$ its decompositiongroup. Take some $\xi \in Sel^{\phi}(E/K)$ and consider the restriction map $res_v: Sel^{\phi}(E/K) \rightarrow H¹(G_v, E[\phi])$. Is it true that $res_v(\xi)=0$ for almost every place $v \in M_K$? I would appreciate every helpful comment! Thanks in advance!
2026-03-29 14:32:27.1774794747
Restricting an Element of the Selmer Group to the Decompositiongroup for some place v
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