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Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v \in M_K} H^1(G_{v_L},E(L_{v_L})), \end{equation} where $M_K$ denotes the set of all places of $K$, $v_L$ denotes a fixed place of $L$ above a place $v \in M_K$, $L_{v_L}$ denotes the completion of $L$ at $v_L$ and $G_{v_L}:=Gal(L_{v_L}/K_v)$. My questions are as follows:

  1. What can say about the kernel and cokernel of $\mathcal{F}$? Are there explicit examples for which $Ker(\mathcal{F})$ and $Coker(\mathcal{F})$ can be determined? (For instance, if we assume that $E(K)$ and $E(L)$ are finite, then what we can say about $Ker(\mathcal{F})$ and $Coker(\mathcal{F})$? )

  2. Assume that $E(K)=0$. Under what conditions one can say $E(L)$ is also trivial?

  3. Is there any relation between the set of primes that E has bad reduction at them and the ones that are ramified in $L$?

  4. Let $\mathfrak{p}$ be a prime of $K$ such that $E$ has good reduction at $\mathfrak{p}$. Is the $\mathfrak{p}$-component of $Image(\mathcal{F})$ necessarily zero? (I mean, does determine the image of $\mathcal{F}$ only by bad reductions and ramifications?)

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  • $\begingroup$ 2.) The shape of the Galois group may help to determine if the torsion part of $E(L)$ is trivial, but there is little hope to finding a general criteria to impose that the rank does not grow. 3.) No if there is no connection between $L$ and $E$, they could be anything. $\endgroup$
    – Chris Wuthrich
    Commented Apr 12, 2022 at 15:01
  • $\begingroup$ 1) For small $G$, one can often calculate all the terms involved in this map explicitly even when the rank is positive. I doubt that the target is $\bigoplus$, I think it is $\prod$. The kernel, like the source is a finite group, often rather small, the target and the cokernel can be quite large. 4) I am quite sure it is not zero usually. But I am on holidays so I won't calculate some examples. Maybe later. $\endgroup$
    – Chris Wuthrich
    Commented Apr 12, 2022 at 17:18
  • $\begingroup$ @Chris Wuthrich $\endgroup$
    – A. Maarefparvar
    Commented Apr 16, 2022 at 8:28
  • $\begingroup$ @ChrisWuthrich Many thanks for the useful comments. Considering a quadratic extension $L/K$ (in the simplest case), and assuming $E(L)_{tors}=0$, can we determine explicitly the kernel and cokernel of $\mathcal{F}$? Based on your comment: "For small G, one can often calculate all the terms involved in this map explicitly ...", would you please hint me how can I do that for the quadratic extensions? $\endgroup$
    – A. Maarefparvar
    Commented Apr 16, 2022 at 8:30
  • $\begingroup$ $E(L)$ is a fin. gen. free $\mathbb{Z}$-module, so as a $\mathbb{Z}[G]$-module it will decompose as a direct sum of $\mathbb{Z}$ with trivial action, $\mathbb{Z}$ with action by $-1$, and $\mathbb{Z}[G]$. It is easy to calculate their cohomology. The local term are only interesting when $v$ isn't split. Now $E(L_v)$ is a fin. gen. $\mathbb{Z}_{\ell}$-module where $\ell$ is below $v$. If $\ell\neq 2$ or if it is torsion-free, the cohomology can be determined easily, otherwise it might be some extra work. $\endgroup$
    – Chris Wuthrich
    Commented Apr 16, 2022 at 9:01

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