Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^\*$ in $K$. Then for each place $v$ of $F$, we have the local Kummer map
$$ \kappa\_v: E(F\_v) \, \otimes \mathbf{Q}\_p/\mathbf{Z}\_p \hookrightarrow H^1(F\_v, E[\pi^{\infty}]). $$
*My question is*: what is the image of $\kappa\_v$ for various places $v$ of $F$? I know that if $v$ does not lie over $\pi$ or $\pi^\*$, then the image of $\kappa\_v$ is zero. But what about the cases $v = \pi$ or $v = \pi^\*$? Is the image of $\kappa\_v$ different in these two cases?



Image of Kummer map for CM Elliptic curves
There are three cases:


1. If $v$ is a finite place of $F$ not lying over $\pi$ or $\overline{\pi}$, then the image of the Kummer map $\kappa\_v$ is zero.
2. If $v$ is a finite place of $F$ lying over $\overline{\pi}$, it turns out that the image of the Kummer map $\kappa\_v$ is *still* zero.
3. The tricky case is when $v$ is a finite place of $F$ lying over $\pi$. Here, the image of the Kummer map $\kappa\_v$ *may be non-zero*. The precise description of the image is more complicated and it seems to involve formal groups in some way.


These results are in pages 7-9 of this paper by Murty and Ouyang:
[http://staff.ustc.edu.cn/~yiouyang/ouyang-murty.pdf](http://staff.ustc.edu.cn/%7Eyiouyang/ouyang-murty.pdf)


