Let $K$ be a quadratic imaginary field and let $\lambda$ a prime of norm $l^2$, for a rational prime $l$.
We consider $E$ to be an elliptic curve such that $E[p](K)$ is trivial, where $p\neq l$ is a rational prime.
Denote with L the number field $K(E[p])$ and assume that $\lambda$ splits completely in $L$. Suppose to have a $K$-point $P$ which is not divisible by $p$ in $E(K)$. I should check that the following are equivalent:
a) $\lambda$ splits completely in $L(\frac{1}{p}P)$,
b) $P$ is divisible by $p$ in $E(K_{\lambda})$, i.e. $\frac{1}{p}P \in E(K_{\lambda})$,
where $K_{\lambda}$ is the completion of $K$ at the finite prime $\lambda$.
The notation $\tfrac{1}{p}P$ is a little bit ambiguous.
I understand (b) as saying that there is a point $Q$ such that $pQ=P$ and $Q$ is defined over $K_{\lambda}=L_w$ for any $w \mid \lambda$. This is clearly equivalent to saying that $L_w(Q)=L_w$, i.e. that all $w$ split in $L(Q)$. Because all the $p$-torsion points are defined over $L$, all the points $Q'$ with $pQ'= P$ are defined over $L(Q)$, hence $L(Q) = L([p]^{-1}(P))$.