In a paper on elliptic curves I see the following notation used in section $4$: let $E/M$ be an elliptic curve over a field $M$ with CM by $\mathcal{O}_K$. Let $E[\lambda]$ by the kernel of the endomorphism given by $\lambda\in \mathcal{O}_K$.
What does $M(E[\lambda])$ stand for?
I assumed it was something simple but became confused when results like the following were stated in the paper (and made to seem immediate): $k(E[\lambda])=k\Leftrightarrow E[\lambda]\subset E(k)$. It doesn't seem like this should be immediate given the naive definition.
The notation was also used in the context of a single point $P$ on an elliptic curve $E$, simply $M(P)$. Is this simply the $M$-algebra generated by $E[\lambda]$ in some respect, and if so how does $k(E[\lambda])=k\Leftrightarrow E[\lambda]\subset E(k)$ hold?
$([\lambda])$ is the smallest extension of $$ over which all points of $[\lambda]$ are defined. So if $([\lambda])=$, then every point of $[\lambda]$ is defined over $$, i.e., $[\lambda]⊂()$, and vice versa.