Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R\_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.


Take $a \in{R\_K}$ and $[a] \in \operatorname{End}E$ . Then there is unique corresponding homomorphism of formal group, $[a](t) \in \operatorname{End}\hat{E}$.


My question is, how can I prove $[a](t)=at+\text{(term higher than degree $2$)}$ ?



Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?
Let $\omega\_E$ be an invaraint differential on $E$. Then $[a]$ satisfies $[a]^\*\omega\_E=a\omega\_E$. That's over $K$, so the same formula holds on the formal group, i.e., $\widehat{[a]}\omega\_{\hat E}=a\omega\_{\hat E}$. On the other hand, if you write $\widehat{[a]}(T)=cT+\text{h.o.t.}$, then $\widehat{[a]}\omega\_{\hat E}=c\omega\_{\hat E}$. Hence $c=a$.


