Let $E$ be an elliptic curve defined over a number field $k$. Let $D$ be a reduced separable divisor on $E$, i.e., we can write $D = p_1+...+p_n$ for $p_i \in E(k)$. Let $C$ be the open complement of $D$ in $E$. It is an affine plane cubic curve.
It is known that for a positive integer $n$, we have a surjective homomorphism $[n] :E(\bar{k}) \rightarrow E(\bar{k})$ and thus an exact sequence $$0 \rightarrow E[n] \rightarrow E \xrightarrow{[n]} E \rightarrow 0.$$
Does this exact sequence hold also for $C$? I know that the Kummer sequence exists for some affine varieties, for example, the torus. I'm just wondering if the cubic law on $C$ would mean that we also have it in this case.
Multiplication by $n$ is never regular on $E-P_1,\ldots,P_r$
(unless $n=0$ or $n$ has trivial kernel, ie. $n=\pm 1$ or $char(k)=p$, $n=p^l$ and $E$ supersingular)
because $\{ Q\in E, \exists m, [n^m]Q \in P_1,\ldots,P_r\}$ is an infinite set so you'd need to remove infinitely many points to make $[n]$ regular on $E-P_1,\ldots,P_r$.