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@Lubin I am specifically interested in the case $\varpi_O=x/y$ and $\varpi_P=x-x(P)$. When $P=O$, I think that $\frac{[n]^*\varpi_O}{\varpi_P}$ should be of the form $n+\text{higher terms in }\varpi_P$.
@R.vanDobbendeBruyn Thank you very much. I am wondering whether this is known or not: if we take $\varpi_O=x/y$ and $\varpi_P=x-x(P)$ where $x,y$ are variables defining $E$ and $x(P)$ is the $x$-coordinates of $P\in E[n]$. Then can we explicitly write down $\frac{[n]^*\varpi_O}{\varpi_P}$?
