I have read in a solution to a problem that I was studying that $t = \frac{x}{y}$ is a uniformizer at the point $(0,1,0)$ of the curve given by $y^2z=x^3+axz^2+bz^3$, and that one can expand $x=t^{-2}+...$ and $y=t^{-3}+...$. I am not sure how one gets this and was hoping someone could clarify?
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The uniformizer is actually $x/y$, not $y/x$, and its derivation is given in my answer here: https://math.stackexchange.com/questions/3335091/ – djao Dec 06 '21 at 00:24
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@djao thanks, that was very helpful. Do you have any idea of a recipe to write down an expansion of a function on the elliptic curve in terms of the uniformizer around $(0,1,0)$. – baltazar Dec 06 '21 at 19:04
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Equations (2) and (4) of https://aghitza.github.io/pdf/other/velu.pdf give you $x$ and $y$ in terms of the uniformizer. For any other rational function, do power series arithmetic on these series. For example if you want $x^2+y$ then take the power series of $x$, square it, and add the power series of $y$. – djao Dec 07 '21 at 21:14