Formal coordinates on a short Weierstrass curve
= Formal coordinates on a short Weierstrass curve
For $y^2=x^3+ax+b$, put $t=-x/y$ and $w=-1/y$. Then $x=t/w$, $y=-1/w$, and the curve equation becomes
$$
w=t^3+atw^2+bw^3.
$$
There is a unique solution $w(t)\in t^3K[[t]]$, and points in the kernel of reduction correspond to parameters of positive valuation.