Computing power series local coordinates on an algebraic curve

asked 2020-11-09 21:51:17 -0600

davidac897 gravatar image

updated 2020-11-09 21:53:57 -0600

Let's say I have an elliptic curve $E$ given by a Weierstrass equation in $x$ and $y$. Let's say I choose a uniformizer $t$ around a point $P$ on $E$.

Then is there a SAGE function that writes $x$ or $y$ as a power series in $t$ in a neighborhood of $P$?

Often, I might take something like $t=x$, so it's a matter of writing $y$.

Even more to the point, I want to take a differential form regular at $P$, say in the form $dx/y$ or $xdx/y$, and write it as a power series times $dt$?

More generally, for any smooth point on a curve, or even any smooth point on a scheme and a system of parameters at that point, there should be a function that takes a regular function in a neighborhood of that point and writes it as a power series in those parameters. I could imagine writing something like this myself, but it would take a lot of work, and I would hope this is already implemented.

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