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Let $K$ be a local field and $E/K$ an elliptic curve such that the set of $N$-torsion points, $E[N]$, is contained in $E(K)$. For $e$ in $E[N]$, I am interested in finding and expression for the function $f_e$ in K(E) such that $div(f_e)=N(e)-N(0)$. More concretely, I am interested in the values of $f_e$ when restricted to the formal group of $E$. I guess using the Weierstrass preparation theorem could gives us an idea of how $f_e$ can be factored, but what about the coefficients of the series that appear on the product?

I know there is an algorithm, Miller's algorithm, that allows one to compute f_e, but the expression is quite complicate. There is also a $p$-adic analogue of the sigma function, due to Tate, for the case in which E has ordinary reduction, which allows one to find an expression for f_e as a product of these functions. What about the case of super-singular reduction? the general case?

Any insights or references would be greatly appreciated.

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    $\begingroup$ If by "concrete" you mean the following: $p$ is an explicit given prime number $K$ is an explicit finite extension of $\mathbf{Q}_p$, $N$ is an explicit positive integer, and $E$ is an explicit elliptic curve in Weierstrass form whose coefficients I know explicitly, and I have an explicit point $e$ with explicit $x$ and $y$ coordinates -- if all this, then your proof that $e$ has order $N$ (which will perhaps be of the form "check $e+e=e_2$, $e_2+e=e_3$,...) can be translated into an explicit construction of $f$ [but the comment box is too small to explain the construction so hang on a second] $\endgroup$
    – user30035
    Commented Feb 17, 2013 at 21:11
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    $\begingroup$ Namely, any assertion of the form $P+Q=R$ is checked by drawing the line $L$ through $P$ and $Q$, which hits $E$ again at $S$, and then the line $M$ through $S$ and the origin hits the curve at $R$, and then $L/M$ has zeros at $P,Q$ and poles at $R$,origin. Now continue; at each stage you get a ratio of linear functions and you can multiply all of them together (the last one being the witness to $e_{N-1}+e=$origin) to get the function you want. $\endgroup$
    – user30035
    Commented Feb 17, 2013 at 21:13

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To expand on wccanard's comment : see my answer to this question for a general method to compute a rational function with given (principal) divisor.

This will give you $f_e$ as a rational function of $x$ and $y$. You then just need to expand this in terms of the standard formal coordinate $z=-x/y$.

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