local parameter z=-x/y of p-adic expansion

Question

My curve is given by ellinit([0,0,0,-3267,45630]) and point P=[-21,324]. Local parameter (z=-x/y) of point 24P = 2*3+3^2+2*3^3 +O(3^5). My concern is how am I to find the local parameter of point 24kP (with the term k appearing in the p-adic expansion). Is there a certain formula that i can use?

Answer 1

For convenience, let’s call $Q=[24](P)$. I’m assuming that your evaluation of the local coordinate of $Q$ is correct, let’s denote it $\xi\in\Bbb Z_3$. The best way I know of to calculate $[k](Q)$ is to use the logarithm and exponential of the formal group of your curve. There are other ways, of course. Because your curve has additive reduction and is defined over $\Bbb Q_3$, the logarithm $L(x)$ will be a $\Bbb Z_3$-series (no denominators!), which is pleasant, ’cause it means that the exponential series $E(x)$ also is in $\Bbb Z_3[[x]]$, and so has good convergence properties. Just take $\xi$, calculate $L(\xi)$, multiply by $k$, and substitute this value into $E(x)$. It works because the logarithm is a homomorphism into the additive formal group, where the $[k]$-map is just multiplication by $k$. I’m sure that pari has methods for getting the logarithm and exponential of the formal group of your curve.