Lazard module structure of rings with formal elliptic curve

Question

Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a module structure of the Lazard-Ring $\mathrm{MU}_*$ (or if simpler $\mathrm{BP}_*$ since I am ultimately interested in prime 2).

My problem now is that I only have an explicit description of the formal coordinate $-x/y=T+...$, and not of the elliptic curve or the formal group law (similarly to the scenario in Rezk notes for math 512). Is there any way I can now get the module structure? I know that the action of the $v_i$'s is given by the coefficients of the $[2]$-series, but I do not know a way to get this out of the coordinate.

I would also be happy if someone could give me a way to describe the formal group law or the Weierstraß equation.

Answer 1

As far as I understand the question (I know elliptic curves, but I don't know what MU and BP are), the task is to express the coefficients of the Weierstrass equation given the series of multiplication by $2$ in its formal group. This can be done by the following: Suppose $[2](t) = 2\,t + c_2\, t^2 + c_3\, t^3 + \cdots$ and $E$ is given by a Weierstrass equation $y^2 + a_1\, xy + a_3\, y = x^3 +a_2\, x^2 + a_4\, x+ a_6$. Then

$\begin{align*} a_1 &= -c_2 \\ a_2 &= -c_3/2 \\ a_3 &= (a_1a_2-c_4)/7 \\ a_4 &= (2a_2^2-6a_1a_3-c_5)/12 \\ a_6 &= (-8a_1^3a_3 - 2a_2^3 - 14a_1a_2a_3 - 8a_1^2a_4 + 4a_3^2 + 4a_2a_4 - c_7)/54\end{align*}$

In particular, it is in short Weierstrass form if and only if $c_2=c_3=c_4=0$, in which case $A=-c_5/12$ and $B=-c_7/54$.

This follows from computing with sage as indicated above in the comment the series $[2](t)$ to get $ 2t - a_1 t^2 - 2a_2t^3 + (a_1a_2 - 7a_3)t^4 + (2a_2^2 - 6a_1a_3 - 12a_4)t^5 + (-a_1a_2^2 - 7a_1^2a_3 - 2a_2a_3 - 6a_1a_4) t^6 + (-8a_1^3a_3 - 2a_2^3 - 14a_1a_2a_3 - 8a_1^2a_4 + 4a_3^2 + 4a_2a_4 - 54a_6)t^7 + O(t^8) $