Is it possible to compute coefficients of the formal group of an elliptic curve?

Question

This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the associated formal group law, assuming this is enough information to coordinatize it. I would assume that the coefficients are rational or at least analytic function of our parametrization, though I am not making any headway. Can anyone enlighten me on this issue?

Edit: When considering the pullback of $Spec(L)\to \mathcal{M}_{FG}$ along $\mathcal{M}_{Ell}\to \mathcal{M}_{FG}$, we get $Spec(A[b_1, b_2, ...])$ so that the zero morphism should give us a map $Spec(A)\to Spec(L)$, which should be what we want. Computation of this map though, is not clear to me.

Answer 1

I assume you mean Weierstrass parametrization. Then $-x/y$ is the standard coordinate near 0, and the relevant formulas can be found in Silverman's book on elliptic curves.

Answer 2

Suppose that $C$ is given by a homogeneous Weierstrass equation $f(x,y,z)=0$. Then there is a unique series $\xi(x)=\sum_{k>0}\xi_kx^k$ such that $\xi(x)=x^3+O(x^4)$ and $f(x,1,\xi(x))=0$. This can be found quite efficiently by successive approximation. Now put $$\chi(x_0,x_1,x_2)=\sum_{i,j,k\geq 0}\xi_{i+j+k+2}x_0^ix_1^jx_2^k.$$ This is $x_0+x_1+x_2\pmod{(x_0,x_1,x_2)^2}$, and one can check that $$ \det\left|\begin{array}{ccc}x_0&x_1&x_2\\1&1&1\\\xi(x_0)&\xi(x_1)&\xi(x_2)\end{array}\right|=(x_0-x_1)(x_0-x_2)(x_1-x_2)\chi(x_0,x_1,x_2).$$ Recall that addition in the elliptic curve is defined in terms of triples of collinear points, and collinearity forces the relevant determinant to vanish. From this we see that the series $G(x_0,x_1)=[-1]_F(x_0+_Fx_1)$ is characterised by $\chi(x_0,x_1,G(x_0,x_1))=0$. This can again be solved quite efficiently by successive approximation. Finally, we have $[-1]_F(x)=-x/(1+a_1x+a_3\xi(x))$, and $x_0+_Fx_1=[-1]_F(G(x_0,x_1))$.

There are a few more details in Appendix B of the paper "Elliptic spectra, the Witten genus and the Theorem of the Cube" by Ando, Hopkins and myself.