= Initial coefficients of the elliptic formal group at infinity
{title2=$F(X,Y)=X+Y-a_1XY-a_2(X^2Y+XY^2)+O((X,Y)^4)$}
For a <Weierstrass equation of an elliptic curve>, use the <local parameter> $t=-x/y$ and put $w=-1/y$. The <formal kernel of a minimal Weierstrass equation> construction gives $w=t^3+a_1t^4+(a_1^2+a_2)t^5+(a_1^3+2a_1a_2+a_3)t^6+O(t^7)$. Substitution into the <elliptic curve group law> yields the displayed initial coefficients and the <formal inverse> $-t/(1-a_1t-a_3w(t))$. In <characteristic zero>, the <invariant differential on an elliptic curve> becomes $(1+a_1t+(a_1^2+a_2)t^2+O(t^3))\,dt$, giving the <formal logarithm> $t+a_1t^2/2+(a_1^2+a_2)t^3/3+O(t^4)$. The addition coefficients are integral polynomial expressions and hence also apply in positive <characteristic of a field>.
Back to article page
