Formal group of an elliptic curve

The local parameter $z=-x/y$ at the identity of a Weierstrass curve expresses addition as a formal group law. Over a local field it describes points reducing to the identity.

For an elliptic curve over a nonarchimedean local field, $E_0$ consists of points with nonsingular reduction, $E_1$ is the kernel of reduction, and $E_r$ for $r\ge 1$ consists of points whose formal parameter has valuation at least $r$. The successive quotients for $r\ge 1$ are copies of the additive residue-field group.