 Solution

A one-dimensional commutative formal group law over ${\mathcal{O}}_K$ is a power series $F(X,Y)\in {\mathcal{O}}_K[[X,Y]]$ satisfyingFor $r\ge 1$, the ideal ${\pi }^r{\mathcal{O}}_K$ becomes a group, denoted $F({\pi }^r{\mathcal{O}}_K)$, under $x{+}_{F}y=F(x,y)$; convergence follows because both inputs lie in the maximal ideal.

Over the characteristic-zero field $K$, there is a unique formal logarithmsatisfying ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$. Its coefficients have bounded denominator growth, so for sufficiently large $r$ both ${\log }_F$ and its inverse formal group exponential converge on ${\pi }^r{\mathcal{O}}_K$ and preserve that ideal. They givewhere the final isomorphism is multiplication by ${\pi }^{-r}$.