Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 3 / e

A one-dimensional commutative formal group law over a ring $R$ is a power series $F(X,Y)\in R[[X,Y]]$ satisfying $F(X,0)=X$, commutativity, and associativity. A morphism $f:F\to G$ is a series $f(T)\in TR[[T]]$ satisfying

Express the isogeny of part b in the formal coordinates of part d and setPart c shows that points approaching $O$ map to points approaching $O^{\prime }$, so $f(t)\in tk[[t]]$. Since $\varphi$ is a group homomorphism, applying the parameter $t^{\prime }$ to $\varphi (P+Q)=\varphi (P)+\varphi (Q)$ givesThus $f$ is the formal-group morphism induced by an isogeny $\hat{E}\to \widehat{E^{\prime }}$.