Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 136 / 3 / a

A Lubin–Tate series for $\pi$ is a power series $f(X)\in {\mathcal{O}}_K[[X]]$ such thatThe key Lubin–Tate lemma is that if $f$ and $g$ are such series and $L(X_1,\dots ,X_r)={\sum }_i{a}_i{X}_i$ with $a_i\in {\mathcal{O}}_K$, there is a unique series $F\in {\mathcal{O}}_K[[X_1,\dots ,X_r]]$ satisfying

Taking $g=f$ and $L=X+Y$ defines ${\mathcal{F}}_f(X,Y)$. Uniqueness applied to the two sides of each identity proves the identity, associativity, and commutativity axioms, so ${\mathcal{F}}_f$ is a formal group law. Taking $L=aX$ defines endomorphisms $[a{]}_f$, and uniqueness givesThus ${\mathcal{F}}_f$ is the Lubin–Tate formal group as a formal ${\mathcal{O}}_K$-module, with $[\pi {]}_f=f$.

For another Lubin–Tate series $g$, apply the lemma with $L=X$ to obtainUniqueness shows that $h$ respects both formal addition and every scalar endomorphism. Its linear coefficient is the unit one, so it has a compositional inverse; equivalently, applying the lemma with $f$ and $g$ reversed supplies the inverse. Hence this Lubin–Tate change of series is an isomorphism ${\mathcal{F}}_f\cong {\mathcal{F}}_g$ of formal ${\mathcal{O}}_K$-modules.