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

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.

Define the Lubin–Tate torsion byThe scalar endomorphisms make this a module over ${\mathcal{O}}_K/{\pi }^n{\mathcal{O}}_K$.

The Newton polygon or Weierstrass preparation theorem applied to the Lubin–Tate congruences shows that $[{\pi }^n{]}_f$ has exactly $q^n$ distinct roots in $\overline{\mathfrak{m}}$. More precisely, the quotient of the distinguished factors for $[{\pi }^n{]}_f$ and $[{\pi }^{n-1}{]}_f$ has degree $q^{n-1}(q-1)$, and its roots are precisely the points killed by ${\pi }^n$ but not by ${\pi }^{n-1}$.

Choose such a point ${\omega }_n$. If $a\in {\mathcal{O}}_K$, write $a={\pi }^{r}u$ with $u$ a unit. Since $[u{]}_f$ is an automorphism, $[a{]}_f({\omega }_n)=0$ exactly when $r\ge n$. Thusis injective. Both sides have $q^n$ elements, so it is an isomorphism of modules. Therefore ${\mu }_{f,n}$ is a free module of rank one.

Over $K={\mathbb{Q}}_p$, consider the two Lubin–Tate seriesBoth have linear term $pX$ and reduce to $X^p$ modulo $p$. The nonzero roots of $f$ are ${\zeta }_p^i-1$, while the nonzero roots of $g$ satisfy

The Lubin–Tate change of series $h:{\mathcal{F}}_f\to {\mathcal{F}}_g$ and its inverse have coefficients in ${\mathbb{Z}}_p$ and converge on the maximal ideal. They therefore give mutually inverse bijections between the first torsion sets without changing the fields generated by them. Hence the first Lubin–Tate torsion fields for the p-adic numbers satisfySince ${\mathbb{Q}}_p({\zeta }_p-1)={\mathbb{Q}}_p({\zeta }_p)$, the required equality follows.