Lubin–Tate formal group (${\mathcal{F}}_f$)

A Lubin–Tate formal group associated with a Lubin–Tate series $f$ is the unique one-dimensional commutative formal group law ${\mathcal{F}}_f$ over the valuation ring ${\mathcal{O}}_K$ for which $f$ is an endomorphism with linear coefficient the chosen uniformizer $\pi$. The Lubin–Tate functional equation lemma constructs the law and the endomorphisms $[a{]}_f$ for every $a\in {\mathcal{O}}_K$, with linear term $aX$. Their addition is given by the law, and their multiplication by composition. Thus the valuation ring acts on the group; $[\pi {]}_f=f$.

For a uniformizer $\pi$ of a non-Archimedean local field $K$ with residue field of cardinality $q$, a Lubin–Tate series is a series $f(T)\in {\mathcal{O}}_K[[T]]$ satisfyingIt determines the associated two-variable Lubin–Tate formal group. A useful example is $f(T)=\pi T+T^q$. For $K={\mathbb{Q}}_p$ and $\pi =p$, the choice $(1+T{)}^p-1$ gives the formal multiplicative group and the Lubin–Tate cyclotomic example.

If $f$ and $g$ are Lubin–Tate series for the same uniformizer, there is a unique strict isomorphism $h:{\mathcal{F}}_f\to {\mathcal{F}}_g$ satisfying $h\circ f=g\circ h$ and $h(X)\equiv X(\mathrm{mod}{X}^2)$.

Given Lubin–Tate series $f,g$ for $\pi$ and a linear form $L=\sum a_i{X}_i$, there is a unique power series $F\equiv L$ modulo terms of degree two satisfyingComparison degree by degree determines each homogeneous term; the Lubin–Tate congruences provide the required divisibility by $\pi$.

The ${\pi }^n$-torsion of a Lubin–Tate formal group isIt is a free rank-one module over ${\mathcal{O}}_K/{\pi }^n{\mathcal{O}}_K$.

Fix a uniformizer $\pi$ of a local field $K$ and a Lubin–Tate formal group $\mathcal{F}$. Its nth torsion field is generated by all its ${\pi }^n$-torsion points. A primitive point ${\lambda }_n$ generates the Lubin–Tate torsion as a free rank-one ${\mathcal{O}}_K/{\pi }^n$-module; therefore $K_{\pi ,n}=K({\lambda }_n)$. If the residue field has size $q$, the Eisenstein layers of Lubin–Tate torsion give $[K_{\pi ,n}:K]=(q-1)q^{n-1}$ and total ramification.

For $K={\mathbb{Q}}_p$ and $\pi =p$, choose the Lubin–Tate series $f(T)=(1+T{)}^p-1$. The associated formal group law is the formal multiplicative group. Its torsion points are $\zeta -1$ for ${\zeta }^{p^n}=1$, so its Lubin–Tate extension is the cyclotomic extension of a p-adic field ${\mathbb{Q}}_p({\mu }_{p^n})$. The Galois action $\zeta \mapsto {\zeta }^a$ gives $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, including the trivial first layer when $p=2$.

Choose compatible primitive Lubin–Tate torsion points with $[\pi ]{\lambda }_{n+1}={\lambda }_n$. Restriction of the Lubin–Tate Galois action on primitive torsion is reduction of unit classes modulo ${\pi }^n$. The inverse limit givesas topological groups. This describes the Galois group without invoking local reciprocity.

For a primitive Lubin–Tate torsion point ${\lambda }_n$, every Galois automorphism has $\sigma ({\lambda }_n)=[a_{\sigma }]({\lambda }_n)$ for a unique unit class $a_{\sigma }(\mathrm{mod}{\pi }^n)$. All torsion points lie in $K({\lambda }_n)$ because the integral endomorphism series converge there; hence the torsion field is Galois. The displayed map is an injective homomorphism. Its domain has order $(q-1)q^{n-1}$ by the primitive Eisenstein polynomial, equal to the number of unit classes, so it is an isomorphism.

For $f(X)=\pi X+X^q$ and its iterates $f_n$, the quotientis Eisenstein. A nonzero root of exact level $n$ therefore generates a totally ramified separable extension.

The two Lubin–Tate series $(1+X{)}^p-1$ and $X^p+pX$ for ${\mathbb{Q}}_p$ have respective nonzero $p$-torsion points ${\zeta }_p-1$ and the roots of $X^{p-1}=-p$. A Lubin–Tate change of series identifies their torsion fields, so