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

For a uniformizer $\pi$ of a local field $K$ with residue-field cardinality $q$, a Lubin–Tate series is a series $f(X)\in {\mathcal{O}}_K[[X]]$ satisfying $f(X)\equiv \pi X(\mathrm{mod}{X}^2)$ and $f(X)\equiv X^q(\mathrm{mod}\pi )$. It determines a unique one-dimensional commutative formal group law ${\mathcal{F}}_f$ on which every $a\in {\mathcal{O}}_K$ acts through an endomorphism $[a{]}_f$ with linear term $aX$.

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 $h(X)\equiv X(\mathrm{mod}{X}^2)$.

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

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