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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.

A one-dimensional commutative formal group law over a ring $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfyingAn isomorphism from $F$ to $G$ is a series $h(T)=uT+O(T^2)$ with $u\in R^{\ast }$ and

Over a characteristic-zero field $K$, every such formal group is isomorphic to the additive formal group. Differentiate the associativity identity and define the invariant differentialTermwise integration is possible in characteristic zero; the formal logarithmhas leading term $T$. Invariance of ${\omega }_F$ givesand evaluation at $(0,0)$ removes the integration constant. Hence ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$. Its unit linear coefficient gives a compositional inverse, so it is an isomorphism to $X+Y$. Therefore any two one-dimensional commutative formal groups over $K$ are isomorphic.

A one-dimensional commutative formal group law over $R$ is a power series $F(X,Y)\in R[[X,Y]]$ satisfyingA homomorphism $\theta :\mathcal{F}\to \mathcal{G}$ is a series $\theta (T)\in TR[[T]]$ such that

Write $\theta (T)=uT+O(T^2)$. If $u={\theta }^{\prime }(0)$ is a unit, recursive comparison of coefficients constructs a unique compositional inverse $\psi (T)\in TR[[T]]$ with $\psi (\theta (T))=T=\theta (\psi (T))$. Apply $\psi$ to the homomorphism identity and substitute $X=\psi (U)$, $Y=\psi (V)$ to obtainThus $\psi$ is a homomorphism from $\mathcal{G}$ to $\mathcal{F}$, so $\theta$ is an isomorphism.

Let $R$ be a complete discrete valuation ring of residue characteristic of a field $p$, with maximal ideal $\mathfrak{m}$, and let $F$ be a one-dimensional formal group law over $R$. Its multiplication series is $[n{]}_F(T)=nT+O(T^2)$. If $p\nmid n$, the invertible morphism criterion for formal group laws constructs its inverse with coefficients in $R$. Both formal power series converge on $\mathfrak{m}$, since their coefficients are integral and the powers of an element of $\mathfrak{m}$ tend to zero. Their compositions are the identity there. Thus multiplication by $n$ is bijective, including when $p=2$ and $n$ is odd. Every finite-order element of $F(\mathfrak{m})$ consequently has order of a group element equal to a prime power with prime $p$.

For a one-dimensional commutative integral formal group law, the formal logarithm identifies $F(4{\mathbb{Z}}_2)$ with $4{\mathbb{Z}}_2$. This is an index-two subgroup of $F(2{\mathbb{Z}}_2)$. Choose a topological generator $h$ of the subgroup and an element $v$ outside it, and write $2v=ch$. If $c$ is odd, $v$ generates the whole group. If $c$ is even, $v-(c/2)h$ has order two and splits off a cyclic factor. The formal additive group and formal multiplicative group realize the two cases.

For odd $p$, a one-dimensional formal group law over ${\mathbb{Z}}_p$ has a formal logarithm ${\log }_F(T)=T+{\sum }_{j\ge 2}{c}_j{T}^j/j$ with $c_j\in {\mathbb{Z}}_p$. This follows by integrating its integral invariant differential. For $t\in p{\mathbb{Z}}_p\setminus \{0\}$, $v_p(c_j{t}^j/j)>v_p(t)$ for $j\ge 2$. Hence the series converges and $v_p({\log }_F(t))=v_p(t)$, so its kernel is zero. As it is a homomorphism into a characteristic-zero additive group, there is no nonzero torsion. In particular, at an odd prime of good reduction over $\mathbb{Q}$, reduction injects the whole rational torsion subgroup, including its $p$-primary part.

For an integral one-dimensional formal group law over a finite extension of the p-adic field, the inverse of ${\partial }_{Y}F(T,0)$ has integral coefficients because its constant term is one. Integrating this invariant differential gives ${\log }_F(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$ with $n{c}_n$ integral. For $v_K(\pi )=1$ and $e=v_K(p)$ the displayed bound follows. The estimate $v_p(n)\le (n-1)/(p-1)$ makes the nonlinear part strictly contracting on ${\pi }^s{\mathcal{O}}_K$ for $s>e/(p-1)$, yielding the deep logarithm subgroup of a formal group.