Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 27 / 3 / i / Solution

In the one-dimensional commutative convention, a formal group law over $R$ is a formal power series $F(X,Y)=X+Y+\text{terms of degree at least two}$ satisfying $F(X,0)=X$, $F(0,Y)=Y$, $F(X,Y)=F(Y,X)$ and $F(F(X,Y),Z)=F(X,F(Y,Z))$. The formal inverse is a series $I(T)=-T+O(T^2)$ with $F(T,I(T))=0$. For $R={\mathbb{Z}}_p$ all these series converge on $p{\mathbb{Z}}_p$, which becomes a group with operation $F$.

Repeated formal addition gives the multiplication isomorphism of a formal group law $[n{]}_F(T)=nT+O(T^2)$. If $p\nmid n$, the coefficient $n$ is a unit in the p-adic integers. We can construct an inverse $H(T)=n^{-1}T+{\sum }_{j\ge 2}{h}_j{T}^j$ by coefficient recursion. At degree $j$, the equation $[n{]}_F(H(T))=T$ has the form $n{h}_j+\text{known integral terms}=0$; dividing by the unit $n$ keeps $h_j$ integral. The inverse is two-sided by uniqueness of formal composition inverses. Since integral coefficients multiplied by successive powers of $t\in p{\mathbb{Z}}_p$ tend to zero, both series converge on that set and their formal identities hold there. This proves prime-to-residue-characteristic multiplication on a formal group:The argument also applies at $p=2$ for odd $n$; it does not require a logarithm or torsion-freeness.