Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 27 / 2

Work with the usual commutative unital ring convention. Write $f(T)=a_{1}T+a_2{T}^2+\cdots$, where $a_1=f^{\prime }(0)$ is a unit, and seek $g(T)={\sum }_{n\ge 1}{b}_n{T}^n$. Since $g$ has zero constant term, every coefficient of the composition of these formal power series involves only finitely many terms.

The linear coefficient forces $b_1=a_1^{-1}$. Suppose $b_1,\dots ,b_{n-1}$ have been chosen. The coefficient of $T^n$ in $f(g(T))$ isIndeed the term involving $b_n$ can only come from $a_{1}g$: in $g^j$ with $j\ge 2$, using a degree-$n$ term forces total degree at least $n+j-1>n$. Choose $b_n=-a_1^{-1}{C}_n$. This recursive construction of a compositional inverse uniquely gives $f\circ g=T$.

To check the other composition rather than assume it, construct $h(T)={\sum }_{n\ge 1}{c}_n{T}^n$ satisfying $h\circ f=T$. At degree $n$, its new coefficient is $c_n{a}_1^n$ plus terms already determined; $a_1^n$ is a unit, so the same recursion works. Composition is associative, as can be checked coefficientwise, or after truncation modulo $T^{N+1}$ for every $N$. ThereforeConsequently the compositional inverse of a formal power series is unique and satisfies

The Weak Mordell-Weil theorem asserts that $E(K)/mE(K)$ is finite for an elliptic curve over a number field $K$ and an integer $m\ge 2$. The role of (i) is to make division by $m$ possible in the local formal group of an elliptic curve whenever the residue characteristic does not divide $m$.

Let $v$ be a finite place with good reduction of an elliptic curve and $v\nmid m$. The kernel of reduction of an elliptic curve is parametrized by $t=-x/y$ in the maximal ideal of the local valuation ring. Multiplication by $m$ is represented by an integral formal power seriesIts linear coefficient is a local unit. Part (i) supplies an integral compositional inverse. Both series converge on the maximal ideal, so multiplication by $m$ is a bijection on the reduction kernel, over the local field and over every finite unramified extension.

This implies unramified division torsors at good primes. For $P\in E(K)$, first divide its reduction by $m$ over the algebraic closure of the residue field. Multiplication by $m$ on the reduced curve is surjective and separable, so the chosen division point is defined over some finite residue extension. Smooth lifting gives a point $Q_0$ over the corresponding unramified local extension with the required reduction. The difference $P-m{Q}_0$ lies in the reduction kernel; the inverse series supplies $Q_1$ with $m{Q}_1=P-m{Q}_0$. Thus $Q=Q_0+Q_1$ divides $P$ over an unramified extension. Applying the same reasoning to all reduced $m$-torsion points lifts all of $E[m]$, so every point of $[m{]}^{-1}P$ lies in the maximal unramified extension. This excludes ramification outside a fixed finite set $S$ consisting of primes of bad reduction and primes over $m$.

For completeness, the remaining global finiteness step in the Kummer-theoretic proof of the weak Mordell-Weil theorem is as follows. Choose $Q\in E(\overline{K})$ with $mQ=P$. The cocycle $\sigma \mapsto \sigma Q-Q$ defines the Kummer map of an elliptic curveChanging $Q$ by an $m$-torsion point changes the cocycle by a coboundary. A zero class makes $Q$ rational after that change, so the map is injective.

Let $L=K(E[m])$, fixed independently of $P$. Since the torsion is rational over $L$, the cocycle restricted to its Galois group is a homomorphism into $E[m]$. Consequently $L(Q)/L$ is Galois of degree at most $m^2$, contains the entire division fibre, and is unramified outside $S$. Hence $[L(Q):K]\le m^2[L:K]$. There are only finitely many extensions of bounded degree unramified outside a fixed finite set: local extensions of bounded degree have bounded field discriminant exponents, and the Hermite–Minkowski theorem then applies to the bounded global field discriminants. Their finite Galois compositum $M$ contains every such division fibre; each $L(Q)$ is stable over $K$ because it contains $E[m]$ and the whole fibre above the rational point $P$. All the Kummer classes consequently factor through the fixed finite group $\mathrm{Gal}(M/K)$ with values in the fixed finite group $E[m]$. There are only finitely many such cocycles, provingThus (i) provides the local division step that bounds ramification; the global arithmetic finiteness completes the argument.