Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 1

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.

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A Minimal Weierstrass equation for $E/{\mathbb{Q}}_p$ is a Weierstrass equation of an elliptic curve with coefficients in ${\mathbb{Z}}_p$ whose discriminant has minimum $p$-adic valuation among all integral equations for $E$ related by admissible changes of variables.

Let $P\in E({\mathbb{Q}}_p)$ and choose projective coordinates $[X:Y:Z]$ with $X,Y,Z\in {\mathbb{Z}}_p$ and at least one coordinate a unit. Reducing the coordinates modulo $p$ givesMultiplying the primitive coordinates by a unit does not alter this point, so this defines the reduction map $E({\mathbb{Q}}_p)\to \bar{E}({\mathbb{F}}_p)$.

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For the filtration of elliptic-curve points over a local field, defineandFor $r\ge 1$, use the formal group of an elliptic curve with parameter $z=-x/y$ and putReduction induceswhile the coefficient of $p^r$ in the formal parameter gives

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For a locally compact group $G$ on which multiplication by $n$ has finite kernel and cokernel, compare a Haar measure with its pushforward under $[n]$. On the one-dimensional $p$-adic Lie group $E({\mathbb{Q}}_p)$, the derivative of $[n]$ at the identity is $n$, henceThis can also be read directly from the successive quotients in the filtration of elliptic-curve points over a local field; factors prime to $p$ act invertibly on a sufficiently small formal-group neighbourhood.

The real Lie group $E(\mathbb{R})$ has one or two circle components. On its identity component, $[n]$ has degree $n$; the component-group kernel and cokernel have the same order. ThereforeOnly primes dividing $n$ contribute to the finite-place product, and unique factorization gives

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