Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 3

A one-dimensional commutative formal group law over ${\mathcal{O}}_K$ is a power series $F(X,Y)\in {\mathcal{O}}_K[[X,Y]]$ satisfyingA morphism $h:\mathcal{F}\to \mathcal{G}$ is a series $h(T)\in T{\mathcal{O}}_K[[T]]$ satisfyingIf $h(T)=uT+O(T^2)$, the invertible morphism criterion for formal group laws says that $h$ is an isomorphism whenever $u\in {\mathcal{O}}_K^{\times }$. Indeed, recursive coefficient comparison constructs a unique compositional inverse $j(T)$; applying $j$ to the morphism identity shows that $j$ is a morphism in the opposite direction.

The multiplication series hasSince $p\nmid n$, its linear coefficient is a unit, so $[n{]}_{\mathcal{F}}$ is an automorphism of the group $\mathcal{F}(\pi {\mathcal{O}}_K)$. Its kernel is therefore zero, and

For a minimal integral Weierstrass equation, let $\tilde{E}$ be the reduced cubic and ${\tilde{E}}_{\mathrm{ns}}(k)$ its nonsingular points, with their induced group law. Define the filtration of elliptic-curve points over a local field byandThe parameter $z=-x/y$ identifies $E_1(K)$ with the formal group of an elliptic curve on $\pi {\mathcal{O}}_K$. Part (a) therefore gives $E_1(K)[n]=0$. Reduction restricts to the exact sequenceIts restriction to $n$-torsion has trivial kernel, yielding the injection

An integral Weierstrass equation has good reduction outside the finitely many primes dividing its nonzero discriminant. This proves finiteness of the set of bad primes. To prove finiteness of rational torsion, choose two distinct good primes. The reduction of torsion points on an elliptic curve injects each primary component at a good prime of different residue characteristic, so the two finite reduced point groups bound every primary component of $E(\mathbb{Q}{)}_{\mathrm{tors}}$.

Forthe displayed equation is minimal andIts bad primes are therefore exactlyThe good reductions at $11$ and $13$ haveTheir coprime orders exclude every rational torsion primary component, including the residue-characteristic components by using the other prime. Hence