Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 2 / b / ii / Solution

Let $E_r({\mathbb{Q}}_2)$ be the $r$th term in the filtration of elliptic-curve points over a local field. Reduction givesof order four, whilehas order two. The formal logarithm is injective on $E_2({\mathbb{Q}}_2)$ and identifies it with an additive subgroup of ${\mathbb{Q}}_2$, so $E_2({\mathbb{Q}}_2)$ is torsion-free. A finite subgroup of $E({\mathbb{Q}}_2)$ therefore injects into $E({\mathbb{Q}}_2)/E_2({\mathbb{Q}}_2)$, whose order is $4\cdot 2=8$. Thus $|E({\mathbb{Q}}_2{)}_{\mathrm{tors}}|$ divides $8$.