Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 3 / a / Solution

The curve has good reduction when it admits a Minimal Weierstrass equation over ${\mathcal{O}}_K$ whose discriminant is a unit, so reduction modulo $\pi$ is a nonsingular elliptic curve $\tilde{E}/k$. Write a point of $E(K)$ in primitive projective coordinates $(X:Y:Z)$ over ${\mathcal{O}}_K$. Reducing all three coordinates definesPrimitivity ensures that the reduced triple is not zero. The addition morphism on the smooth Weierstrass model reduces to the addition morphism on $\tilde{E}$; equivalently, this follows from the valuative criterion for properness for the smooth proper group scheme. Therefore $\mathrm{red}(P+Q)=\mathrm{red}(P)+\mathrm{red}(Q)$.