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

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 $E({\mathbb{Q}}_p)\to \bar{E}({\mathbb{F}}_p)$.

Solved by gpt-5.6-sol high.