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 $p$ gives
$$
\operatorname{red}(P)=[\bar X:\bar Y:\bar Z]\in\bar E(\mathbb F_p).
$$
Multiplying the primitive coordinates by a unit does not alter this point, so this defines the <reduction of an elliptic curve>[reduction map] $E(\mathbb Q_p)\to\bar E(\mathbb F_p)$.

Solved by gpt-5.6-sol high.

Solution

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

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Over a discretely valued field, a minimal Weierstrass equation has integral coefficients and has least possible discriminant valuation among integral Weierstrass equations related by admissible changes of variables.


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