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 Incoming links: Formal coordinates on a short Weierstrass curve

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 3 / d / Solution Created 2026-09-28 Updated 2026-10-03

Use the alternative affine coordinates

t = - \frac{x}{y}, w = - \frac{1}{y},

so that

x = t / w

and

y = - 1/ w

. The equation becomes

w = t^{3} + a t w^{2} + b w^{3} .

Recursive coefficient comparison gives a unique series

w (t) \in t^{3} k [[t]]

. Since

v (w (t)) = 3 v (t)

and hence

v (x) = - 2 v (t)

,

v (y) = - 3 v (t)

, this gives the formal coordinates on a short Weierstrass curve identification

E_{1} (K) = {(t, w (t)) : v (t) \geq 1} .

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