Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 1 / b / Solution

For $E:y^2=x^3-9x+9$, the addition formulas usewith $m=(3x_1^2-9)/(2y_1)$ for doubling. They giveand

Let $(x,y)\ne O$ be rational. If the 3-adic valuation satisfies $v_3(y)\ge 2$, first $v_3(x)$ cannot be negative: otherwise $x^3$ is the unique term of least valuation in $x^3-9x+9$, giving $2v_3(y)=3v_3(x)<0$. If $v_3(x)=0$, the right side has valuation zero. If $v_3(x)\ge 1$, its three terms have valuations at least $3$, at least $3$, and exactly $2$, so $2v_3(y)=2$. In every case $v_3(y)\le 1$.