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

HereBy the Lutz–Nagell theorem, a nonzero rational torsion point has integral coordinates and either $y=0$ or $y^2\mid 3^6$. Part b gives $v_3(y)\le 1$, so $y\in \{0,\pm 1,\pm 3\}$. The cubic $x^3-9x+9$ has no integral zero. Substitution of $y=\pm 1,\pm 3$ gives exactlyThis proves the required inclusion.

Since $2Q=-Q$, the point $Q$ has order three. The point $P$ is not torsion because $2P=(9/4,3/8)$ is nonintegral, contradicting Nagell–Lutz. If $P+Q$ or $P-Q$ were torsion, adding the torsion point $\mp Q$ would make $P$ torsion; their negatives are excluded in the same way. Hence