 5

We first prove all three general torsion assertions before treating the lettered cases. If a finite point has $|x{|}_p>1$, the cubic term dominates the integral lower terms, so $v_p(x)=-2m$, $v_p(y)=-3m$ for an integer $m\ge 1$. Its parameter $z=-x/y$ then lies in $p{\mathbb{Z}}_p$, with norm at most $p^{-1}$. The integral Weierstrass model defines its formal group at infinity over ${\mathbb{Z}}_p$: using $w=-1/y$ gives $w=z^3+Az{w}^2+B{w}^3$, an integral power series starting in degree three.

The supplied torsion estimate excludes a nonzero torsion parameter for $p\ge 3$, since $p^{-1}<p^{-1/(p-1)}$, and the lower bound increases for higher $p$-powers. For $p=2$ it excludes orders $2^n$ with $n\ge 2$, because their bound is strictly above $1/2$. The remaining possibility is order two, which would imply the finite point equals its negative and hence $y=0$. A nonintegral $x$ cannot solve the monic cubic with integral coefficients, so this is impossible too. Thus every finite torsion point has integral $x$, and the equation then forces integral $y$.

For good reduction the reduction map is a group homomorphism whose kernel is this formal group at infinity. Its torsion-free kernel proves injectivity on torsion, giving a subgroup of the finite reduced group. Applying local integrality to a rational torsion point at every prime gives $x,y\in \mathbb{Z}$.

If $y\ne 0$, the double of a rational torsion point is finite and integral. The doubling formulatherefore implies $y^2\mid (3x^2+a{)}^2$. Also $y^2=x^3+ax+b$. Evaluating the supplied polynomial identity at the integral $x$ now givesThis establishes the needed Nagell–Lutz theorem conclusions without assuming them in the lettered applications.