Nagell–Lutz theorem

Remarks

The integrality comes from a study of the denominators of points in the $p$-adic filtration: points whose coordinates have $p$ in the denominator form a torsion-free subgroup. Divisibility follows next: if $P$ has finite order, so does $2P$, and both have integer coordinates; the duplication formula then forces $y^2 \mid D$. The theorem gives an algorithm for the torsion subgroup: list the finitely many candidates and test each for finite order. Example: on $y^2 = x^3 + 1$ one has $D = 27$, so a torsion point with $y \ne 0$ has $y \in \set{\pm 1, \pm 3}$. This gives the points $(0, \pm 1)$ and $(2, \pm 3)$. With $(-1, 0)$ and $O$ they form a cyclic group of order $6$. The converse fails: on $y^2 = x^3 + 2$ the point $(-1, 1)$ has integer coordinates and $1 \mid 108$, but it has infinite order. The hypothesis that $a$ and $b$ are integers is needed; any curve can be scaled to this form by $(x, y) \mapsto (u^2 x, u^3 y)$.

Going deeper

For a student

The theorem was found independently by Nagell and Lutz in the 1930s. It is the standard tool, together with reduction modulo good primes, to compute the torsion subgroup $T$ of $E(\Q)$ in Mordell's theorem. Mazur's torsion theorem then lists all groups that can occur.