Mordell's theorem

Remarks

By the structure theorem for finitely generated abelian groups, $E(\Q) \cong \Z^r \oplus T$, where $T$ is the finite torsion subgroup and $r \ge 0$ is the rank of $E$. The proof has two steps. The weak Mordell theorem says that $E(\Q)/2E(\Q)$ is finite; it is proved with a descent map into a finite group built from square classes of rationals. Then a height function $h$ on $E(\Q)$, with finitely many points of bounded height and $h(2P) \approx 4h(P)$, lets one show by descent that finitely many points together with $2E(\Q)$ generate everything. Example: for $y^2 = x^3 - x$ one has $E(\Q) = \set{O, (0,0), (1,0), (-1,0)}$, of rank $0$. Weil extended the theorem to abelian varieties over number fields; the general result is the Mordell–Weil theorem.

Going deeper

For a student

The theorem gives the two basic invariants of an elliptic curve over $\Q$. The torsion subgroup is described by Mazur's torsion theorem, and the rank is the subject of the Birch and Swinnerton-Dyer conjecture. The descent step uses height functions, and the obstruction to making descent effective is measured by the Selmer group and the Tate–Shafarevich group. No algorithm is known that provably computes the rank in all cases.