Mordell–Weil theorem

Remarks

The proof has two parts. The weak Mordell–Weil theorem says that $A(K)/mA(K)$ is finite for an integer $m \ge 2$. It is proved with Kummer theory and the finiteness of the class group and of the unit group modulo $m$-th powers. Then a height function on $A(K)$ has finitely many points of bounded height and satisfies $h(mP) \approx m^2 h(P)$. A descent with it shows that representatives of $A(K)/mA(K)$ and the points of small height generate $A(K)$. Mordell proved the case of an elliptic curve over $\Q$, which is Mordell's theorem; Weil proved the general case. So $A(K) \cong \Z^r \oplus T$ with $T$ finite; the integer $r$ is the rank of $A$ over $K$. The hypothesis on $K$ matters. Over $\overline{\Q}$ the group $E(\overline{\Q})$ of an elliptic curve contains points of every finite order, so its torsion subgroup is infinite and it is not finitely generated. Over $\R$ the group $E(\R)$ is uncountable.

Going deeper

For a student

The theorem gives the rank and the torsion subgroup as the basic invariants of $A(K)$. The rank is hard to compute: the descent gives only an upper bound through the Selmer group, and the gap is measured by the Tate–Shafarevich group. The Birch and Swinnerton-Dyer conjecture predicts the rank from an $L$-function. Let $C$ be a curve of genus at least $2$ embedded in its Jacobian. The theorem places $C(K)$ inside a finitely generated group, and Faltings's theorem shows that $C(K)$ is finite.