Faltings's theorem

Remarks

The statement was conjectured by Mordell in 1922 and proved by Faltings in 1983; it was long called the Mordell conjecture. Faltings's proof goes through the Tate conjecture for abelian varieties over number fields. It also proves the Shafarevich conjecture: there are finitely many abelian varieties of given dimension with good reduction outside a given finite set of primes, up to isomorphism. A construction of Parshin passes from curves to abelian varieties. Vojta later gave a different proof by Diophantine approximation, simplified by Bombieri. The genus hypothesis is necessary. A curve of genus $0$ with one rational point is isomorphic to $\mathbb{P}^1$ and has infinitely many rational points. An elliptic curve, of genus $1$, has infinitely many rational points when the rank in Mordell's theorem is positive. Example: for $n \ge 4$, the Fermat curve $x^n + y^n = z^n$ has genus $(n-1)(n-2)/2 \ge 3$, so it has only finitely many rational points. The proofs are not effective: they bound the number of points but give no algorithm to find them all.

Going deeper

For a student

A curve of genus $0$ has no rational point or infinitely many, and for genus $1$ with a point the points form a finitely generated group. Faltings's theorem completes this qualitative picture of rational points on curves. For integral points on affine curves, the analogous finiteness is Siegel's theorem on integral points, which already applies from genus $1$.