---
tag: H0DB07WM20
kind: statement
role: theorem
name: Faltings's theorem
msc: [11G30, 14G05, 14H25]
uses: [MWBDFS2KJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Faltings%27s_theorem
same_as:
  wikidata: Q240950
  wikipedia: Faltings's theorem
---

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]], and let $C$ be a smooth projective geometrically irreducible curve over $K$ of genus $g \ge 2$.
Then the set $C(K)$ of points of $C$ defined over $K$ is finite.

## 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|BZHGKN6Q3K]] 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.

## Explanations

### L1 human

A polynomial equation in two variables describes a curve, and the shape of the curve over the complex numbers has a number of holes, its genus.
Faltings's theorem says that once the genus is at least two, the equation has only finitely many rational solutions.
So the shape of a curve controls how many rational points it can have.

### L2 human

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|4KTBRP27P6]], which already applies from genus $1$.