Let $K$ be a number field and let $C/K$ be a non-singular curve defined over $K$ and genus $g$ . When the genus is $0$ , the curve is isomorphic to $\mathbb{P}^1$ (over an algebraic closure $\overline{K}$ ) and therefore $C(K)$ is either empty or equal to $\mathbb{P}^1(K)$ (in particular $C(K)$ is infinite). If the genus of $C$ is $1$ and $C(K)$ contains at least one point over $K$ then $C/K$ is an elliptic curve and the Mordell-Weil theorem shows that $C(K)$ is a finitely generated abelian group (in particular, $C(K)$ may be finite or infinite). However, if $g\geq 2$ , Mordell conjectured in $1922$ that $C(K)$ cannot be infinite. This was first proven by Faltings in $1983$ .

The reader may also be interested in Siegel's theorem.