Faltings’ theorem

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 $1983$ .

Theorem (Faltings’ Theorem (Mordell’s conjecture)).

Let $K$ be a number field and let $C/K$ be a non-singular curve defined over $K$ of genus $g\\geq 2$ . Then $C(K)$ is finite.

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

单词	FaltingsTheorem
释义	Faltings’ theorem 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 $1983$ . Theorem (Faltings’ Theorem (Mordell’s conjecture)). Let $K$ be a number field and let $C/K$ be a non-singular curve defined over $K$ of genus $g\\geq 2$ . Then $C(K)$ is finite. The reader may also be interested in Siegel’s theorem.
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