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[parent] Faltings' theorem (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 1 (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.




"Faltings' theorem" is owned by alozano.
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See Also: Siegel's theorem

Other names:  Mordell's conjecture

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Cross-references: Siegel's theorem, theorem, finite, abelian group, finitely generated, Mordell-Weil theorem, elliptic curve, point, contains, infinite, algebraic closure, isomorphic, genus, curve, non-singular, number field
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This is version 2 of Faltings' theorem, born on 2006-06-07, modified 2006-06-07.
Object id is 7968, canonical name is FaltingsTheorem.
Accessed 2566 times total.

Classification:
AMS MSC14H99 (Algebraic geometry :: Curves :: Miscellaneous)
 14G05 (Algebraic geometry :: Arithmetic problems. Diophantine geometry :: Rational points)

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