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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, 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 1705 times total.

Classification:
AMS MSC: 14H99 (Algebraic geometry :: Curves :: Miscellaneous)
 14G05 (Algebraic geometry :: Arithmetic problems. Diophantine geometry :: Rational points)
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