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[parent] Siegel's theorem (Theorem)

The following theorem is a very deep application of Roth's theorem. Let $ K$ be a number field and let $ S$ be a finite set of places of $ K$. Let $ R_S$ be the ring of $ S$-integers in $ K$. Let $ C/K$ be a smooth projective curve of genus $ g$ defined over $ K$ and let $ f$ be a non-constant function in the function field of $ C/K$, i.e. $ f\in K(C)$.

Theorem 1 (Siegel's Theorem)   Assume that $ C/K$ has genus $ g\geq 1$. Then the set $ \{P\in C(K) : f(P) \in R_S\}$ is finite.

In particular, when $ f$ is the coordinate functions $ x(P)$ and $ y(P)$, Siegel's theorem implies that a curve of genus $ \geq 1$ has only finitely many integral points. For example, this shows that an elliptic curve defined over $ \mathbb{Q}$ can only have finitely many points defined over $ \mathbb{Z}$.



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


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Cross-references: elliptic curve, points, integral, curve, implies, coordinate, finite, function field, function, genus, projective curve, smooth, places, finite set, number field, Roth's theorem, application
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This is version 3 of Siegel's theorem, born on 2006-06-07, modified 2006-06-07.
Object id is 7969, canonical name is SiegelsTheorem.
Accessed 997 times total.

Classification:
AMS MSC: 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
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