Siegel’s 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 http://planetmath.org/node/RingOfSIntegersring 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 (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 $\\mathbb{Z}$ .

单词	SiegelsTheorem
释义	Siegel’s 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 http://planetmath.org/node/RingOfSIntegersring 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 (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 $\\mathbb{Z}$ .
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