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 $\\mathbb{Z}$ .
随便看	the top 10 most beautiful theorems TheTopologistsSineCurveHasTheFixedPointProperty the topologist’s sine curve has the fixed point property TheTorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve the torsion subgroup of an elliptic curve injects in the reduction of the curve The type of booleans TheUnionOfALocallyFiniteCollectionOfClosedSetsIsClosed the union of a locally finite collection of closed sets is closed The unit type The universal cover as an identity system The universal cover in type theory The van Kampen theorem The van Kampen theorem with a set of basepoints The Yoneda lemma ThinAlgebraicSet thin algebraic set ThinDoubleTrack thin double track ThirdIsomorphismTheorem third isomorphism theorem Thirteen thirteen This is my sample file ThisIsMySampleFile1 ThomassensTheoremOn3connectedGraphs