In mathematics, Siegel's theorem on integral points states that a curve of genus greater than zero has only finitely many integral points over any given number field.
The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For g > 1 it was superseded by Faltings's theorem in 1983.
Siegel's theorem on integral points: For a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0.
In 1926, Siegel proved the theorem effectively in the special case
g=1
In 1929, Siegel proved the theorem unconditionally by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from diophantine geometry (required in Weil's version, to apply to the Jacobian variety of C).
In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new method based on the subspace theorem.[1]
Siegel's result was ineffective for
g\ge2
d\ge5
g=1
. Serge Lang . Elliptic curves: Diophantine analysis . Grundlehren der mathematischen Wissenschaften . 231 . 1978 . 3-540-08489-4 . 128–153 . 0388.10001 .