Siegel's theorem on integral points

# Siegel's theorem on integral points

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Siegel's theorem on integral points is the 1929 result of Carl Ludwig Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...

, that for a smooth
Singular point of an algebraic variety
In mathematics, a singular point of an algebraic variety V is a point P that is 'special' , in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curveexhibits at , cannot simply be parametrized near the...

algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

C of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

g defined over a number field K, presented in affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

O of K, provided g > 0. This result covers the Mordell curve
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is an integer and where n ≠ 0. In Mordell curves, if is a solution, it therefore follows that is as well....

, for example.

This was proved by combining a version of the Thue–Siegel–Roth theorem
Thue–Siegel–Roth theorem
In mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very...

, from diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....

, with the Mordell–Weil theorem
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group...

from diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...

(required in Weil's version, to apply to the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

of C). It was the first major result on diophantine equations that depended only on the genus, not any special algebraic form of the equations. For g > 1 it was in the end superseded by Faltings' theorem
Faltings' theorem
In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension...

.

Siegel's result was ineffective (see effective results in number theory
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...

), since Thue
Axel Thue
Axel Thue was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics....

's method in diophantine approximation also is ineffective in describing possible very good rational approximations to algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s. Effective results in some cases derive from Baker's method.