In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Siegel's theorem on integral points states that a curve of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
greater than zero has only finitely many integral points over any given
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
.
The theorem was first proved in 1929 by
Carl Ludwig Siegel
Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
and was the first major result on
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s 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 1 it was superseded by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
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field \mathbb of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and know ...
in 1983.
Statement
Siegel's theorem on integral points: For a smooth algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
''C'' of genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'' defined over a number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
''K'', presented in affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
in a given coordinate system, there are only finitely many points on ''C'' with coordinates in the ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
''O'' of ''K'', provided ''g'' > 0.
History
In 1926, Siegel proved the theorem effectively in the special case
, so that he proved this theorem conditionally, provided the
Mordell's conjecture is true.
In 1929, Siegel proved the theorem unconditionally by combining a version of the
Thue–Siegel–Roth theorem, from
diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
, with the
Mordell–Weil theorem from
diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
(required in Weil's version, to apply to the
Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...
of ''C'').
In 2002,
Umberto Zannier and
Pietro Corvaja gave a new proof by using a new method based on the
subspace theorem.
[Corvaja, P. and Zannier, U. "A subspace theorem approach to integral points on curves", Compte Rendu Acad. Sci., 334, 2002, pp. 267–271 ]
Effective versions
Siegel's result was ineffective for
(see
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. Where it ...
), since
Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to almost all
algebraic number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s of degree
. Siegel proved it effectively only in the special case
in 1926. Effective results in some cases derive from
Baker's method
A baker is someone who primarily bakes bread.
Baker or Bakers may also refer to:
Business
* Bakers Coaches, trading name BakerBus, a bus and coach operator in Staffordshire, England
* Baker Hughes, an oilfield services company
* Baker McKenzie, ...
.
See also
*
Diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
References
*
*
* {{cite journal , last=Siegel , first=Carl Ludwig , authorlink=Carl Ludwig Siegel , title=Über einige Anwendungen diophantischer Approximationen , lang=de , journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften , year=1929
Diophantine equations
Theorems in number theory