In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Siegel's theorem on integral points states that for a
smooth algebraic curve ''C'' of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
''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
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 deno ...
''O'' of ''K'', provided ''g'' > 0.
The theorem was first proved in 1929 by
Carl Ludwig Siegel and was the first major result on
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
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 1 it was superseded by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
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by ...
in 1983.
History
In 1929, Siegel proved the theorem by combining a version of the
Thue–Siegel–Roth theorem, from
diophantine approximation, with the
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elli ...
from
diophantine geometry (required in Weil's version, to apply to the
Jacobian variety of ''C'').
In 2002,
Umberto Zannier
Umberto Zannier (born 25 May 1957, in Spilimbergo, Italy) is an Italian mathematician, specializing in number theory and Diophantine geometry.
Education
Zannier earned a Laurea degree from University of Pisa and studied at the Scuola Normale Sup ...
and
Pietro Corvaja
Pietro Corvaja (born 19 July 1967 in Padua, Italy
Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory ...
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 (see
effective results in number theory), since
Thue Thue may refer to:
* Axel Thue, a Norwegian mathematician
* Thue (food) Thue is a delicacy in Tibetan cuisine made with dri cheese (or sometimes parmesan or other hard cheeses), brown sugar (usually porang) and unsalted sweet cream butter. These in ...
's method in diophantine approximation also is ineffective in describing possible very good rational approximations to
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. Effective results in some cases derive from
Baker's method
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendenta ...
.
See also
*
Diophantine geometry
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