In
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
...
, the Bombieri–Lang conjecture is an unsolved problem conjectured by
Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathe ...
and
Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
about the
Zariski density of the set of
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fie ...
s of an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
of
general type.
Statement
The weak Bombieri–Lang conjecture for surfaces states that if
is a smooth
surface of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in ...
defined over a number field
, then the points of
do not form a
dense set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
on
.
The general form of the Bombieri–Lang conjecture states that if
is a positive-dimensional algebraic variety of general type defined over a number field
, then the points of
do not form a dense set in the Zariski topology.
The refined form of the Bombieri–Lang conjecture states that if
is an algebraic variety of general type defined over a number field
, then there is a dense open subset
of
such that for all number field extensions
over
, the set of points in
is finite.
History
The Bombieri–Lang conjecture was independently posed by Enrico Bombieri and Serge Lang. In a 1980 lecture at the
University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
, Enrico Bombieri posed a problem about the degeneracy of rational points for surfaces of general type.
Independently in a series of papers starting in 1971, Serge Lang conjectured a more general relation between the distribution of rational points and
algebraic hyperbolicity In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to ...
,
formulated in the "refined form" of the Bombieri–Lang conjecture.
Generalizations and implications
The Bombieri–Lang conjecture is an analogue for surfaces of
Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points.
If true, the Bombieri–Lang conjecture would resolve the
Erdős–Ulam problem, as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational.
In 1997,
Lucia Caporaso,
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
,
Joe Harris, and Patricia Pacelli showed that the Bombieri–Lang conjecture implies a
uniform boundedness conjecture for rational points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g \geq 2 that there exists a number N(K,g) depending only on K and g such that for any algebraic curve C ...
: there is a constant
depending only on
and
such that the number of rational points of any
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
curve
over any
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
number field is at most
.
References
Diophantine geometry
Unsolved problems in geometry
Conjectures
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