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 variety, alg ...

, 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 Mathem ...

and Serge Lang 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 field ...

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. Mo ...

general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...

Statement

The weak Bombieri–Lang conjecture for surfaces states that if

X

is a smooth surface of general type defined over a number field

k

, then the points of

X

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 r ...

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 n ...

X

. The general form of the Bombieri–Lang conjecture states that if

X

is a positive-dimensional algebraic variety of general type defined over a number field

k

, then the points of

X

do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if

X

is an algebraic variety of general type defined over a number field

k

, then there is a dense open subset

U

X

such that for all number field extensions

k'

over

k

, the set of points in

U

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 research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

, 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, 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 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 ...

, 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 In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Er ...

, 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, 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

B_

depending only on

g

and

d

such that the number of rational points of any

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

curve

X

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 mathematics ...

d

number field is at most

B_

References

Diophantine geometry Unsolved problems in geometry Conjectures {{algebraic-geometry-stub