Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface

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 ...

, the Manin conjecture describes the conjectural distribution of rational points on 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 ...

relative to a suitable

height function A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algeb ...

. It was proposed by

Yuri I. Manin Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical log ...

and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.

Conjecture

Their main conjecture is as follows. Let

V

be a

Fano variety In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has ...

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 f ...

K

, let

H

be a height function which is relative to the anticanonical divisor and assume that

V(K)

is Zariski dense in

V

. Then there exists a non-empty Zariski open subset

U \subset V

such that the counting function of

K

-rational points of bounded height, defined by :

N_(B)=\#\

for

B \geq 1

, satisfies :

N_(B) \sim c B (\log B)^,

B \to \infty.

Here

\rho

is the rank of the

Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...

V

and

c

is a positive constant which later received a conjectural interpretation by Peyre. Manin's conjecture has been decided for special families of varieties, but is still open in general.

References

{{reflist Conjectures Diophantine geometry Unsolved problems in number theory