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 ...

, 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 solution set, set of solutions of a system of polynomial equations over the real number, ...

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 algebr ...

. It was proposed by Yuri I. Manin 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 , is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient proje ...

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

, let

H

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

V(K)

Zariski dense In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not H ...

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 ver ...

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