Manin conjecture

File:Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface.png diagonal cubic surface.]]

In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators

{{cite journal

| title = Rational points of bounded height on Fano varieties

| journal = Inventiones Mathematicae

| volume = 95

| year = 1989

| issue = 2

| pages = 421–435

| mr = 974910

| zbl = 0674.14012 | doi=10.1007/bf01393904| bibcode = 1989InMat..95..421F }}

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 defined

over a number field $K$ ,

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

: $N_{U,H}(B)=\#\{x \in U(K):H(x)\leq B\}$

for $B \geq 1$ ,

satisfies

: $N_{U,H}(B) \sim c B (\log B)^{\rho-1},$

as $B \to \infty.$

Here

$\rho$

is the rank of the Picard group of $V$

and $c$

is a positive constant which

later received a conjectural interpretation by Peyre.{{cite journal

| last = Peyre | first = E.

| title = Hauteurs et mesures de Tamagawa sur les variétés de Fano

| journal = Duke Mathematical Journal

| volume = 79

| year = 1995

| issue = 1

| pages = 101–218

| mr = 1340296

| doi = 10.1215/S0012-7094-95-07904-6

| zbl = 0901.14025}}

Manin's conjecture has been decided for special families of varieties,

{{cite book | editor1-last=Duke | editor1-first=William | title=Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005 | location=Providence, RI | publisher=American Mathematical Society | isbn=978-0-8218-4307-9 | series=Clay Mathematics Proceedings | volume=7

| last = Browning | first = T. D.

| chapter = An overview of Manin's conjecture for del Pezzo surfaces

| year = 2007

| pages = 39–55

| mr = 2362193

| zbl = 1134.14017}}

but is still open in general.

References

Category:Conjectures

Category:Diophantine geometry

Category:Unsolved problems in number theory