Manin conjecture explained
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[1] 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
be a
Fano variety definedover a
number field
,let
be a height function which is relative to the
anticanonical divisorand assume that
is
Zariski dense in
. Then there existsa non-empty
Zariski open subset
such that the counting functionof
-rational points of bounded height, defined by
NU,H(B)=\#\{x\inU(K):H(x)\leqB\}
for
, satisfies
NU,H(B)\simcB(logB)\rho-1,
as
Here
is the rank of the
Picard group of
and
is a positive constant whichlater received a conjectural interpretation by Peyre.
[2] Manin's conjecture has been decided for special families of varieties,[3] but is still open in general.
Notes and References
- Franke . J. . Manin . Y. I. . Yuri I. Manin . Tschinkel . Y. . Rational points of bounded height on Fano varieties . . 95 . 1989 . 2 . 421–435 . 974910 . 0674.14012 . 10.1007/bf01393904.
- Peyre . E. . Hauteurs et mesures de Tamagawa sur les variétés de Fano . . 79 . 1995 . 1 . 101–218 . 1340296 . 10.1215/S0012-7094-95-07904-6 . 0901.14025.
- Book: Browning, T. D.
. Duke . William . Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005 . Providence, RI . . 978-0-8218-4307-9 . Clay Mathematics Proceedings . 7 . An overview of Manin's conjecture for del Pezzo surfaces . 2007 . 39–55 . 2362193 . 1134.14017.
