20:
330:
444:
476:
214:
359:
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161:
540:
520:
234:
185:
128:
108:
81:
55:
and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.
19:
658:
Analytic number theory. A tribute to Gauss and
Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005
717:
669:
242:
367:
712:
661:
656:
Browning, T. D. (2007). "An overview of Manin's conjecture for del Pezzo surfaces". In Duke, William (ed.).
613:
545:
Manin's conjecture has been decided for special families of varieties, but is still open in general.
568:
452:
193:
338:
707:
679:
634:
589:
481:
687:
642:
597:
137:
8:
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219:
170:
113:
93:
66:
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44:
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is a positive constant which later received a conjectural interpretation by Peyre.
188:
164:
131:
626:
675:
630:
585:
48:
24:
566:; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties".
611:
Peyre, E. (1995). "Hauteurs et mesures de
Tamagawa sur les variétés de Fano".
701:
563:
52:
28:
499:
88:
84:
36:
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43:
describes the conjectural distribution of rational points on an
660:. Clay Mathematics Proceedings. Vol. 7. Providence, RI:
23:
Rational points of bounded height outside the 27 lines on
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508:
484:
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370:
341:
325:{\displaystyle N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}}
245:
222:
196:
173:
140:
116:
96:
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561:
439:{\displaystyle N_{U,H}(B)\sim cB(\log B)^{\rho -1},}
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353:
324:
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236:-rational points of bounded height, defined by
130:be a height function which is relative to the
319:
277:
63:Their main conjecture is as follows. Let
655:
18:
700:
610:
216:such that the counting function of
13:
718:Unsolved problems in number theory
462:
274:
14:
729:
187:. Then there exists a non-empty
16:Unsolved problem in number theory
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604:
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459:
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1:
662:American Mathematical Society
627:10.1215/S0012-7094-95-07904-6
548:
471:{\displaystyle B\to \infty .}
58:
7:
10:
734:
209:{\displaystyle U\subset V}
614:Duke Mathematical Journal
569:Inventiones Mathematicae
354:{\displaystyle B\geq 1}
47:relative to a suitable
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440:
355:
326:
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491:{\displaystyle \rho }
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441:
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132:anticanonical divisor
125:
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78:
51:. It was proposed by
22:
713:Diophantine geometry
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506:
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171:
156:{\displaystyle V(K)}
138:
114:
94:
67:
498:is the rank of the
189:Zariski open subset
664:. pp. 39–55.
582:10.1007/bf01393904
532:
512:
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120:
100:
73:
33:
671:978-0-8218-4307-9
535:{\displaystyle c}
515:{\displaystyle V}
229:{\displaystyle K}
180:{\displaystyle V}
134:and assume that
123:{\displaystyle H}
103:{\displaystyle K}
76:{\displaystyle V}
45:algebraic variety
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129:
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109:
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101:
82:
80:
79:
74:
41:Manin conjecture
733:
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728:
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724:
723:
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698:
697:
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139:
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135:
115:
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95:
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87:defined over a
68:
65:
64:
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49:height function
17:
12:
11:
5:
731:
721:
720:
715:
710:
694:
693:
670:
648:
621:(1): 101–218.
603:
576:(2): 421–435.
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552:
550:
547:
531:
511:
487:
467:
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461:
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176:
152:
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99:
72:
60:
57:
15:
9:
6:
4:
3:
2:
730:
719:
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703:
689:
685:
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379:
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364:
363:
362:
361:, satisfies
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316:
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237:
223:
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174:
166:
165:Zariski dense
147:
141:
133:
117:
97:
90:
86:
70:
56:
54:
53:Yuri I. Manin
50:
46:
42:
38:
30:
29:cubic surface
26:
21:
657:
651:
618:
612:
606:
573:
567:
564:Manin, Y. I.
562:Franke, J.;
557:
544:
500:Picard group
448:
334:
89:number field
85:Fano variety
62:
40:
34:
708:Conjectures
37:mathematics
702:Categories
688:1134.14017
643:0901.14025
598:0674.14012
549:References
59:Conjecture
486:ρ
463:∞
460:→
426:−
423:ρ
412:
397:∼
346:≥
314:≤
284:∈
275:#
201:⊂
27:diagonal
25:Clebsch's
680:2362193
635:1340296
590:0974910
686:
678:
668:
641:
633:
596:
588:
110:, let
39:, the
478:Here
83:be a
666:ISBN
522:and
335:for
684:Zbl
639:Zbl
623:doi
594:Zbl
578:doi
502:of
449:as
409:log
167:in
163:is
35:In
704::
682:.
676:MR
674:.
637:.
631:MR
629:.
619:79
617:.
592:.
586:MR
584:.
574:95
572:.
690:.
645:.
625::
600:.
580::
530:c
510:V
466:.
457:B
434:,
429:1
419:)
415:B
406:(
403:B
400:c
394:)
391:B
388:(
383:H
380:,
377:U
373:N
349:1
343:B
320:}
317:B
311:)
308:x
305:(
302:H
299::
296:)
293:K
290:(
287:U
281:x
278:{
272:=
269:)
266:B
263:(
258:H
255:,
252:U
248:N
224:K
204:V
198:U
175:V
151:)
148:K
145:(
142:V
118:H
98:K
71:V
31:.
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