20:
330:
444:
19:
476:
214:
359:
496:
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.
717:
669:
658:
Analytic number theory. A tribute to Gauss and
Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005
242:
712:
661:
367:
613:
568:
656:
Browning, T. D. (2007). "An overview of Manin's conjecture for del Pezzo surfaces". In Duke, William (ed.).
707:
545:
Manin's conjecture has been decided for special families of varieties, but is still open in general.
452:
193:
665:
44:
338:
683:
638:
622:
593:
577:
188:
164:
131:
679:
634:
589:
481:
687:
675:
642:
630:
597:
585:
137:
48:
24:
525:
505:
219:
170:
113:
93:
66:
701:
563:
52:
28:
542:
is a positive constant which later received a conjectural interpretation by Peyre.
499:
88:
84:
626:
36:
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".
581:
18:
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
528:
508:
484:
455:
370:
341:
325:{\displaystyle N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}}
245:
222:
196:
173:
140:
116:
96:
69:
439:{\displaystyle N_{U,H}(B)\sim cB(\log B)^{\rho -1},}
534:
514:
490:
470:
438:
353:
324:
228:
208:
179:
155:
122:
102:
75:
236:-rational points of bounded height, defined by
130:be a height function which is relative to the
8:
319:
277:
527:
507:
483:
454:
421:
375:
369:
340:
250:
244:
221:
195:
172:
139:
115:
95:
68:
63:Their main conjecture is as follows. Let
554:
7:
216:such that the counting function of
718:Unsolved problems in number theory
462:
274:
14:
187:. Then there exists a non-empty
16:Unsolved problem in number theory
459:
418:
405:
393:
387:
310:
304:
295:
289:
268:
262:
150:
144:
1:
662:American Mathematical Society
627:10.1215/S0012-7094-95-07904-6
471:{\displaystyle B\to \infty .}
734:
209:{\displaystyle U\subset V}
614:Duke Mathematical Journal
569:Inventiones Mathematicae
354:{\displaystyle B\geq 1}
47:relative to a suitable
536:
516:
492:
472:
440:
355:
326:
230:
210:
181:
157:
124:
104:
77:
32:
537:
517:
493:
491:{\displaystyle \rho }
473:
441:
356:
327:
231:
211:
182:
158:
132:anticanonical divisor
125:
105:
78:
51:. It was proposed by
22:
713:Diophantine geometry
526:
506:
482:
453:
368:
339:
243:
220:
194:
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:
488:
468:
436:
351:
322:
226:
206:
177:
153:
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
725:
692:
691:
653:
647:
646:
608:
602:
601:
559:
541:
539:
538:
533:
521:
519:
518:
513:
497:
495:
494:
489:
477:
475:
474:
469:
445:
443:
442:
437:
432:
431:
386:
385:
360:
358:
357:
352:
331:
329:
328:
323:
261:
260:
235:
233:
232:
227:
215:
213:
212:
207:
186:
184:
183:
178:
162:
160:
159:
154:
129:
127:
126:
121:
109:
107:
106:
101:
82:
80:
79:
74:
41:Manin conjecture
733:
732:
728:
727:
726:
724:
723:
722:
698:
697:
696:
695:
672:
655:
654:
650:
610:
609:
605:
561:
560:
556:
551:
524:
523:
504:
503:
480:
479:
451:
450:
417:
371:
366:
365:
337:
336:
246:
241:
240:
218:
217:
192:
191:
169:
168:
136:
135:
112:
111:
92:
91:
87:defined over a
65:
64:
61:
49:height function
17:
12:
11:
5:
731:
729:
721:
720:
715:
710:
700:
699:
694:
693:
670:
648:
621:(1): 101–218.
603:
576:(2): 421–435.
553:
552:
550:
547:
531:
511:
487:
467:
464:
461:
458:
447:
446:
435:
430:
427:
424:
420:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
384:
381:
378:
374:
350:
347:
344:
333:
332:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
259:
256:
253:
249:
225:
205:
202:
199:
176:
152:
149:
146:
143:
119:
99:
72:
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
730:
719:
716:
714:
711:
709:
706:
705:
703:
689:
685:
681:
677:
673:
667:
663:
659:
652:
649:
644:
640:
636:
632:
628:
624:
620:
616:
615:
607:
604:
599:
595:
591:
587:
583:
579:
575:
571:
570:
565:
558:
555:
548:
546:
543:
529:
509:
501:
485:
465:
456:
433:
428:
425:
422:
414:
411:
408:
402:
399:
396:
390:
382:
379:
376:
372:
364:
363:
362:
361:, satisfies
348:
345:
342:
316:
313:
307:
301:
298:
292:
286:
283:
280:
271:
265:
257:
254:
251:
247:
239:
238:
237:
223:
203:
200:
197:
190:
174:
166:
165:Zariski dense
147:
141:
133:
117:
97:
90:
86:
70:
58:
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:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
