790:
contains only finitely many maximal special subvarieties. This is the Manin–Mumford conjecture in the semiabelian setting and the André–Oort conjecture in the
Shimura setting. Both are now theorems; the former has been known for several decades, while the latter was proven in full generality only
672:
The abelian and modular versions of the Zilber–Pink conjecture are special cases of the conjecture for
Shimura varieties, while in general the semiabelian case is not. However, special subvarieties of semiabelian and Shimura varieties share many formal properties which makes the same formulation
75:. The general version is now known as the Zilber–Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.
440:. For example, in semiabelian varieties special points are torsion points and special varieties are translates of irreducible algebraic subgroups by torsion points. In the modular setting special points are the
67:
in the early 2000's. For semiabelian varieties the conjecture implies the
Mordell–Lang and Manin–Mumford conjectures. Richard Pink proposed (again independently) a more general conjecture for
250:
382:
1173:
326:
1050:
959:
Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical
Heights on Shimura Varieties and the André-Oort Conjecture".
122:
708:
647:
621:
571:
495:
794:
Many partial results have been proven on the Zilber–Pink conjecture. An example in the modular setting is the result that any variety contains only finitely many maximal
1019:
545:
276:
168:
428:. For example, if in a 3-dimensional space two lines intersect, then it is an unlikely intersection, for two randomly chosen lines would almost never intersect.
788:
768:
748:
728:
667:
599:
519:
469:
422:
402:
188:
142:
1209:
1183:
885:
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Special varieties of a
Shimura variety are certain arithmetically defined subvarieties. They are higher dimensional versions of
36:
930:
Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic
1228:
32:
40:
193:
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atypical subvarieties, where a strongly atypical subvariety is an atypical subvariety with no constant coordinate.
925:
331:
445:
863:
441:
281:
1024:
681:
While the Zilber–Pink conjecture is wide open, many special cases and weak versions have been proven.
1233:
28:
868:
95:
48:
687:
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604:
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474:
71:
which also implies the André–Oort conjecture. In the case of algebraic tori, Zilber called it the
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960:
1205:
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881:
1199:
524:
255:
147:
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1100:
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873:
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937:
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945:
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932:. Progress in Mathematics (in French). Vol. 35. Birkhäuser-Boston. pp. 327–352.
437:
68:
64:
56:
44:
858:
Pink, Richard (2005). "A Combination of the
Conjectures of Mordell–Lang and André–Oort".
917:
773:
753:
733:
713:
652:
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407:
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127:
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are not expected to intersect at all, so when they do, the intersection is said to be
1222:
1169:
1152:
1112:
921:
328:, atypical intersections are "atypically large" and are not expected to occur. When
60:
52:
815:
Zilber, Boris (2002), "Exponential sums equations and the
Schanuel conjecture",
92:
if its dimension is larger than expected. More precisely, given three varieties
20:
1144:
828:
1104:
877:
1081:
Pila, Jonathan; Tsimerman, Jacob (2016), "Ax-Schanuel for the j-function",
1053:, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 22, pp. 1705–1745
920:(1983). "Sous-variétés d'une variété abélienne et points de torsion". In
444:
and special varieties are irreducible components of varieties defined by
1125:
Aslanyan, Vahagn (2021), "Weak
Modular Zilber–Pink with Derivatives",
1135:
965:
1201:
Some
Problems of Unlikely Intersections in Arithmetic and Geometry
1095:
601:
be a mixed
Shimura variety or a semiabelian variety defined over
1031:
862:. Progress in Mathematics. Vol. 235. pp. 251–282.
846:, International Mathematics Research Notices, vol. 2007
905:, Ann. Sci. Éc. Norm. Supér, vol. 49, pp. 813–858
669:
contains only finitely many maximal atypical subvarieties.
844:
Anomalous Subvarieties—Structure Theorems and Applications
842:
Bombieri, Enrico; Masser, David; Zannier, Umberto (2007),
1067:
Quantitative Reduction Theory and Unlikely Intersections
88:
The intersection of two algebraic varieties is called
1027:
1004:
776:
756:
736:
716:
690:
655:
629:
607:
587:
553:
527:
507:
477:
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130:
98:
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782:
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539:
513:
489:
463:
416:
396:
376:
320:
270:
244:
182:
162:
136:
116:
770:. Hence, the Zilber–Pink conjecture implies that
984:, Compositio Math., vol. 148, pp. 1–27
27:is a far-reaching generalisation of many famous
903:o-minimality and certain atypical intersections
860:Geometric Methods in Algebra and Number Theory
245:{\displaystyle \dim Z>\dim X+\dim Y-\dim U}
1175:Point-Counting and the Zilber–Pink Conjecture
982:Some unlikely intersections beyond André-Oort
8:
1069:, IMRN, vol. 2022, pp. 16138–16195
521:is an atypical component of an intersection
980:Habegger, Philipp; Pila, Jonathan (2012),
901:Habegger, Philipp; Pila, Jonathan (2016),
73:Conjecture on Intersection with Tori (CIT)
1204:. Princeton: Princeton University Press.
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377:{\displaystyle \dim X+\dim Y-\dim U<0}
333:
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195:
175:
149:
129:
97:
807:
1065:Daw, Christopher; Orr, Martin (2022),
996:Daw, Christopher; Orr, Martin (2021),
7:
321:{\displaystyle \dim X+\dim Y-\dim U}
31:conjectures and statements, such as
84:Atypical and unlikely intersections
1045:{\displaystyle {\mathcal {A}}_{2}}
252:. Since the expected dimension of
14:
677:Partial results and special cases
1178:. Cambridge University Press.
451:Given a mixed Shimura variety
117:{\displaystyle X,Y\subseteq U}
1:
750:is an atypical subvariety of
998:Unlikely intersections with
703:{\displaystyle V\subseteq X}
642:{\displaystyle V\subseteq X}
616:{\displaystyle \mathbb {C} }
566:{\displaystyle T\subseteq X}
490:{\displaystyle V\subseteq X}
710:contains a special variety
1250:
1145:10.1007/s00208-021-02213-7
577:The Zilber–Pink conjecture
170:is said to be atypical in
829:10.1112/S0024610701002861
673:valid in both settings.
573:is a special subvariety.
1105:10.1215/00127094-3620005
1014:{\displaystyle E\times }
878:10.1007/0-8176-4417-2_11
540:{\displaystyle V\cap T}
271:{\displaystyle X\cap Y}
163:{\displaystyle X\cap Y}
16:Mathematical conjecture
1046:
1015:
784:
764:
744:
724:
704:
663:
649:be a subvariety. Then
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25:Zilber–Pink conjecture
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55:and independently by
49:semiabelian varieties
1229:Diophantine geometry
1025:
1002:
817:J. London Math. Soc.
774:
754:
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688:
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627:
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551:
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408:
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332:
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256:
194:
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144:of the intersection
128:
96:
730:then by definition
499:atypical subvariety
51:it was proposed by
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114:
1211:978-0-691-15370-4
1089:(13): 2587–2605,
783:{\displaystyle V}
763:{\displaystyle V}
743:{\displaystyle T}
723:{\displaystyle T}
662:{\displaystyle V}
594:{\displaystyle X}
514:{\displaystyle V}
471:and a subvariety
464:{\displaystyle X}
446:modular equations
432:Special varieties
417:{\displaystyle Y}
397:{\displaystyle X}
183:{\displaystyle U}
137:{\displaystyle Z}
69:Shimura varieties
1241:
1215:
1196:Zannier, Umberto
1189:
1156:
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384:, the varieties
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1163:Further reading
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978:
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918:Raynaud, Michel
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869:10.1.1.499.3023
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473:
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442:singular moduli
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65:Umberto Zannier
57:Enrico Bombieri
17:
12:
11:
5:
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922:Artin, Michael
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438:special points
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124:, a component
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45:algebraic tori
15:
13:
10:
9:
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4:
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1185:9781009170321
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1084:
1083:Duke Math. J.
1077:
1074:
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1037:
1021:CM curves in
1008:
1005:
992:
989:
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943:
939:
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887:0-8176-4349-4
883:
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717:
697:
694:
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684:If a variety
682:
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670:
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588:
576:
574:
560:
557:
554:
534:
531:
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83:
78:
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58:
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50:
46:
42:
38:
37:Manin–Mumford
34:
30:
26:
22:
1200:
1174:
1126:
1120:
1086:
1082:
1076:
1066:
1060:
997:
991:
981:
975:
954:
929:
912:
902:
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859:
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843:
837:
823:(2): 27–44,
820:
816:
810:
795:
793:
683:
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671:
580:
498:
450:
435:
425:
89:
87:
72:
61:David Masser
53:Boris Zilber
41:Mordell–Lang
24:
18:
1234:Conjectures
29:Diophantine
21:mathematics
1223:Categories
1136:1803.05895
1127:Math. Ann.
966:2109.08788
946:0581.14031
926:Tate, John
802:References
791:recently.
623:, and let
33:André–Oort
1153:119654268
1113:118973278
1096:1412.8255
1009:×
864:CiteSeerX
695:⊆
634:⊆
558:⊆
532:∩
482:⊆
363:
357:−
351:
339:
313:
307:−
301:
289:
263:∩
237:
231:−
225:
213:
201:
155:∩
109:⊆
79:Statement
1198:(2012).
1172:(2022).
928:(eds.).
796:strongly
426:unlikely
90:atypical
938:0717600
1208:
1182:
1151:
1111:
944:
936:
884:
866:
547:where
43:. For
39:, and
23:, the
1149:S2CID
1131:arXiv
1109:S2CID
1091:arXiv
961:arXiv
497:, an
1206:ISBN
1180:ISBN
882:ISBN
581:Let
404:and
369:<
207:>
47:and
1141:doi
1101:doi
1087:165
942:Zbl
874:doi
825:doi
501:of
360:dim
348:dim
336:dim
310:dim
298:dim
286:dim
278:is
234:dim
222:dim
210:dim
198:dim
190:if
19:In
1225::
1147:,
1139:,
1129:,
1107:,
1099:,
1085:,
940:.
934:MR
924:;
880:.
872:.
821:65
819:,
448:.
63:,
59:,
35:,
1214:.
1188:.
1143::
1133::
1103::
1093::
1071:.
1055:.
1038:2
1032:A
1006:E
986:.
969:.
963::
948:.
907:.
890:.
876::
848:.
832:.
827::
778:V
758:V
738:T
718:T
698:X
692:V
657:V
637:X
631:V
610:C
589:X
561:X
555:T
535:T
529:V
509:V
485:X
479:V
459:X
412:Y
392:X
372:0
366:U
354:Y
345:+
342:X
316:U
304:Y
295:+
292:X
266:Y
260:X
240:U
228:Y
219:+
216:X
204:Z
178:U
158:Y
152:X
132:Z
112:U
106:Y
103:,
100:X
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