160:
proved, in 2014, the functional transcendence result needed for the general Pila-Zannier approach and
Emmanuel Ullmo has deduced from it a technical result needed for the induction step in the strategy. The remaining technical ingredient was the problem of bounding below the Galois degrees of special
191:
In 2019-2020, Gal
Biniyamini, Harry Schmidt and Andrei Yafaev, building on previous work and ideas of Harry Schmidt on torsion points in tori and abelian varieties and Gal Biniyamini's point counting results, have formulated a conjecture on bounds of heights of special points and deduced from its
274:
Manin-Mumford and André–Oort conjectures can be generalized in many directions, for example by relaxing the properties of points being `special' (and considering the so-called `unlikely locus' instead) or looking at more general ambient varieties: abelian or semi-abelian schemes, mixed
Shimura
70:
André's first version of the conjecture was just for one dimensional irreducible components, while Oort proposed that it should be true for irreducible components of arbitrary dimension in the moduli space of
488:
Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical
Heights on Shimura Varieties and the André-Oort Conjecture".
207:
and
Michael Groechenig) a proof of the Biniyamini-Schmidt-Yafaev height conjecture, thus completing the proof of the André-Oort conjecture using the Pila-Zannier strategy.
138:
to develop an approach to the Manin-Mumford-André-Oort type of problems. In 2009, Jonathan Pila proved the André-Oort conjecture unconditionally for arbitrary products of
39:, which is now a theorem (proven in several different ways). The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in
255:
118:(GRH) being true. In fact, the proof of the full conjecture under GRH was published by Bruno Klingler, Emmanuel Ullmo and Andrei Yafaev in 2014 in the
688:
657:
82:. It seems that André was motivated by applications to transcendence theory while Oort by the analogy with the Manin-Mumford conjecture.
545:
510:
36:
115:
706:
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in 1989 and a more general statement (albeit with a restriction on the type of the
Shimura variety) was conjectured by
445:
292:
276:
192:
validity the bounds for the Galois degrees of special points needed for the proof of the full André-Oort conjecture.
247:
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Pila, Jonathan (2009), "Rational points of definable sets and results of André–Oort–Manin–Mumford type",
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119:
173:
177:
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A related conjecture that has two forms, equivalent if the André–Oort conjecture is assumed, is the
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107:
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157:
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95:
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Oort, Frans (1997), "Canonical liftings and dense sets of CM points", in
Fabrizio Catanese (ed.),
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131:
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200:
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123:
75:
72:
64:
60:
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Pink, Richard (2005), "A combination of the conjectures of
Mordell–Lang and André–Oort",
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127:
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The conjecture in its modern form is as follows. Each irreducible component of the
348:
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in 1995. The modern version is a natural generalization of these two conjectures.
680:
428:
283:. Most of these questions are open and are a subject of current active research.
90:
Various results have been established towards the full conjecture by Ben Moonen,
577:
Zilber, Boris (2002), "Exponential sums equations and the
Schanuel conjecture",
370:
20:
590:
181:
48:
474:
378:
262:
no special subvariety of dimension > 0 that intersects the image of the
667:
Yafaev, Andrei (2007), Burns, David; Buzzard, Kevin; Nekovar, Jan (eds.),
442:
408:
Pila, Jonathan (2011), "O-minimality and the André–Oort conjecture for
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varieties etc.... These generalizations are colloquially known as the
565:, Progress in Mathematics, vol. 253, Birkhauser, pp. 251–282
622:
494:
650:
Some
Problems of Unlikely Intersections in Arithmetic and Geometry
361:
188:
and independently by Andreatta, Goren, Howard and Madapusi-Pera.
114:, among others. Some of these results were conditional upon the
279:
because problems of this type were proposed by Richard Pink and
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isogeny estimates. The averaged Colmez conjecture was proved by
534:
Carlson, James; Müller-Stach, Stefan; Peters, Chris (2017).
675:, Cambridge: Cambridge University Press, pp. 381–406,
652:. Princeton: Princeton University Press. pp. 96–127.
511:"Mathematicians Prove 30-Year-Old André-Oort Conjecture"
227:, there are only finitely many smooth projective curves
203:
claimed in a paper (featuring an appendix written by
35:, that can be seen as a non-abelian analogue of the
43:. A special case of the conjecture was stated by
256:moduli space of abelian varieties of dimension g
319:, Aspects of Mathematics, vol. E13, Vieweg
563:Geometric methods in algebra and number theory
347:Klingler, Bruno; Yafaev, Andrei (2014-11-01).
8:
463:Notices of the American Mathematical Society
540:. Cambridge University Press. p. 285.
621:
493:
427:
360:
606:"Autour de la conjecture de Zilber–Pink"
223:conjectured that for sufficiently large
335:, Cambridge: Cambridge University Press
304:
673:L-functions and Galois Representations
669:"The André-Oort conjecture - a survey"
142:, a result which earned him the 2011
7:
537:Period Mappings and Period Domains
14:
646:"About the André–Oort conjecture"
250:. Oort then conjectured that the
63:of a set of special points in a
116:generalized Riemann hypothesis
1:
258:– has for sufficiently large
681:10.1017/cbo9780511721267.011
509:Sloman, Leila (2022-02-03).
429:10.4007/annals.2011.173.3.11
168:, this bound was deduced by
136:transcendental number theory
443:Clay Research Award website
371:10.4007/annals.2014.180.3.2
349:"The André-Oort conjecture"
728:
610:J. Théor. Nombres Bordeaux
248:abelian variety of CM-type
174:averaged Colmez conjecture
591:10.1112/S0024610701002861
394:Int. Math. Res. Not. IMRN
67:is a special subvariety.
266:in a dense open subset.
37:Manin–Mumford conjecture
317:-functions and geometry
277:Zilber–Pink conjectures
217:Coleman–Oort conjecture
211:Coleman–Oort conjecture
16:Mathematical conjecture
293:Zilber–Pink conjecture
199:, Ananth Shankar, and
166:Siegel modular variety
604:Rémond, Gaël (2009),
415:Annals of Mathematics
353:Annals of Mathematics
130:used techniques from
120:Annals of Mathematics
73:principally polarised
25:André–Oort conjecture
707:Diophantine geometry
579:J. London Math. Soc.
312:André, Yves (1989),
164:For the case of the
29:Diophantine geometry
333:Arithmetic Geometry
195:In September 2021,
144:Clay Research Award
448:2011-06-26 at the
690:978-0-511-72126-7
659:978-0-691-15370-4
461:"February 2018".
172:in 2015 from the
76:Abelian varieties
41:Shimura varieties
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642:Zannier, Umberto
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237:Jacobian variety
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635:Further reading
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270:Generalizations
264:Torelli mapping
213:
201:Jacob Tsimerman
170:Jacob Tsimerman
124:Umberto Zannier
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65:Shimura variety
61:Zariski closure
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12:
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221:Robert Coleman
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205:Hélène Esnault
154:Emmanuel Ullmo
150:Bruno Klingler
140:modular curves
112:Emmanuel Ullmo
108:Bruno Klingler
104:Laurent Clozel
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84:
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31:, a branch of
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158:Andrei Yafaev
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134:geometry and
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128:Jonathan Pila
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100:Bas Edixhoven
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96:Andrei Yafaev
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78:of dimension
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33:number theory
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585:(2): 27–44,
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712:Conjectures
355:: 867–925.
122:. In 2006,
21:mathematics
701:Categories
520:2022-02-04
495:2109.08788
299:References
182:Xinyi Yuan
92:Yves André
49:Frans Oort
45:Yves André
475:1088-9477
379:0003-486X
362:1209.0936
254:– of the
231:of genus
132:o-minimal
55:Statement
644:(2012).
446:Archived
287:See also
246:) is an
176:and the
161:points.
86:Results
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23:, the
490:arXiv
357:arXiv
685:ISBN
654:ISBN
542:ISBN
471:ISSN
375:ISSN
184:and
156:and
126:and
110:and
677:doi
618:doi
587:doi
424:doi
420:173
412:",
367:doi
19:In
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648:.
614:21
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315:G
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240:J
233:g
229:C
225:g
80:g
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