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Galois orbits and equidistribution: Manin-Mumford and André-Oort. (English) Zbl 1209.11055
In this review article, the author outlines a similar proof of the Manin-Mumford conjecture and the André-Oort conjecture. This proof is based on a “Galois theory – ergodic theory alternative”. In the case of the Manin-Mumford conjecture, it is due to N. Ratazzi and E. Ullmo [“Galois + equidistribution = Manin-Mumford”, Clay Math. Proc. 8, 419–430 (2009; Zbl 1250.11062)] and in the case of the André-Oort conjecture, it is due to B. Klingler, E. Ullmo and A. Yafaev [“On the André-Oort conjecture for products of modular curves.”, Clay Math. Proc. 8, 431–439 (2009; Zbl 1254.11062); B. Klingler and A. Yafaev, “Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture”, preprint
http:www.institut.math.jussieu.fr/ klingler/papiers/KY12.pdf].

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37P99 Arithmetic and non-Archimedean dynamical systems
14K12 Subvarieties of abelian varieties
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References:
[1] F. Breuer, Special subvarieties of Drinfeld modular varieties. Preprint, 2009. Available on author’s web-page.
[2] L. Clozel, E. Ullmo, Equidistribution de sous-variétés spéciales. Annals of Mathematics 161 (2005), 1571-1588. · Zbl 1099.11031
[3] P. Deligne, Variétés de Shimura : interpretation modulaire et techniques de construction de modeles canoniques. In Automorphic Forms, Representations and \(L\)-functions. Part II, Vol 33 of Proc. of Symp. in Pure Math., 247-290, AMS. · Zbl 0437.14012
[4] B. Klingler, A. Yafaev, The André-Oort conjecture. Preprint, submitted. Available on Klingler’s web-page.
[5] N. Ratazzi, E. Ullmo, Galois+Equidistribution=Manin-Mumford. Preprint. To appear in the Proceedings of Clay summer school on Arithmetic Geometry, Goettingen, 2007. Available on Ullmo’s web-page.
[6] E. Ullmo, A. Yafaev, The André-Oort conjecture for products of modular curves. Preprint. To appear in the Proceedings of Clay summer school on Arithmetic Geometry, Goettingen, 2007. Available on Ullmo’s web-page. · Zbl 1254.11062
[7] E. Ullmo, A. Yafaev, Galois orbits and equidistribution : towards the André-Oort conjecture. Preprint, submitted. Available on Ullmo’s web-page.
[8] R. Pink, A Combination of the Conjectures of Mordell-Lang and André-Oort. In Geometric Methods in Algebra and Number Theory (Bogomolov, F., Tschinkel, Y., Eds.), Progress in Mathematics 253, Birkhäuser, 2005, 251-282. · Zbl 1200.11041
[9] R. Pink, A Common Generalization of the Conjectures of André-Oort, Manin-Mumford, and Mordell-Lang. Preprint available on author’s web-page.
[10] P. Tzermias, The Manin-Mumford conjecture: a brief survey. Bull. London Math. Soc. 32 (2000), no. 6, 641-652. · Zbl 1073.14525
[11] A. Yafaev, A conjecture of Yves André’s. Duke Mathematical Journal. 132 (2006), no. 3, 393-407. · Zbl 1097.11032
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