# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.