We overview a unified approach to the André-Oort and Manin-Mumford conjectures based on a combination of Galois-theoretic and ergodic techniques. This paper is based on recent work of Klingler, Ullmo and Yafaev on the André-Oort conjecture, and of Ratazzi and Ullmo on the Manin-Mumford conjecture.
On passe en revue une approche unifiée aux conjectures de Manin-Mumford et d’André-Oort basée sur la combinaison de techniques galoisiennes et ergodiques. Ce texte est basé sur les travaux récents de Klingler, Ullmo et Yafaev sur la conjecture de André-Oort, et de Ratazzi et Ullmo sur la conjecture de Manin-Mumford.
@article{JTNB_2009__21_2_491_0, author = {Yafaev, Andrei}, title = {Galois orbits and equidistribution: {Manin-Mumford} and {Andr\'e-Oort.}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {491--500}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {2}, year = {2009}, doi = {10.5802/jtnb.684}, zbl = {1209.11055}, mrnumber = {2541439}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.684/} }
TY - JOUR AU - Yafaev, Andrei TI - Galois orbits and equidistribution: Manin-Mumford and André-Oort. JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 491 EP - 500 VL - 21 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.684/ DO - 10.5802/jtnb.684 LA - en ID - JTNB_2009__21_2_491_0 ER -
%0 Journal Article %A Yafaev, Andrei %T Galois orbits and equidistribution: Manin-Mumford and André-Oort. %J Journal de théorie des nombres de Bordeaux %D 2009 %P 491-500 %V 21 %N 2 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.684/ %R 10.5802/jtnb.684 %G en %F JTNB_2009__21_2_491_0
Yafaev, Andrei. Galois orbits and equidistribution: Manin-Mumford and André-Oort.. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 491-500. doi : 10.5802/jtnb.684. http://archive.numdam.org/articles/10.5802/jtnb.684/
[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. | MR | Zbl
[3] P. Deligne, Variétés de Shimura : interpretation modulaire et techniques de construction de modeles canoniques. In Automorphic Forms, Representations and -functions. Part II, Vol 33 of Proc. of Symp. in Pure Math., 247–290, AMS. | MR | Zbl
[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.
[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. | MR
[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. | MR | Zbl
[11] A. Yafaev, A conjecture of Yves André’s. Duke Mathematical Journal. 132 (2006), no. 3, 393–407. | MR | Zbl
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