Comptes Rendus
Number theory/Algebraic geometry
Topological and equidistributional refinement of the André–Pink–Zannier conjecture at finitely many places
[Raffinements topologiques et équidistributionnels de la conjecture d'André–Pink–Zannier en un nombre fini de places]
Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 231-235.

On présente quelques applications des résultats récents en dynamique homogène à un problème d'intersections atypiques dans les variétés de Shimura (la conjecture de André–Pink–Zannier) et ses raffinements.

We present some applications of recent results in homogeneous dynamics to an unlikely intersections problem in Shimura varieties (the André–Pink–Zannier conjecture) and its refinements.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.01.013

Rodolphe Richard 1 ; Andrei Yafaev 2

1 DPMMS, University of Cambridge, United Kingdom
2 UCL, Department of Mathematics, United Kingdom
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Rodolphe Richard; Andrei Yafaev. Topological and equidistributional refinement of the André–Pink–Zannier conjecture at finitely many places. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 231-235. doi : 10.1016/j.crma.2019.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.01.013/

[1] P. Habegger; G. Rémond; T. Scanlon; E. Ullmo; A. Yafaev Around the Zilber–Pink Conjecture, Panoramas et Synthèses, vol. 52, 2017

[2] B. Klingler; E. Ullmo; A. Yafaev The hyperbolic Ax–Lindemann–Weierstrass conjecture, Publ. Math. Inst. Hautes Études Sci., Volume 123 (2016), pp. 333-360

[3] J. Milne, Shimura varieties, available on author's web page.

[4] B. Moonen Linearity properties of Shimura varieties. I, J. Algebraic Geom., Volume 7 (1998), pp. 539-567

[5] M. Orr Families of abelian varieties with many isogenous fibres, J. Reine Angew. Math., Volume 705 (2015), pp. 211-231

[6] R. Pink A combination of the conjectures of Mordell-Lang and André-Oort, Geometric Methods in Algebra and Number Theory, Progr. Math., vol. 235, 2005, pp. 251-282

[7] R. Richard; T. Zamojski Stabilité analytique et convergence locale de translatées en dynamique homogène S-arithmétique, C. R. Acad. Sci. Paris, Ser. I, Volume 357 (2019) no. 3, pp. 241-246

[8] R. Richard, T. Zamojski, Limit distribution of translated pieces of possibly irrational leaves in S-arithmetic homogeneous spaces, in: R. Richard, A. Yafaev, T. Zamojski (Eds.), Homogeneous Dynamics and Unlikely Intersections, available on ArXiv.

[9] R. Richard, A. Yafaev, Inner Galois equidistribution in S-Hecke orbits, in: R. Richard, A. Yafaev, T. Zamojski (Eds.), Homogeneous Dynamics and Unlikely Intersections, available on ArXiv.

[10] E. Ullmo Quelques applications du theorème d'Ax–Lindemann hyperbolique, Compos. Math., Volume 150 (2014) no. 2, pp. 175-190

[11] U. Zannier Some Problems of Unlikely Intersections in Arithmetic and Geometry. With Appendixes by David Masser, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, USA, 2012 (xiv+160 p)

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