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A reformulation of the dynamical Manin-Mumford conjecture. (English) Zbl 1459.37099

Summary: We advance a new conjecture in the spirit of the dynamical Manin-Mumford conjecture. We show that our conjecture holds for all polarisable endomorphisms of abelian varieties and for all polarisable endomorphisms of \((\mathbb{P}^1)^N\). Furthermore, we show various examples which highlight the restrictions one would need to consider in formulating any general conclusion in the dynamical Manin-Mumford conjecture.

MSC:

37P55 Arithmetic dynamics on general algebraic varieties
37P35 Arithmetic properties of periodic points
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[1] Baker, M. H. and Hsia, L.-C., ‘Canonical heights, transfinite diameters, and polynomial dynamics’, J. reine angew. Math.585 (2005), 61-92. · Zbl 1071.11040
[2] Call, G. S. and Silverman, J. H., ‘Canonical heights on varieties with morphisms’, Compos. Math.89 (1993), 163-205. · Zbl 0826.14015
[3] Dujardin, R. and Favre, C., ‘The dynamical Manin-Mumford problem for plane polynomial automorphisms’, J. Eur. Math. Soc. (JEMS)19(11) (2017), 3421-3465. · Zbl 1392.37111
[4] Fakhruddin, N., ‘Questions on self maps of algebraic varieties’, J. Ramanujan Math. Soc.18(2) (2003), 109-122. · Zbl 1053.14025
[5] Ghioca, D., ‘The Mordell-Lang theorem for Drinfeld modules’, Int. Math. Res. Not. IMRN2005(53) (2005), 3273-3307. · Zbl 1158.11030
[6] Ghioca, D. and Nguyen, K. D., ‘Dynamical anomalous subvarieties: structure and bounded height theorems’, Adv. Math.288 (2016), 1433-1462. · Zbl 1402.37091
[7] Ghioca, D., Nguyen, K. D. and Ye, H., ‘The dynamical Manin-Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of (ℙ^1)^n’, Compos. Math.154(7) (2018), 1441-1472. · Zbl 1432.37122
[8] Ghioca, D., Nguyen, K. D. and Ye, H., ‘The dynamical Manin-Mumford conjecture and the dynamical Bogomolov conjecture for split rational maps’, J. Eur. Math. Soc. (JEMS)21(5) (2019), 1571-1594. · Zbl 1426.37064
[9] Ghioca, D. and Tucker, T. J., ‘Proof of a dynamical Bogomolov conjecture for lines under polynomial actions’, Proc. Amer. Math. Soc.138(3) (2010), 937-942. · Zbl 1187.37134
[10] Ghioca, D., Tucker, T. J. and Zhang, S., ‘Towards a dynamical Manin-Mumford conjecture’, Int. Math. Res. Not. IMRN2011 (2011), 5109-5122. · Zbl 1267.37110
[11] Medvedev, A. and Scanlon, T., ‘Invariant varieties for polynomial dynamical systems’, Ann. of Math. (2)179 (2014), 81-177. · Zbl 1347.37145
[12] Pink, R., ‘The Galois representations associated to a Drinfeld module in special characteristic. II. Openness’, J. Number Theory116(2) (2006), 348-372. · Zbl 1173.11037
[13] Raynaud, M., ‘Sous-variétés d’une variété abélienne et points de torsion’, in: Arithmetic and Geometry, Vol. I, , 35 (Birkhauser, Boston, MA, 1983), 327-352. · Zbl 0581.14031
[14] Yuan, X. and Zhang, S., ‘The arithmetic Hodge index theorem for adelic line bundles’, Math. Ann.367(3-4) (2017), 1123-1171. · Zbl 1372.14017
[15] Zhang, S., ‘Distributions in algebraic dynamics’, in: Surveys in Differential Geometry, Vol. 10 (International Press, Somerville, MA, 2006), 381-430. · Zbl 1207.37057
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