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A REFORMULATION OF THE DYNAMICAL MANIN–MUMFORD CONJECTURE

Part of: Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  01 June 2020

DRAGOS GHIOCA Open the ORCID record for DRAGOS GHIOCA [Opens in a new window]  and
THOMAS J. TUCKER Open the ORCID record for THOMAS J. TUCKER [Opens in a new window]
DRAGOS GHIOCA*
Affiliation:
Department of Mathematics,University of British Columbia, Vancouver, BC, V6T 1Z2, Canada email [email protected]
THOMAS J. TUCKER
Affiliation:
Department of Mathematics,University of Rochester, Rochester, NY, 14620, USA email [email protected]
*
✉[email protected]
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Abstract

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.

Keywords

dynamical Manin–Mumford conjectureabelian varieties

MSC classification

Primary: 37P35: Arithmetic properties of periodic points
Secondary: 37P55: Arithmetic dynamics on general algebraic varieties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 154 - 161
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was partially supported by a Discovery Grant from the National Science and Engineering Research Council of Canada.

References

Baker, M. H. and Hsia, L.-C., ‘Canonical heights, transfinite diameters, and polynomial dynamics’, J. reine angew. Math. 585 (2005), 61–92.CrossRefGoogle Scholar
Call, G. S. and Silverman, J. H., ‘Canonical heights on varieties with morphisms’, Compos. Math. 89 (1993), 163–205.Google Scholar
Dujardin, R. and Favre, C., ‘The dynamical Manin–Mumford problem for plane polynomial automorphisms’, J. Eur. Math. Soc. (JEMS) 19(11) (2017), 3421–3465.CrossRefGoogle Scholar
Fakhruddin, N., ‘Questions on self maps of algebraic varieties’, J. Ramanujan Math. Soc. 18(2) (2003), 109–122.Google Scholar
Ghioca, D., ‘The Mordell–Lang theorem for Drinfeld modules’, Int. Math. Res. Not. IMRN 2005(53) (2005), 3273–3307.CrossRefGoogle Scholar
Ghioca, D. and Nguyen, K. D., ‘Dynamical anomalous subvarieties: structure and bounded height theorems’, Adv. Math. 288 (2016), 1433–1462.CrossRefGoogle Scholar
Ghioca, D., Nguyen, K. D. and Ye, H., ‘The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of (ℙ1)n’, Compos. Math. 154(7) (2018), 1441–1472.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Ghioca, D., Tucker, T. J. and Zhang, S., ‘Towards a dynamical Manin–Mumford conjecture’, Int. Math. Res. Not. IMRN 2011 (2011), 5109–5122.Google Scholar
Medvedev, A. and Scanlon, T., ‘Invariant varieties for polynomial dynamical systems’, Ann. of Math. (2) 179 (2014), 81–177.CrossRefGoogle Scholar
Pink, R., ‘The Galois representations associated to a Drinfeld module in special characteristic. II. Openness’, J. Number Theory 116(2) (2006), 348–372.CrossRefGoogle Scholar
Raynaud, M., ‘Sous-variétés d’une variété abélienne et points de torsion’, in: Arithmetic and Geometry, Vol. I, Progress in Mathematics, 35 (Birkhauser, Boston, MA, 1983), 327–352.CrossRefGoogle Scholar
Yuan, X. and Zhang, S., ‘The arithmetic Hodge index theorem for adelic line bundles’, Math. Ann. 367(3–4) (2017), 1123–1171.Google Scholar
Zhang, S., ‘Distributions in algebraic dynamics’, in: Surveys in Differential Geometry, Vol. 10 (International Press, Somerville, MA, 2006), 381–430.Google Scholar