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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 dghioca@math.ubc.ca
THOMAS J. TUCKER
Affiliation:
Department of Mathematics,University of Rochester, Rochester, NY, 14620, USA email thomas.tucker@rochester.edu
*
✉dghioca@math.ubc.ca
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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.

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