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Rationality of dynamical canonical height

Published online by Cambridge University Press:  18 January 2018

LAURA DE MARCO  and
DRAGOS GHIOCA
LAURA DE MARCO
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA email demarco@northwestern.edu
DRAGOS GHIOCA
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 email dghioca@math.ubc.ca
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Abstract

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 9 , September 2019 , pp. 2507 - 2540
Copyright
© Cambridge University Press, 2018 

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