Rationality of dynamical canonical height

TL;DR

This paper provides a dynamical proof that the canonical height on elliptic curves over function fields takes rational values at rational points and explores conditions under which local canonical heights can be irrational, with applications to orbit intersections.

Contribution

It introduces a dynamical approach to rationality of canonical heights, characterizes irrational local heights for quadratic maps, and applies these results to orbit intersection problems.

Findings

Canonical heights on elliptic curves over function fields are rational at rational points.

Explicit examples of irrational local canonical heights in degrees greater than 1.

Finiteness of orbit intersections when canonical heights are positive rational numbers and degrees are multiplicatively independent.

Abstract

We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps, we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application of our results, we prove that if the canonical heights of two points a and b under the action of two rational functions f and g (defined over k) are positive rational numbers, and if the degrees of f and g are multiplicatively independent, then the orbit of a under f intersects the orbit of b under g in at most finitely many points, complementing the results of Ghioca-Tucker-Zieve.