Heights of points with bounded ramification

Abstract

Let $E$ be an elliptic curve defined over a number field $K$ with fixed non-archimedean absolute value $v$ of split-multiplicative reduction, and let $f$ be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on $E$ is either zero or bounded from below by a positive constant, for all points of bounded ramification over $v$. In this paper we make this bound effective and prove an analogue result for the canonical height associated to $f$. We also study variations of this result by changing the reduction type of $E$ at $v$. This will lead to examples of fields $F$ such that the N\'eron-Tate height on non-torsion points in $E(F)$ is bounded from below by a positive constant and the height associated to $f$ gets arbitrarily small on $F$. The same example shows, that the existence of such a lower bound for the N\'eron-Tate height is in general not preserved under finite field extensions.