Heights of points with bounded ramification

TL;DR

This paper extends Baker's 2003 results by providing effective bounds for heights of points with bounded ramification on elliptic curves and associated Lattès maps, exploring how these bounds vary with reduction types.

Contribution

It makes Baker's height bounds effective for Lattès maps and analyzes how these bounds change with different reduction types of elliptic curves.

Findings

Effective lower bounds for canonical heights on elliptic curves.

Existence of fields where heights are bounded below or arbitrarily small.

Non-invariance of height bounds under finite field extensions.

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.