Lower bounds for heights on some algebraic dynamical systems

TL;DR

This paper establishes lower bounds for heights of points on certain algebraic dynamical systems, generalizing previous results for elliptic curves and providing new examples where heights are either zero or bounded away from zero.

Contribution

It generalizes height lower bounds from elliptic curves to simple abelian varieties and introduces new polynomial examples with height gaps in algebraic dynamical systems.

Findings

Lower bounds for Néron-Tate heights on elliptic curves with split multiplicative reduction.

Refined height bounds for degenerate simple abelian varieties.

First examples of polynomials with height gaps in their canonical heights.

Abstract

Let $v$ be a finite place of a number field $K$ and write $K^{nr,v}$ for the maximal field extension of $K$ in which $v$ is unramified. The purpose of this paper is split up into two parts. The first one generalizes a theorem of Pottmeyer: If $E$ is an elliptic curve defined over $K$ with split multiplicative reduction at $v$, then the N\'eron-Tate height of a non-torsion point $P\in E(\bar{K})$ is bounded from below by $C / e_v(P)^{2 e_v(P)+1}$, where $C>0$ is an absolute constant and $e_v(P)$ is the maximum of all ramification indices $e_w(K(P) \vert K)$ with $w\vert v$. Among other things, we refine this result by showing that given a simple abelian variety $A$ defined over $K$ that is degenerate at $v$, the N\'eron-Tate height of a non-torsion point $P\in A(\bar{K})$ is at least $C / \mathrm{lcm}_{w\vert v} \{e_w(K(P)\vert K)\}^2$, where $C>0$ is an absolute constant. We then give applications towards Lehmer's conjecture. Next, we provide the first examples of polynomials $\phi\in K[X]$ of degree at least $2$ so that the canonical height $\hat{h}_\phi$ of any point in $\bbP^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.