Lower bounds for heights on some algebraic dynamical systems

TL;DR

The article develops lower bounds for heights in algebraic dynamical systems, connecting three main settings: the Weil height on P^1, the Néron–Tate height on abelian varieties, and dynamical heights. It generalizes Pottmeyer’s elliptic-curve bound to simple abelian varieties degenerate at a place, yielding a $\\hat{h}$-lower bound in terms of $l_v(P)$ and highlighting Lehmer-type consequences; it then proves a tropical version of a height bound via Yamaki’s lemma and Poincaré reducibility, providing a broad criterion requiring nontrivial tropicalizations of simple factors. Building on this, the paper gives a dynamical Bogomolov theorem: if the Julia set contains an atypical element, then $\\mathbb{P}^1(K^{nr,v})$ has the strong Bogomolov property for $\\hat{h}_φ$, with explicit Newton-polygon criteria enabling concrete polynomial families to satisfy the hypothesis. The results combine tropical geometry, Berkovich dynamics, and equidistribution to yield new Lehmer-type bounds and dynamical Bogomolov statements with potential applications to arithmetic dynamics and Diophantine geometry.

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 Néron-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 Néron-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 $φ\in K[X]$ of degree at least $2$ so that the canonical height $\hat{h}_φ$ of any point in $\bbP^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.

Theorems & Definitions (32)

• Theorem 1.1: Pottmeyer, P15, Theorem 4.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Conjecture 1.7: Lehmer's conjecture
• Corollary 1.8
• Remark 1.9
• Remark 1.10