Lower bounds for heights on some algebraic dynamical systems

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 $C/e_v(P{)}^{2e_v(P)+1}$, where $C>0$ is an absolute constant and $e_v(P)$ is the maximum of all ramification indices $e_w(K(P)|K)$ with $w|v$. 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 $C/{\mathrm{lcm}}_{w|v}\{e_w(K(P)|K){\}}^2$, where $C>0$ is an absolute constant. We then give applications towards Lehmer's conjecture. Next, we provide the first examples of polynomials $\varphi \in K[X]$ of degree at least $2$ so that the canonical height ${\hat{h}}_{\varphi }$ of any point in $\bbP^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.

arXiv commentTheorem 1.2 has been greatly improved. The first version dealt with lower bounds for the Néron-tate height on abelian varieties that are totally degenerate at some finite place $v$ of a number field $K$. In v2, we obtain the same lower bound for all abelian varieties over K whose simple abelian subvarieties of $A_{\bar{K}}$ are degenerate at some finite place of $\bar{K}$ lying over $v$