Lower bounds for heights on some algebraic dynamical systems

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 $C\mathrm{/}{e}_v(P{)}^{2e_v(P)+1}$, where $C>0$ is an absolute constant and $e_v(P)$ is the maximum of all ramification indices $e_w(K(P)\mathrm{\mid }K)$ with $w\mathrm{\mid }v$. 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 $C\mathrm{/}{\mathrm{l}\mathrm{c}\mathrm{m}}_{w\mathrm{\mid }v}\{e_w(K(P)\mathrm{\mid }K){\}}^2$, 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 ${\widehat{h}}_{\phi }$ of any point in ${\text{bbP}}^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.