Arakelov-Green's functions for dynamical systems on projective varieties

Abstract

We introduce functions associated to polarized dynamical systems that generalize averages of the dynamical Arakelov-Green's functions for rational functions due to Baker and Rumely. For a polarized dynamical system $X\to X$ over a product formula field, we prove an Elkies-style lower bound for these functions evaluated on the adelic points of $X$. As an application, we prove a Lehmer-type lower bound on the canonical height of a non-torsion point $P$ on an abelian variety $A/K$, where $K$ is a product formula field having perfect residue fields at its completions (for instance, $K$ may be a number field or the function field of a curve over $\mathbb{C}$ or ${\mathbb{F}}_p$). For $A$ of dimension $g$, the lower bound has the form \[\widehat{h}(P)\ge\frac{C}{D^{2g+3}(\log D)^{2g}},\] where $C=C(A,K,\hat{h})>0$, $D=[K(P):K]\ge 2$, and $P\in A(\bar{K})\setminus A(\bar{K}{)}_{\text{tors}}$ is not contained in a torsion translate of an abelian subvariety of $A$ having everywhere potential good reduction.