2019/06/06 by Bridy, Andrew, Larson, Matt · 1 citation
#11G50 #14G40 #37P15 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1906.02654
The Arakelov-Zhang pairing ⟨ψ,ϕ⟩ is a measure of the "dynamical distance" between two rational maps ψ and ϕ defined over a number field K. It is defined in terms of local integrals on Berkovich space at each completion of K. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height hϕ and the standard Weil height h, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.