A Bogomolov property for the canonical height of maps with superattracting periodic points

2021/06/24 by Looper, Nicole R.
#11G50 #37P05 #37P35 #37P40 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.13003

Abstract

We prove that if f is a polynomial over a number field K with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an ε>0 such that only finitely many P∈ K^ab have canonical height less than ε with respect to f. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the K-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in K. We further prove unconditional analogues of these results in the function field setting.

Related

• A lower bound on the canonical height for polynomials
• Canonical height functions on the affine plane associated with polynomial automorphisms
• Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity
• Computing points of small height for cubic polynomials
• Good reduction and canonical heights of subvarieties