A Dynamical Bogomolov Property

TL;DR

This paper proves that certain algebraic extensions of number fields have a positive lower bound for heights, using Berkovich space analysis, which advances understanding of the Bogomolov Property in arithmetic dynamics.

Contribution

It establishes the Bogomolov Property for maximal unramified algebraic extensions of number fields with respect to canonical heights from Lattès maps.

Findings

Maximal unramified extensions have the Bogomolov Property for these heights.

Uses Berkovich space analysis to prove algebraic properties.

Connects arithmetic dynamics with non-Archimedean analytic methods.

Abstract

A field F is said to have the Bogomolov Property related to a height function h, if h(a) is either zero or bounded from below by a positive constant for all a in F. In this paper we prove that the maximal algebraic extension of a number field K, which is unramified at a place v|p, has the Bogomolov Property related to all canonical heights coming from a Latt\`es map related to a Tate elliptic curve. To prove this algebraical statement we use analytic methods on the related Berkovich spaces.