Relative Bogomolov extensions

TL;DR

This paper investigates the Bogomolov property in algebraic extensions of the rationals, providing criteria for when extensions possess this property and constructing examples with specific ramification characteristics.

Contribution

It introduces the concept of relative Bogomolov extensions, establishes a ramification criterion, and constructs examples based on ramification conditions in Galois extensions.

Findings

Extensions with bounded ramification index can be Bogomolov.

Constructed examples of both Bogomolov and non-Bogomolov extensions.

Ramification criterion determines the Bogomolov property in extensions.

Abstract

An algebraic extension K of the rationals has the Bogomolov property if the absolute logarithmic height of non-torsion points of K* is bounded away from 0. We define a relative extension L/K to be Bogomolov if this holds for points of L\K. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K, when K is Galois over Q.